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Description and prediction of mortality in some coastal douglas fir stands Paillé, Gilbert 1970

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DESCRIPTION AND PREDICTION OF MORTALITY IN SOME COASTAL DOUGLAS FIR STANDS by GILBERT PAILLE B,A.Sc, Lav a l U n i v e r s i t y , Quebec, 1965 M.Sc, Lav a l U n i v e r s i t y , Quebec, 1968 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Fa c u l t y of FORESTRY We accept t h i s t h e s i s as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August, 1970 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the r e q u i r e -ments f o r an advanced degree a t the U n i v e r s i t y of B r i t i s h C o l -umbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g of t h i s t h e s i s f o r s c h o l a r l y purposes may be gr a n t e d by the Head of my Department or by. h i s r e p r e s e n t a t i v e s . I t i s un d e r s t o o d t h a t c o p y i n g or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p ermis-s i o n . Department The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8 , Canada ABSTRACT i i T h i s study i s based on 68 permanent sample p l o t s l o n g es-t a b l i s h e d throughout the C o a s t a l Douglas f i r Region of the Pa-c i f i c Northwest, i n b o t h n a t u r a l and p l a n t e d s t a n d s of Douglas f i r ( Pseudotsuga m e n z i e s i i ( M i r b . ) F r a n c o ) . These p l o t s , es-t a b l i s h e d by seven a g e n c i e s , c o v e r 27 a c r e s of l a n d and c o n t a i n 13 thousand t r e e s , of which 10 thousand had been l o c a t e d i n the C a r t e s i a n system o f c o o r d i n a t e s . Trees were i n d i v i d u a l l y measured, u s u a l l y every 5 y e a r s , d u r i n g p e r i o d s v a r y i n g between 10 and 30 ye Four computer programs have been w r i t t e n i n F o r t r a n I V t o compile d e t a i l e d i n f o r m a t i o n about f o r e s t growth and y i e l d , s t o c k i n g and s t a n d d e n s i t y , frequency d i s t r i b u t i o n s of t r e e parameters, t h e i r s p a t i a l arrangements and p r o b a b i l i t i e s of mor-t a l i t y . M u l t i p l e r e g r e s s i o n t e c h n i q u e s were used t o determine r e l a t i o n s h i p s among m o r t a l i t y , and stand and s i t e c h a r a c t e r i s -t i c s . The o b j e c t i v e s of the study were t o f u l l y d e s c r i b e "regu-l a r " m o r t a l i t y caused p r i m a r i l y by c o m p e t i t i o n i n second-growth s t a n d s , and t o d e v e l o p some methods t o f a c i l i t a t e and improve i t s p r e d i c t i o n . The problem has been t a c k l e d by u s i n g b o t h the sta n d approach and the t r e e approach. R e s u l t s show t h a t r e g u l a r m o r t a l i t y can be d e s c r i b e d and p r e d i c t e d by making use of the s t a n d parameters e n t e r i n g growth f u n c t i o n s , i . e . d e n s i t y , age and s i t e i n d e x . I t i s b e s t i i i e x p r e s s e d i n number of sterns per a c r e . However, f i r s t - and second-order' l i n e a r models i n c l u d i n g these v a r i a b l e s c o u l d not account f o r more than W5 p e r c e n t of the v a r i a t i o n i n mor-t a l i t y i n n a t u r a l s t a n d s . S i x groups of m o r t a l i t y t a b l e s are p r e s e n t e d , i n d i c a t i n g the a n n u a l p r o b a b i l i t y of i n d i v i d u a l t r e e m o r t a l i t y by age c l a s s e s , based on t h e i r r e l a t i v e s i z e , i n c r e m e n t , or p o s i t i o n w i t h r e g a r d t o the s t a n d i n which they grow. These p r o b a b i l i -t i e s c o u l d be used t o d e v i s e marking r u l e s f o r t h i n n i n g opera-t i o n s , and t o r e p l a c e or supplement c o m p e t i t i o n formulae b u i l t i n t o most f o r e s t growth s i m u l a t o r s . As such, they s e r v e i n q u a n t i f y i n g changes i n the r e l a t i v e growth c a p a b i l i t y o f each t r e e w i t h t i m e . A l l p e r c e n t a g e d i a m e t e r d i s t r i b u t i o n s of dead t r e e s s t u d i e d f i t t o n e g a t i v e b i n o m i a l p r o b a b i l i t y d i s t r i b u t i o n s . S p a t i a l arrangements of dead t r e e s were clumpy o n l y i n v e r y dense s t a n d s , i n mixed s t a n d s o f Douglas f i r and o t h e r c o n i f e r s , o r i n s t a n d s a f f e c t e d by i r r e g u l a r m o r t a l i t y . F i v e methods a r e g i v e n t o p r e d i c t m o r t a l i t y on a stand ba-s i s . A new s e m i - s t o c h a s t i c s t a n d model, based on m o r t a l i t y t a b l e s , i s p r e s e n t e d as a t o o l t o i n v e s t i g a t e growth and m o r t a l -i t y i n a c t u a l or h y p o t h e t i c a l f o r e s t s t a n d s . I n a d d i t i o n t o p r o v i d i n g much i n f o r m a t i o n about m o r t a l i t y , i t s e s t i m a t e s are as p r e c i s e as those from o t h e r c u r r e n t methods f o r y i e l d p r e -d i c t i o n . The i n p u t c o n s i s t s of a t a l l y of t r e e d i a m e t e r s and a h e i g h t - d i a m e t e r e q u a t i o n ; the output i s composed of h i s t o g r a m s of dead t r e e d i a m e t e r s , maps of t h e i r most l i k e l y s p a t i a l i v arrangement, and stand growth and y i e l d i n f o r m a t i o n f o r 1 0 - y e a r p r e d i c t i o n p e r i o d s . Moreover, the model al l o w s p r o b a b i l i t i e s of i r r e g u l a r and c a t a s t r o p h i c m o r t a l i t y to be taken i n t o account. RESUME Cette etude est basee sur 68 placettes echantillons per-manentes eta b l i e s depuis longtemps dans l a Re'gion du sapin de Douglas de l a Cote du Pacifique. E l l e s representent des peuplements naturels de Douglas (Pseudotsuga menziesii (Mirb.) Franco). E l l e s couvrent une superficie de 27 acres et con-tiennent 13 mi l l e arbres, dont 10 mille ont ete l o c a l i s e s dans l e systeme de coordonnees cartesiennes et mesures durant des periodes de 10 a 30 annees. Quatre programmes furent e'crits en langage Fortran IV en vue de compiler des informations detaille'es concernant l a croissance et l e rendement f o r e s t i e r , l e degre d'occupation et l a densite des peuplements, l e s d i s t r i b u t i o n s de frequences des parametres caracterisant l e s tiges i n d i v i d u e l l e s , leur arrange-ment s p a t i a l et leur probabilite de mortalite. Des techniques de regressions multiples furent appliquees a f i n de developper des r e l a t i o n s entre l a mortalite' et l e s caracteristiques des stations et des peuplements. L'etude avait pour buts l a description complete de l a mortalite reguliere obsorvee en peuplements de seconde venue et l e developpement de quelques methodes aptes a en f a c i l i t e r et a en ameliorer l a prediction. Les r e s u l t a t s montrent que l a mortalite re'guliere peut etre adequatement decrite et predite en faisant usage des fac-teurs de peuplements u t i l i s e s dans l e s fonctions d'accroissement, v i c ' e s t - a - d i r e densite, age et i n d i c e de q u a l i t e de s t a t i o n . E l l e s'exprime l e mieux en nombre de t i g e s a 1'acre. Cependant, des equations l i n e a i r e s du premier et du second degre' contenant ces v a r i a b l e s n'ont pas pu ex p l i q u e r p l u s de 2+3 pour cent de l a v a r i a t i o n dans l a m o r t a l i t e ' en peuplements n a t u r e l s . S i x groupes de t a b l e s de m o r t a l i t e ' sont pre'sentees. E l l e s indiquent l a p r o b a b i l i t e " annuelle de m o r t a l i t e par c l a s s e s d'age, basees sur l a dimension r e l a t i v e , 1'accroissement r e l a -t i f et l a p o s i t i o n de t i g e s i n d i v i d u e l l e s par rapport aux peuplements dans l e s q u e l l e s e l l e s c r o i s s e n t . Ces p r o b a b i l i t e s pourraient etre u t i l i s e ' e s dans 1 1 e l a b o r a t i o n de r e g i e s de mar-telage et pour remplacer ou completer l e s formules de competi-t i o n f a i s a n t p a r t i e de l a plupart des modeles de s i m u l a t i o n des peuplements f o r e s t i e r s . Comme t e l l e s , e l l e s servent a quanti-f i e r l e s changements dans l a capacite de competition r e l a t i v e de chaque t i g e suivant son age. Toutes l e s d i s t r i b u t i o n s de frequences des diametres des arbres morts (exprimees en pourcentage) s'apparentaient assez bien a des d i s t r i b u t i o n s de p r o b a b i l i t e s negatives binomiales. Les arbres morts presentaient des arrangements spatiaux en bouquets seulement dans l e s peuplements t r e s denses, en peuple-ments melanges et dans l e s peuplements a f f e c t e s par l a m o r t a l i t e i r r e g u l i e r e . Cinq methodes de p r e d i c t i o n de l a m o r t a l i t e sur une base de peuplement sont d e c r i t e s . De p l u s , un modele de peuplement semi-stochastique, base sur l e s t a b l e s de m o r t a l i t e , est pre-sente comme un nouvel o u t i l pour 1'etude de l a croissance et de v i i l a m o r t a l i t e ' en peuplements r e e l s et h y p o t h e t i q u e s . Une l i s t e de d i a m e t r e s e t une f o n c t i o n d i a m e t r e - h a u t e u r d o i v e n t e t r e f o u r n i e s en entre'e; l a s o r t i e se compose d 1 histogrammes des d i a m e t r e s d ' a r b r e s morts, de c a r t e s montrant l e u r arrangement s p a t i a l p r o b a b l e e t d 1 i n f o r m a t i o n s sur l ' a c c r o i s s e m e n t e t l e rendement du peuplement par pe'riodes de 10 annees. De p l u s , l e modele iaermet de t e n i r compte de l a p r o b a b i l i t e " de m o r t a l i t e i r r e ' g u l i e r e et c a t a s t r o p h i q u e ; sa p r e c i s i o n e s t comparable a c e l l e des a u t r e s methodes de p r e d i c t i o n usite'es. INDEX OF CONTENTS Page TITLE PAGE i ABSTRACT i i RESUME v INDEX OF CONTENTS v i i i INDEX OF TABLES x i v INDEX OF FIGURES x v i i ACKNOWLEDGEMENTS x i x CHAPTER 1 INTRODUCTION 1 CHAPTER 2 A LITERATURE REVIEW ON MORTALITY OF DOUGLAS FIR 5 2.1 DEFINITION OF MORTALITY 5 2.2 CAUSES OF MORTALITY 8 2.2.1 COMPETITION 8 2.2.2 INJURIES 12 2.2.3 DISEASES 13 2.3 AMOUNT AND TIMING OF MORTALITY 14 2.3.1 MORTALITY IN NATURAL FORESTS 14 2.3.1.1 DURING STAND ESTABLISHMENT PERIOD 14 2.3.1.2 IN YOUNG FORESTS 15 2.3.1.3 IN MATURE FORESTS.... 17 2.3.2 MORTALITY IN PLANTATIONS 20 2.4 DISTRIBUTION OF MORTALITY 21 2.4.1 FREQUENCY DISTRIBUTIONS OF TREE PARAMETERS 21 i x Page 2.Z+.2 MORTALITY SPATIAL DISTRIBUTIONS 23 2 .5 PREDICTION OF GROWTH, YIELD AND MORTALITY 2k 2 . 5 . 1 DIRECT METHODS 2k 2.5.2 INDIRECT METHODS 25 2 . 5 . 2 . 1 NORMAL YIELD TABLES 25 2 . 5 . 2 . 2 STAND-TABLE PROJECTION 26 2 . 5 . 2 . 3 TWO-WAY GROWTH PREDICTION 27 2 . 5 . 2 . i f GROWTH AND YIELD FUNCTIONS 27 2 . 5 . 3 SIMULATION TECHNIQUES 28 2 . 5 . 3 . 1 REGULAR MORTALITY 28 2 . 5 . 3 . 2 IRREGULAR AND CATASTROPHIC MORTALITY.. 28 " 2 .6 CHAPTER SUMMARY 29 CHAPTER 3 OBJECTIVES, DATA AND METHODS 31 3 .1 DEFINITION OF OBJECTIVES 31 3 . 2 DESCRIPTION OF DATA 32 3 . 2 . 1 PERMANENT SAMPLE PLOTS 32 3 . 2 . 2 STEM MAPPED RECORDS 36 3 .3 METHODS 37 3 . 3 . 1 CODING AND SORTING THE INFORMATION 37 3 . 3 . 2 COMPUTER PROGRAMMING 37 3 . 3 . 3 SAMPLING AND CURVE FITTING 38 3 . 3 . 3 . 1 SAMPLING FOR TREE ATTRIBUTES 38 3 . 3 . 3 . 2 SAMPLING FOR SPATIAL ARRANGEMENTS 39 3 . 3 . 3 . 3 CURVE FITTING. kl X Page 3.3.4 SUMMARY OF THE INFORMATION 42 3.4 CHAPTER SUMMARY 47 CHAPTER 4 AMOUNT AND TIMING OF MORTALITY 49 4.1 MORTALITY RELATED TO STAND CHARACTERISTICS A9 4.1.1 PERIODIC ANNUAL MORTALITY IN NORMAL STANDS 49 4.1.2 PERIODIC ANNUAL MORTALITY IN SOME NATURAL AND PLANTED STANDS 63 4.1.2.1 CORRELATIONS BETWEEN PERIODIC ANNUAL MORTALITY AND STAND PARAMETERS 65 4.1.2.2 MULTIPLE REGRESSIONS OF PERIODIC ANNUAL MORTALITY ON STAND PARAMETERS 71 .1. FOR PURE NATURAL STANDS 72 i i . FOR MIXED NATURAL STANDS 73 i i i . FOR PLANTATIONS 73 4.1.3 GROSS AND NET GROWTH AND YIELD 80 4.1.3.1 GROSS GROWTH 82 4.1.3.2 NET GROWTH AND YIELD 85 4.1.3.3 CUMULATIVE MORTALITY 89 4.2 MORTALITY RELATED TO TREE CHARACTERISTICS 94 4.2.1 MORTALITY AND RELATIVE TREE SIZE 97 4.2.2 MORTALITY AND RELATIVE HEIGHT 101 4.3.2 MORTALITY AND CROWN CLASS 105 x i Page i+.2.2+ MORTALITY AND RELATIVE INCREMENT IN SIZE AND HEIGHT 105 i f . 2 , 5 MORTALITY AND RELATIVE CROWN WIDTH/DIAMETER RATIO 109 Zf .5 CHAPTER SUMMARY I l l CHAPTER 5 DISTRIBUTION OF MORTALITY .113 5 . 1 DISTRIBUTION OF DEAD TREES 112+ 5 . 1 . 1 DIAMETER DISTRIBUTIONS 112+ 5 . 1 . 2 SPATIAL DISTRIBUTIONS 122 5 . 2 DISTRIBUTION OF LIVING TREES 131 5 . 2 . 1 DIAMETER DISTRIBUTIONS 13-1 5 . 2 . 2 HEIGHT DISTRIBUTIONS 133 5 . 2 . 3 SPATIAL DISTRIBUTIONS 13*+ 5 . 3 RELATIONSHIPS BETWEEN DISTRIBUTIONS OF DEAD TREES AND LIVING TREES 1L.0 5.2+ EFFECTS OF MORTALITY DISTRIBUTIONS ON SITE OCCUPANCY AND USE OF SITE CAPACITY .12+if 5 .2+.1 ANALYSIS OF SITE OCCUPANCY INDICATORS . I i f5 5.2+.2 ANALYSIS OF EMPTY QUADRATS .12+9 5.2+.3 USE OF SITE CAPACITY J .50 5 . 5 CHAPTER SUMMARY J. 5 6 CHAPTER 6 PREDICTION OF MORTALITY a58 6 . 1 THE STAND APPROACH ..158 6 . 1 . 1 YIELD TABLES 1 5 8 6 . 1 . 2 PREDICTION EQUATIONS 1 5 9 x i i Page 6.1.3 RELATIONSHIP BETWEEN LIVE AND DEAD TREES 159 6.1.4 PERCENTAGE DIAMETER DISTRIBUTIONS 160 6.2 THE TREE APPROACH 160 6.2.1 SEMI-STOCHASTIC STAND MODEL 160 6.2.1.1 ANNUAL DIAMETER INCREMENT FUNCTIONS 160 6.2.1.2 HEIGHT FUNCTIONS 165 6.2.1.3 VOLUME FUNCTIONS 165 6.2.1.4 SPATIAL PATTERN FUNCTIONS 166 6.2.1.5 MORTALITY ALLOCATION 169 i . MORTALITY GENERATOR MODEL 1 170 i i . MORTALITY GENERATOR MODEL II....173 6.2.1.6 DESCRIPTION OF THE STAND MODEL 175 6.2.1.7 TESTING OF THE STAND MODEL 177 6.2.2 THE IDEAL STOCHASTIC STAND MODEL 180 6.3 CHAPTER SUMMARY 182 CHAPTER 7 DISCUSSION 184 7.1 BASIC DATA 184 7.2 NATURE OF MORTALITY 186 7.3 AMOUNT, TIMING AND DISTRIBUTION OF MORTALITY 138 7.4 SPATIAL PATTERNS 192 7.5 SITE OCCUPANCY 196 7.6 PREDICTION OF MORTALITY 197 x i i i Page 7.6.1 MORTALITY TABLES 198 7.6.2 DIAMETER INCREMENT I98 7.6.3 MORTALITY GENERATORS 200 7.6.4 PRECISION OF PREDICTIONS 201 CHAPTER 8 CONCLUSION 203 LITERATURE CITED. 207 APPENDIX I COMPUTER PROGRAMS 223 I I REGRESSION EQUATIONS FROM ORIGINAL DATA 236 I I I MEASUREiiSNT OF SITE OCCUPANCY 250 IV IDENTIFICATION OF SYMBOLS 265 V MORTALITY TABLES 268 VI IRREGULAR AND CATASTROPHIC MORTALITY 28l V I I STAND MODEL - DESCRIPTION, FLOW CHART, OUTPUT 288 V I I I ACTUAL AND SIMULATED STAND CHARACTERISTICS 294 Biographical Information x i v INDEX OF TABLES TABLE Page I DECADAL PERCENTAGE MORTALITY IN NUMBER OF TREES 16 I I CLASSIFICATION OF PLOTS STUDIED BY ORIGIN, AGE AND STAND COMPOSITION 33 I I I STATISTICS FOR PURE NATURAL STANDS 43 IV STATISTICS FOR MIXED NATURAL STANDS 45 V STATISTICS FOR PLANTATIONS 46 VI PERIODIC ANNUAL PERCENT MORTALITY AND ANNUAL MORTALITY IN NUMBER OF STEMS IN NORMAL STANDS 51 VII PERIODIC ANNUAL PERCENT MORTALITY AND ANNUAL MORTALITY IN NUMBER OF STEMS IN PLANTATIONS 52 V I I I AVERAGE PROBABILITY OF INDIVIDUAL TREE DEATH BASED ON TOTAL STAND AGE AND BASAL AREA PER ACRE 60 IX AVERAGE PROBABILITY OF INDIVIDUAL TREE DEATH BASED ON TOTAL STAND AGE AND SITE QUALITY 61 X AVERAGE PROBABILITY OF INDIVIDUAL TREE DEATH BASED ON STAND AGE AND AVERAGE SIZE 62 XI MORTALITY CHARACTERISTICS MEASURED IN SAMPLE PLOTS 64 XV TABLE Page XII SIMPLE CORRELATION BETWEEN SOME EXPRESSIONS , OF MORTALITY AND STAND CHARACTERISTICS (PURE NATURAL DOUGLAS FIR STANDS). 66 X I I I SIMPLE CORRELATION BETWEEN SOME EXPRESSIONS OF MORTALITY AND STAND CHARACTERISTICS (MIXED DOUGLAS FIR STANDS) 68 XIV SIMPLE CORRELATION BETWEEN SOME EXPRESSIONS OF MORTALITY AND STAND CHARACTERISTICS (PLANTATIONS) 69 XV MULTIPLE REGRESSION EQUATIONS OF PERIODIC ANNUAL MORTALITY ON STAND CHARACTERISTICS (PURE STANDS-GROUP I) 7 k XVI MULTIPLE REGRESSION EQUATIONS OF PERIODIC-ANNUAL MORTALITY ON STAND CHARACTERISTICS (MIXED STANDS-GROUPS I I , I I I , IV) f 75 XVII MULTIPLE REGRESSION EQUATIONS OF PERIODIC ANNUAL PERCENT MORTALITY ON STAND CHARACTERISTICS (PLANTATIONS) 77 XVIII MULTIPLE REGRESSION EQUATIONS OF PERIODIC ANNUAL MORTALITY ON STAND CHARACTERISTICS (PLANTATIONS) 78 IXX GROWTH AND YIELD CIIARACTERISTICS MEASURED IN SAMPLE PLOTS 8 l XX MULTIPLE REGRESSION EQUATIONS OF PERIODIC ANNUAL GROSS INCREMENT IN BASAL .AREA AND CUBIC VOLUME ON STAND AND SITE CHARACTERISTICS... 8k x v i TABLE Page XXI VARIABLE DENSITY YIELD TABLE-GROUPS I AND I I 86 XXII MULTIPLE REGRESSION EQUATIONS OF MEAN ANNUAL INCREMENT ON STAND AND SITE CHARACTERISTICS 87 X X I I I MULTIPLE REGRESSION EQUATIONS OF NET YIELD OH STAND AND SITE CHARACTERISTICS 90 XXIV PREDICTED CUMULATIVE MORTALITY ON SITE CLASS I I I DOUGLAS FIR-GROUP 1 93 XXV SPATIAL DISTRIBUTION OF DEAD TREES IN NATURAL STANDS 124 XXVI SPATIAL DISTRIBUTION OF DEAD TREES 14 YEARS AFTER PLANTATION IN A SQUARE LATTICE 127 XXVII INDEX OF NON-RANDOMNESS IN POPULATIONS OF DEAD TREES 128 XXVIII RELATIONSHIPS BETWEEN SITE OCCUPANCY MEASUREMENTS IN PURE NATURAL STANDS 147 XXIX RELATIONSHIPS BETWEEN SITE OCCUPANCY MEASUREMENTS IN PLANTATIONS 148 XXX SIZE OF OPENINGS IN QUADRAT SAMPLING CONSISTENT WITH RANDOuTIESS 151 XXXI AVERAGE USE OF SITE CAPACITY IN NATURAL STANDS AND PLANTATIONS 153 XXXII ANNUAL DIAMETER INCREMENT FUNCTIONS FOR DOUGLAS FIR 163 X X X I I I BREAK DOWN OF REGULAR MORTALITY INTO INGROWTH, GROWING STOCK AND SAWTIMBER 189 x v i i INDEX OF FIGURES FIGURE Page 1 REGULAR MORTALITY OF DOUGLAS FIR IN NORMAL STANDS 18 2 CLASSIFICATION OF DATA BY AGE AND SITE QUALITY 3k 3 PERIODIC ANNUAL PERCENT MORTALITY RELATED TO AGE 53 k PERIODIC ANNUAL PERCENT MORTALITY RELATED TO BASAL AREA ?k 5 PERIODIC ANNUAL PERCENT MORTALITY RELATED TO NUMBER OF TREES 55 6 PERIODIC ANNUAL PERCENT MORTALITY RELATED TO STAND DIAMETER 56 7 PERIODIC ANNUAL MORTALITY RELATED TO NUMBER OF TREES 57 8 PERIODIC PROBABILITY OF INDIVIDUAL TREE MORTALITY BY RELATIVE SIZE CLASSES 98 9 PERIODIC PROBABILITY OF INDIVIDUAL TREE MORTALITY FOR THE WIND RIVER SPACING PLANTATION 100 10 PERIODIC PROBABILITY OF INDIVIDUAL TREE MORTALITY BY RELATIVE HEIGHT CLASSES 102 11 PERIODIC PROBABILITY OF INDIVIDUAL TREE MORTALITY BY CROWN CLASSES 106 12 PERIODIC PROBABILITY OF INDIVIDUAL TREE MORTALITY ON RELATIVE INCREMENT IN SIZE .110 13 PERCENTAGE DIAMETER DISTRIBUTIONS OF DEAD TREES 115 lk TREND IN SPATIAL PATTERN OF LIVING TREES IN GROUP 1 136 x v i i i FIGURE Page 15 TREND IN SPATIAL PATTERN OF LIVING TREES IN GROUPS I I , I I I , I V 137 16 TREND IN SPATIAL PATTERN OF LIVING TREES IN PLANTATIONS 138 17 MORTALITY DBH RELATED TO STAND DBH IN NATURAL STANDS 141 18 MORTALITY DBH RELATED TO STAND DBH IN PLANTATIONS 142 19 MORTALITY GENERATOR MODEL I 172 20 MORTALITY GENERATOR MODEL I I 174 x i x A CKNOV/LED GEMEN T S In the preparation of t h i s d i s s e r t a t i o n , the author has received a considerable amount of i n t e l l e c t u a l s t i m u l a t i o n , encouragement and help from h i s supervisor, Dr. J.H.G. Smith, to whom he i s most t h a n k f u l . Assistance and c o n s t r u c t i v e c r i t i c i s m were given to the author by the members of h i s Graduate Committee, who also r e -viewed and commented on the manuscript. They are: Drs. IC. Graham, C.S. R o l l i n g , B.J. van der Kamp, A. Kozak, and D.D. Hunro. A l a r g e part of the data was made a v a i l a b l e by several agencies, i n c l u d i n g the B r i t i s h Columbia Forest Service, the Canada Department of F i s h e r i e s and F o r e s t r y , the U.S. Forest Service, the Washington State Department of Natural Resources, and the U n i v e r s i t y of B r i t i s h Columbia Research Forest; and a number of f o r e s t Companies, namely B.C. Forest Products L t d . , Crown Z e l l e r b a c h Corporation, Weyerhaeuser Company, Weldwood of Canada L t d . , Columbia C e l l u l o s e L t d . , and Rayonier Canada Lt d . Many people, belonging to these and other agencies, o f f e r e d an appreciable c o n t r i b u t i o n at the planning stage, and during the execution; s p e c i a l mention s h a l l be made of Drs. II.W.F. Bunce, J.E. King, Y. Lee, J.Y. L i n , R.E. M i l l e r , K.J. M i t c h e l l , D.L. Reukema, R.F. Strand, G. W a l l i s , and of Messrs. G.L. Ainscough, P.W. Appleby, M.W. Bradshaw, P.A. B r i e g l e b , W.G. Burch, H.N. C l i f f , G.E. Hoyer, I . McRae, G.R. Staebler, XX J . Walters, and R.L. Williamson. F i n a n c i a l a s s i s t a n c e was granted to the' author i n the form of f e l l o w s h i p s by Laval U n i v e r s i t y , Quebec, by the Quebec De-partment of Lands and Forests, and by Van Dusen F o r e s t r y f e l l o w -ships, without which the r e a l i z a t i o n of t h i s p r o j e c t would have been impossible. F i e l d work was financed through grants by the N a t i o n a l Research Council of Canada and the Canadian Forestry Service (Extramural Research), made a v a i l a b l e to the author's main a d v i s o r . S p e c i a l thanks go to K i s s L. Cowdell f o r her help i n com-municating with the computer, to Mrs. M. Lambden f o r her d r a f t i n g work, to Mr. G. Young, academic s t a f f member, f o r t e c h n i c a l a s s i s t a n c e , and to f e l l o w graduate students M. McGreevy and C. Goulding f o r u s e f u l comments and d i s c u s s i o n s . Above a l l , the author wishes to acknowledge with sincere g r a t i t u d e the f i r m support, constant a i d and great confidence received from h i s wife, L i s e , i n the r e a l i z a t i o n of t h i s endeavor. 1 CHAPTER 1 INTRODUCTION I n 1967, a m u l t i - d i s c i p l i n e group composed o f s i x s p e c i a l -i s t s was a s s i g n e d the t a s k o f a n a l y z i n g the t i m b e r m o r t a l i t y problem i n the P a c i f i c Northwest Reg i o n , and t o recommend a bal< anced program o f a c t i o n and r e s e a r c h t o c a p t u r e the p o t e n t i a l p r e s e n t l y b e i n g l o s t . I n i t s r e p o r t (Anon. 1967), t h i s com-m i t t e e r e v i e w e d the r o l e o f m o r t a l i t y i n f o r e s t management, p r o t e c t i o n and u t i l i z a t i o n , a n a l y z e d c u r r e n t s o u r c e s o f i n f o r -mation, and s t r e s s e d the need f o r more d a t a to ser v e as a b a s i s f o r broad programs o f management, as w e l l as f o r s a l v a g e and c o n t r o l o p e r a t i o n s . I t was s t a t e d t h a t , by 1970, a l l m o r t a l i t j ' - d a t a i n the P a c i f i c Northwest would come from remeasured ( s u r v e y o r growth and y i e l d ) p l o t s , i n s i z e and number l a r g e enough t o s u b s t a n -t i a l l y reduce the e r r o r i n e s t i m a t e s o f m o r t a l i t y caused by i t s v a r i a b i l i t y . A f t e r r e c o g n i t i o n o f t h i s approach, the p r e s e n t study was s e t up t o u t i l i z e m o r t a l i t y i n f o r m a t i o n a l r e a d y accumulated i n some p l o t r e c o r d s , i n an attempt t o d e s c r i b e and p r e d i c t the p r o c e s s i n as many d e t a i l s , and w i t h as much a c c u r a c y a s pos-s i b l e . I n so d o i n g , i t was d e c i d e d t o put the emphasis on r e g u l a r m o r t a l i t y which i s the most i m p o r t a n t of t h r e e t y p e s c u r r e n t l y r e c o g n i z e d . The aims were 1) t o e s t a b l i s h r e l a t i o n s between r e g u l a r I n t r o d u c t i o n 2 l o s s e s and some s t a n d c h a r a c t e r i s t i c s ; 2) t o a n a l y z e f r e q u e n c y d i s t r i b u t i o n s of dead and l i v e t r e e parameters; 3) t o i n v e s t i -gate s p a t i a l arrangements of dead and l i v e t r e e s , and d e t e r -mine how they a f f e c t s i t e occupancy, hoy/ they a r e r e l a t e d , and how they change w i t h t i m e ; and Zf) t o f i n d a s i m p l e approach t o m o r t a l i t y p r e d i c t i o n on a t r e e b a s i s t h a t would be r e l i a b l e , and a l l o w f o r a c c u m u l a t i o n o f d a t a c o l l e c t e d by v a r i o u s agen-c i e s . D e s c r i p t i o n and p r e d i c t i o n of i r r e g u l a r and c a t a s t r o p h i c l o s s e s were n e g l e c t e d because they would have r e q u i r e d e x p e n d i -t u r e s i n t i m e , c a p i t a l , and man-power beyond the r e a c h o f our means. The d a t a c o l l e c t e d throughout the Douglas f i r R e g i o n were judged adequate to p e r m i t s t u d i e s of amount, t i m i n g and d i s t r i -b u t i o n o f m o r t a l i t y i n young-growth f o r e s t s , 20 t o 80 y e a r s o f age. Searches f o r comparable i n f o r m a t i o n i n younger s t a n d s proved o n l y p a r t l y s u c c e s s f u l , because c u r r e n t methods o f de-s c r i p t i o n of m o r t a l i t y a t t h a t stage a re d i f f e r e n t (more q u a l i -t a t i v e ) ; m o r t a l i t y d i s t r i b u t i o n i s the f a c t o r most s e r i o u s l y n e g l e c t e d , a l t h o u g h c r u c i a l i n f u r t h e r s t a n d development. S i m i l a r h a n d i c a p s must have been f e l t i n the b u i l d i n g o f g r o s s y i e l d t a b l e s f o r Douglas f i r 1 by S t a e b l e r (1955a) and C u r t i s (1967); b o t h i n c l u d e d m o r t a l i t y o n l y from age 20 onward. U n l e s s o t h e r w i s e s p e c i f i e d , C o a s t a l Douglas f i r , Douglas f i r , o r f i r r e f e r r e d t o i n t h i s d i s s e r t a t i o n i s Pseudotsuga men-z i e s i i ( M i r b . ) Franco v a r . m e n z i e s i i . I n t r o d u c t i o n 3 The p r e s u m p t i o n t h a t e x t e n s i v e m o r t a l i t y i n f o r m a t i o n w i l l be needed i n the f u t u r e , e s p e c i a l l y i n i n t e n s i v e l y managed f o r -e s t a r e a s , u n d e r l i e s t h i s s t u d y . T h i s stems from the f a c t t h a t , u n l i k e c a t a s t r o p h i c m o r t a l i t y which o f t e n w a r r a n t s major c a p i -t a l e x p e n d i t u r e s and changes i n e s t a b l i s h e d p l a n s because o f i t s c o n c e n t r a t i o n and v a l u e , r e g u l a r m o r t a l i t y i s not worth c o n t r o l l i n g or s a v i n g u n l e s s r o a d s e x i s t and c u l t u r a l t r e a t -ments a r e c o ntemplated. I n a r e a s where such f a c i l i t i e s a r e a t hand, and where such a c t i o n s are to be t a k e n , i t i s of i n t e r e s t t o know: How much t i m b e r w i l l be l o s t ? When? and Where? Answers t o the f i r s t two q u e s t i o n s have been g i v e n i n p u b l i s h e d s t u d i e s and i n l o c a l e x p e r i m e n t s aimed a t growth and y i e l d e s t i m a t e s . They were almost e x c l u s i v e l y based on a stand approach t o the problem, which i s u s e f u l f o r i n d i c a t i n g the magnitude of the phenomenon but h a r d l y h e l p s i n d e c i s i o n making " a t the stump." F o r t h a t purpose, the t r e e approach i s s u i t a b l e e s p e c i a l l y when e n v i s a g e d i n a p r o b a b i l i s t i c framework. I f i t i s l o g i c a l and customary t o t h i n k of i n d i v i d u a l t r e e d e a t h i n terms of odds, i t s h o u l d be f e a s i b l e t o e x p r e s s them mathemati-c a l l y w i t h r e g a r d to time and some e a s i l y measured t r e e c h a r a c -t e r i s t i c s . Thus, e v a l u a t i o n of m o r t a l i t y chances c o u l d become p a r t of marking r u l e s , f o r i n s t a n c e , i n t h i n n i n g o p e r a t i o n s . The t h i r d q u e s t i o n , as t o where m o r t a l i t y w i l l o c c u r , can b e s t be answered by t a k i n g the stand ( o r an even l a r g e r a rea) approach. I n the case of c a t a s t r o p h i c m o r t a l i t y , p r o b a b i l i t i e s o f o c c u r r e n c e of w i l d f i r e s and windstorms f o r example can be I n t r o d u c t i o n k computed by r e g i o n and s u b - r e g i o n , based on h i s t o r i c a l know-l e d g e . I n the case o f i r r e g u l a r m o r t a l i t y , caused by i n s e c t s and f u n g i , b e h a v i o r i s t i c knowledge w i l l i n d i c a t e p r o b a b l e d i s -p e r s i o n p a t t e r n s . As f o r r e g u l a r m o r t a l i t y , s tand c h a r a c t e r -i s t i c s s h o u l d be the b e s t i n d i c a t o r s o f i t s s p a t i a l arrangement. The i n t e g r a t i o n o f t h i s i n f o r m a t i o n s h o u l d o r i e n t the f o r e s t manager i n i t s c h o i c e o f c u l t u r a l t r e a t m e n t s . An o b v i o u s advantage o f t a k i n g b o t h the s t a n d approach and the t r e e approach i n s t u d y i n g growth and m o r t a l i t y i s t h a t i t a l l o w s e l a b o r a t e computer s i m u l a t i o n . The most r e c e n t models (Newham and M a l o l e y , 1970) p r o v i d e f a c i l i t i e s f o r f o r e s t e r s and s i l v i c u l t u r i s t s t o generate a l m o s t any k i n d o f s t a n d , t o grow them and ten d them i n s e v e r a l ways, and t o choose the b e s t a l t e r n a t i v e f o r i m p l e m e n t a t i o n . I n the s e models, however com-p l i c a t e d t hey may be, one of the most d i f f i c u l t p r o c e s s e s t o reprod u c e remains m o r t a l i t y . T h i s s t u d y s h a l l o f f e r some pos-s i b l e s o l u t i o n s t o i t s a l l o c a t i o n , based on accumulated e x p e r i -ence i n the form o f some l o n g - e s t a b l i s h e d permanent sample p l o t s . CHAPTER 2 5 A LITERATURE REVIEW ON MORTALITY OF DOUGLAS FIR 2.1 DEFINITION OF MORTALITY Competition for, l i g h t , water and space i s the most impor-tant f a c t o r responsible f o r the b u i l d up of c o n d i t i o n s l e a d i n g to the s o - c a l l e d REGULAR MORTALITY, whereas endemic diseases, i n s e c t s , and i n j u r i e s u s u a l l y lead to IRREGULAR MORTALITY. Taken i n t h i s sense, r e g u l a r m o r t a l i t y would be caused by fac-t o r s a c t i n g continuously on i n d i v i d u a l t r e e s , u l t i m a t e l y c r e a t i n g a s t r e s s that can no longer be t o l e r a t e d by l i v i n g organisms. On the other hand, i r r e g u l a r m o r t a l i t y would be introduced p e r i o d i c a l l y i n t o stands of t r e e s by e x t e r n a l and i d e n t i f i a b l e agents, d i s t u r b i n g s e r i o u s l y the community as a whole (ecosystem), However, except i n case of catastrophe ( l a r g e scale blow-downs, widespread epidemics, w i l d or a c c i d e n t a l f i r e s , and the l i k e ) and i n case of i n j u r i e s provoked by men or animals, most of the time i t i s quite d i f f i c u l t to diagnose whether i n d i v i d u a l t r e e s have a c t u a l l y been k i l l e d by r e g u l a r or i r r e g u l a r agents. The e f f o r t to determine the prime l e t h a l f a c t o r i n v o l v e d i s often f u t i l e because the process has already begun whenever signs of decadence do appear, and because many d i f f e r e n t symptoms show up on trees already dead. However, new tech-niques have been developed to help i d e n t i f y p h y s i o l o g i c a l l y a f f e c t e d t r e e s ; remote sensing, a e r i a l c o l o r photography, A l i t e r a t u r e r e v i e w 6 v i d e o t a p e r e c o r d i n g and t e l e v i s i o n cameras are used f o r t h a t purpose (Wear et a l . , 1964, A l d r i c h e t _ a l . , "1969, Anon., 1969). I n p r a c t i c e , r e g u l a r and i r r e g u l a r t y p e s of m o r t a l i t y are b e i n g e v a l u a t e d a t a d i f f e r e n t s c a l e . T h i s s u b t l e t y does f a c i l i t a t e the t a s k of p r e s e n t i n g the r e s u l t s , but does not h e l p i n d i f f e r e n t i a t i n g whether t r e e s are d y i n g as a r e s u l t of i n t e n s e p h y s i o l o g i c a l s t r e s s i n d u c e d by c o m p e t i t i o n or under the temporary a t t a c k o f e x t e r n a l a g e n t s . I n most c a s e s , the d i l e m n a i s s o l v e d i n the f o l l o w i n g crude but p r a c t i c a l way: On the one hand, the amount ( t i m b e r volume) of m o r t a l i t y due t o i r r e g u l a r and i d e n t i f i a b l e causes i s e v a l u a t e d by means of l a r g e s c a l e s u r v e y s . I n these e s t i m a t e s , the r e s u l t i s com-posed of t i m b e r a c t u a l l y l o s t i n a g i v e n r e g i o n to f i r e s , d i s e a s e s , i n s e c t a t t a c k s , and m i s c e l l a n e o u s causes, and of an e s t i m a t e d growth l o s s comprised e i t h e r of the amount of wood t h a t would have been produced or of the amount by which the normal r a t e of growth has been impeded by t h e s e a g e n t s . T h i s i s the concept of t o t a l growth impact put f o r w a r d by H e p t i n g and Jemison (Anon., 1958). On the o t h e r hand, r e g u l a r m o r t a l i t y i s c o n s i d e r e d t o be the d i f f e r e n c e between g r o s s and net f o r e s t p r o d u c t i o n as e s t i -mated on a per a c r e b a s i s , by means of sample p l o t s e s t a b l i s h e d i n s t a n d s t h a t have not s u f f e r e d i r r e g u l a r m o r t a l i t y (as de-f i n e d a b ove). These f i g u r e s are summarized i n y i e l d t a b l e s . T h i s d i f f e r e n c e , which i s a l s o c a l l e d "normal 1 1 m o r t a l i t y , i s by f a r g r e a t e r t h a n the amount of i r r e g u l a r m o r t a l i t y because A l i t e r a t u r e r e v i e w 7 i t does o c c u r c o n t i n u o u s l y on every a c r e o f every f o r e s t . One can see t h a t , f o r a l l p r a c t i c a l p u rposes, the a c t u a l d i f f e r e n c e between r e g u l a r and i r r e g u l a r m o r t a l i t y i s i n the s c a l e o f e v a l u a t i o n . F o l l o w i n g t h a t l i n e of thought, s u c c e s -s i v e rerneasurements o f a permanent sample p l o t would p e r m i t the e v a l u a t i o n o f r e g u l a r m o r t a l i t y as l o n g as no i r r e g u l a r causes a r e i d e n t i f i e d (on a stand b a s i s ) . When t h i s o c c u r s , a f f e c t e d p l o t s a r e n o t , o r s h o u l d not be used any l o n g e r i n the b u i l d i n g of y i e l d summaries. More e n l i g h t e n m e n t on how m o r t a l i t y has been e s t i m a t e d and c l a s s i f i e d i n the Douglas f i r R e gion can be found i n Wear and L a u t e r b a c k (1955), W o r t h i n g t o n (1955), Anon. (1958, 1965a, 1965b), McMahon (1961), and C h i l d s and Shea (1967). D e s c r i p t i o n s o f methods and a r e a s i n which r e g u l a r m o r t a l i t y has been measured a r e g i v e n i n Hervey (1936), Hunger (1946), Johnson (1953), S t a e b l e r (1953, 1955a, 1955*0, S t e e l e and W o r t h i n g t o n (1955), S.A.F. (1956), Wright and L a u t e r b a c h (1958), W i l l i a m s o n (1963), H e p t i n g (1964), C u r t i s (1965), Anon. (1967), Lee (1969), and Reukema (1969). Throughout the p r e s e n t paper, t h e f o l l o w i n g t h r e e t y p e s of m o r t a l i t y s h a l l be d i f f e r e n t i a t e d : 1. CATASTROPHIC MORTALITY: widesp r e a d m o r t a l i t y , i n c u r r e d by c a t a s t r o p h e s l i k e f i r e s , windstorms, e x t r e m e l y heavy r a i n - , snow-, s l e e t - , o r h a i l - s t o r m s , and i n s e c t e p i d e m i c s . 2. IRREGULAR MORTALITY: s p o r a d i c m o r t a l i t y observed on s c a t -t e r e d i n d i v i d u a l t r e e s o r s m a l l groups o f t r e e s as caused by A l i t e r a t u r e r e v i e w 8 i d e n t i f i a b l e a g e n t s or causes l i k e endemic d i s e a s e s and i n -j u r i e s provoked e i t h e r by i n s e c t s , a n i m a l s , f l o o d s , c l i m a t i c v a r i a t i o n s ( e x t r e m e s ) , or a c c i d e n t s (wounds, s c a r s , b r e a k a g e s ) , 3. REGULAR MORTALITY: m o r t a l i t y i n d u c e d by the i n d i v i d u a l t r e e ' s i n a b i l i t y t o compete s u c c e s s f u l l y v / i t h i t s n e i g h b o u r s t o get s u f f i c i e n t amounts and/or q u a l i t y of l i g h t , water and nu-t r i e n t s t o s t a y a l i v e . I t must be n o t i c e d t h a t t h i s c l a s s i f i c a t i o n , based on s p a t i a l d i s t r i b u t i o n and causes, i s f a r from b e i n g r i g i d be-cause of v a r i a t i o n s i n d i s t r i b u t i o n s and i n t e r r e l a t i o n s between causes, e s p e c i a l l y when d i a g n o s e s are made l o n g a f t e r the a c t u a l d e a t h . I t i s , however, c o n v e n i e n t t o use f o r q u a l i f y i n g c u r r e n t e s t i m a t e s o f m o r t a l i t y , made i n most growth and y i e l d sample p l o t s . 2.2 CAUSES OF MORTALITY 2 . 2 . 1 COMPETITION I n a p l a n t s o c i e t y , i n d i v i d u a l s m u t u a l l y attempt t o modify t h e i r s u r r o u n d i n g s , and t h u s compete a g a i n s t each o t h e r m a i n l y f o r growth f a c t o r s . I n r e c e n t y e a r s , c o m p e t i t i o n has a t t r a c t e d c o n s i d e r a b l e a t t e n t i o n from s c i e n t i s t s ( K i r a . e t a l . , 1953, Kramer and K o z l o w s k i , I 9 6 0 , Harper, 1961, K o z l o w s k i , 1962, 1968, R O h r i g , 19Gk, M i l l e r , 1967, K r u e g e r , I 9 6 0 , 1967, Osborn, 1 9 6 8 a ) . The a u t h o r made a thorough r e v i e w o f t h i s p r o c e s s and of the v a r i o u s methods deve l o p e d t o measure i t , as A l i t e r a t u r e r e v i e w 9 background i n f o r m a t i o n f o r the p r e s e n t s t u d y ( P a i l l e " , 1969a). U n t i l the 1950's, most measurements o f c o m p e t i t i o n i n f o r e s t s were made on a stand, b a s i s . However, w i t h the advent of e l e c t r o n i c computer f a c i l i t i e s , and a f t e r t he p u b l i c a t i o n o f B i c k f o r d e t a l . (1957) on the s t a t u s o f s t o c k i n g , n o r m a l i t y , and s t a n d d e n s i t y measurements, a g r e a t d e a l o f emphasis was put on the e v a l u a t i o n o f the i n d i v i d u a l t r e e c o m p e t i t i v e p o s i -t i o n . I n f a c t , t he t r e n d was s e t e a r l i e r by Chisman and Schumacher (1940) who determined the minimum a r e a t h a t a t r e e would need t o s u r v i v e , as a f u n c t i o n of i t s s i z e . S t a e b l e r , i n 1951> developed a c o m p e t i t i o n i n d e x on the assumption t h a t the c o m p e t i t i o n e x e r t e d upon a t r e e was d i r e c t l y p r o p o r t i o n a l t o the o v e r l a p of i t s c o m p e t i t i o n c i r c l e ( a f u n c -t i o n o f i t s s i z e ) by those o f i t s n e i g h b o u r s . I n 1953j Ker suggested t h a t the number o f s i d e s f r e e of crown o v e r l a p c o u l d be c o n s i d e r e d as an i n d e x o f c o m p e t i t i o n f o r i n d i v i d u a l t r e e s . K r a j i c e k e t a l . (1961) p e r f e c t e d an i n d i c a t o r o f crown c o m p e t i t i o n f o r growing space, by c a l c u l a t i n g the a r e a a v a i l -a b l e t o the average t r e e i n the s t a n d i n r e l a t i o n t o the maxi-mum a r e a i t c o u l d use i f i t were open-grov/n. Spurr (1962) proposed an angle-summation method aimed a t e v a l u a t i n g t he i n f l u e n c e of competing b a s a l a r e a on the b a s a l a r e a i n c r e m e n t o f i n d i v i d u a l t r e e s . Newnham (1964) used the a n g l e of crown i n t e r s e c t i o n t o e v a l u a t e c o m p e t i t i o n . H i s i n d e x was based on the p r o p o r t i o n o f A l i t e r a t u r e r e v i e w 10 a s u b j e c t t r e e crown c i r c u m f e r e n c e e n c l o s e d by those o f ad-j a c e n t t r e e s . T h i s i n d e x was s l i g h t l y m o d i f i e d by Lee (1967) w i t h r e g a r d t o c o m p e t i t o r ' s d i s t a n c e s from the competing t r e e . Brown (1965), assuming t h a t each t r e e has a v a i l a b l e f o r i t s e l f h a l f the d i s t a n c e t o each of i t s n e i g h b o u r s , d e f i n e d c o m p e t i t i o n i n terms o f Area P o t e n t i a l l y A v a i l a b l e . G e r r a r d (1968) h y p o t h e s i z e d t h a t the c o m p e t i t i o n s t r e s s s u s t a i n e d by a t r e e would be d i r e c t l y p r o p o r t i o n a l t o the over -l a p o f i t s c o m p e t i t i o n c i r c l e w i t h those of i t s n e i g h b o u r s , and i n v e r s e l y p r o p o r t i o n a l t o the a r e a of i t s own c o m p e t i t i o n c i r c l e . The c o m p e t i t i o n q u o t i e n t developed was based on the assumption t h a t l a r g e r t r e e s c o u l d endure a more i n t e n s i v e com-p e t i t i o n . A s i m i l a r approach was t a k e n by Opie (1968) i n the development o f a n z o n e count" model. M i t c h e l l (1969) took h e i g h t growth and crown e x p a n s i o n i n t o account t o determine the degree of i n d i v i d u a l t r e e over-t o p p i n g . H i s p r e m i s e s were t h a t the c o m p e t i t i v e s t r e s s on an i n d i v i d u a l t r e e depends on a v a i l a b l e l i g h t as determined by i t s h e i g h t and crown s i z e r e l a t i v e t o competing t r e e s , and t h a t h e i g h t and crown growth a re i n f l u e n c e d by the c o m p e t i t i v e p o s i -t i o n o f the t r e e . L i n (1969) c a l c u l a t e d t h a t a wes t e r n hemlock t r e e had ample space t o grow when the view a n g l e from i t s stem t o t h a t o f a ne i g h b o r was l e s s than 2.15 degrees; i t s minimum growing space was re a c h e d when i t stopped e n l a r g i n g i t s d i a m e t e r , or when the view a n g l e was e q u a l o r l a r g e r than 5.25 degrees. A l i t e r a t u r e r e v i e w 2.1 He gave a v a l u e of 25 t o each of f o u r q u a d r a n t s around a sub-j e c t t r e e i n which the sura of the a n g l e s subtended by compe t i -t o r s was l e s s than 2.15°, t h i s v a l u e d e c r e a s i n g t o aero as the sum of a n g l e s reached 5.25°. T h i s i n d e x proved t o be h i g h l y c o r r e l a t e d w i t h t r e e d i a m e t e r i n c r e m e n t . F i n a l l y , B e l l a (1969), t r y i n g t o get away from the l i n e -a r l y a d d i t i v e type o f c o m p e t i t i o n e f f e c t between competing t r e e s and c o m p e t i t o r s , promoted the c o m p e t i t i v e i n f l u e n c e - z o n e o v e r l a p concept. The two p r i n c i p l e s i n v o l v e d were t h a t the c o m p e t i t i o n e f f e c t on a competing t r e e i s p r o p o r t i o n a l t o the amount o f zone o v e r l a p o f i t s c o m p e t i t o r s , whereas the i n d i v i -d u a l c o n t r i b u t i o n of a c o m p e t i t o r depends on i t s r e l a t i v e s i z e and t h a t o f the competing t r e e , e x p o n e n t i a l l y weighted. S e v e r a l c h a r a c t e r i s t i c s a r e common to thes e c o m p e t i t i o n models. F i r s t l y , most o f them a r e based on the assumption t h a t t r e e d i a m e t e r o r t r e e d i a m e t e r i n c r e m e n t a t b r e a s t h e i g h t i s a parameter r e f l e c t i n g p a s t and c u r r e n t c o m p e t i t i v e s t r e s s e s s u s t a i n e d by t r e e s , i n the ground a s w e l l as i n the canopy above, due t o the c l o s e r e l a t i o n s h i p between crown spread, r o o t spread and t r e e s i z e ( S m i t h , 196.3, 1964b, 1966b). A second common assumption i s t h a t t r e e s grow a t t h e i r f a s t e s t r a t e and spread t h e i r branches out t o a maximum e x t e n t i n the open, presumed t o be a c o m p e t i t i o n l e s s environment. However, a l l t h e s e c o m p e t i t i o n models d i f f e r i n t h e i r d e f i n i t i o n of a minimum v i t a l space, and i n t h e i r procedure t o reduce growth as c o m p e t i t i o n l e v e l s v a r y w i t h t i m e . None of A l i t e r a t u r e r e v i e w ]_? them i s c o m p l e t e l y s a t i s f a c t o r y f o r p r e d i c t i n g m o r t a l i t y be-cause t r e e s do not always d i e even a f t e r a c o n s i d e r a b l e r e d u c -t i o n i n t h e i r g i r t h i n c r e m e n t . Newnham (1966) and N i s h i z a w a (1968) have e s t i m a t e d the r e l a t i v e m e r i t s o f s e v e r a l c o m p e t i t i o n formulae i n e s t i m a t i n g t r e e growth. Angle-summation methods were r a t e d as b e s t among 8 measures by N i s h i z a w a to e v a l u a t e r e c e n t d i a m e t e r growth, whereas Newnham chose h i s c o m p e t i t i o n i n d e x among 18 o t h e r s . 2.2.2 INJURIES I n j u r i e s to t r e e s by i n s e c t s , a n i m a l s , weather con-d i t i o n s , and f i r e s account f o r 88 p e r c e n t o f the m o r t a l i t y a t t r i b u t e d t o n a t u r a l causes i n the P a c i f i c Northwest ( M e t c a l f , 1968). H a l f of t h i s l o s s i s clue t o i n s e c t s a l o n e . A l i t e r a t u r e r e v i e w made by the a u t h o r ( P a i l l e ' , 1969b) has shown t h a t a t l e a s t 300 i n s e c t s f e e d on Douglas f i r cones, seeds, s e e d l i n g s , l e a v e s , bark and wood. Of t h i s number, o n l y about a dozen have been r a t e d as d e a d l y . The most damaging of them a l l i s the Douglas f i r b ark b e e t l e , Dendroctonus pseudo-tsugae Hopk., which can d e v e l o p i n t o epidemic p r o p o r t i o n s a f t e r b e i n g f a v o r e d by windthrows, snowbreaks, or f i r e s (Wright and L a u t e r b a c h , 1958, W r i g h t and Harvey, 1967). Animal damage t o young Douglas f i r f o r e s t s has been a t t r i b u t e d to 19 s p e c i e s , to w h i c h deer c o n t r i b u t e d more than 50 p e r c e n t (Anon., 1968, Crouch, 1969). However, a c c o r d i n g t o M i t c h e l l (1964), even i n a r e a s where widespread and r e p e a t e d A l i t e r a t u r e r e v i e w ]_3 b r o w s i n g by deer i s observed, growth and s u r v i v a l a r e not s e r i o u s l y a f f e c t e d . Heat, f r o s t , d rought, snow, i c e , s l e e t , and. wind a l t o -g e ther cause a g r e a t amount o f damage e s p e c i a l l y t o s m a l l p l a n t s and t o young f o r e s t s . However, except f o r snow- and wind-breakages, weather-caused i n j u r i e s provoke l i t t l e d i r e c t mor-t a l i t y but f a v o r the development o f p e s t s and d i s e a s e s which are much more damaging. F i r e s cause r e l a t i v e l y l i t t l e damage t o f o r e s t s i n the P a c i f i c Northwest. I n 1962, f o r i n s t a n c e , 30 MM c u b i c f e e t of wood were b u r n t i n Oregon and Washington, which r e p r e s e n t e d about t h r e e p e r c e n t o f a l l l o s s e s (Anon., 1967)* 2.2,3 DISEASES Tre e s i n j u r e d a r e almost i n v a r i a b l y i n v a d e d by d i s e a s e s which slow down t h e i r r a t e o f growth, k i l l them, and c r e a t e c u l l . West o f the Cascades, i n Oregon and Washington, 91 MM c u b i c f e e t of wood a r e l o s t i n t h i s manner each y e a r i n Douglas f i r f o r e s t s ( C h i l d s and Shea, 1967). F o r t y p e r c e n t of t h i s amount i s withdrawn from second-growth f o r e s t s , m a i n l y as a growth l o s s . The a c t u a l m o r t a l i t y i n the s e s t a n d s r e p r e s e n t s o n l y 10 p e r c e n t o f the t o t a l l o s s o f 90 MM c u b i c f e e t . Root-r o t i s the major s i n g l e agent r e s p o n s i b l e f o r d i s e a s e - m o r t a l i t y i n young f o r e s t s . I t has been s t u d i e d e x t e n s i v e l y by W a l l i s and R e y n o l d s (1965), and F o s t e r and Johnson (1963a) i n B r i t i s h Columbia, and by C h i l d s (i960) i n the U n i t e d S t a t e s . T h i s A l i t e r a t u r e r e v i e w li+ d i s e a s e i s p a r t i c u l a r l y damaging i n t h a t i t shows c o n t a g i o u s d i s p e r s i o n , c r e a t i n g i n c r e a s i n g l y l a r g e openings i n the f o r e s t canopy; i t o c c u r s a l s o on i s o l a t e d i n d i v i d u a l s , where i t i s much more d i f f i c u l t , t o r e c o g n i z e . 2.3 AMOUNT AND TIMING OF MORTALITY 2.3.1 MORTALITY IN NATURAL FORESTS 2.3.1.1 DURING STAND ESTABLISHMENT PERIOD A f t e r c l e a r c u t t i n g s ( f o l l o w e d or not by br o a d c a s t b u r n i n g s ) , n a t u r a l Douglas f i r s e e d l i n g s s u f f e r s e r i o u s m o r t a l i t y caiised by heat i n j u r y , drought, f r o s t , i n -s e c t , r o d e n t s , r o o t - r o t , damping-off, o r c o m p e t i t i o n from h e r -baceous v e g e t a t i o n . The r e l a t i v e i n f l u e n c e o f f a c t o r s a f f e c -t i n g s e e d l i n g e s t a b l i s h m e n t was d i s c u s s e d a t l e n g t h by Is a a c (1943), A l l e n (1942), Garman (1955), Anon. (1958), Bunce (I960), H u n t l y (I960), Spurr (1961), V/ommack (1964), and Smith e t a l . (1966). A f t e r the t a k e - o f f , s a p l i n g s u s u a l l y grow i n dense t h i c k e t s , w i t h more t h a n 1,000 stems per a c r e (Anon., 1965c). I n f a c t , Douglas f i r y i e l d t a b l e s i n d i c a t e t h a t , a t age 10 i n w e l l s t o c k e d s t a n d s , the number o f stems per a c r e v a r i e s be-tween 1500 and 3500 on s i t e s 170 and 80 r e s p e c t i v e l y (Anon., 1959); and, on the average s i t e , t he number p r e s e n t a t age 10 would be reduced by 30 p e r c e n t b e f o r e age 20. T h i s s i g n i f i e s t h a t , by t h a t t i m e , c o m p e t i t i o n amongst t r e e s f o r l i g h t , nu-t r i e n t and water has a l r e a d y s t a r t e d . A l i t e r a t u r e r e v i e w ^5 The " s u c c e s s " of r e g e n e r a t i o n i s e s t i m a t e d by v a r i o u s s t o c k e d - q u a d r a t s u r v e y s c a r r i e d out up t o t h r e e t i m e s between age 1 and 10. F o u r - m i l a c r e q u a d r a t s , spaced 2 or more c h a i n s a p a r t on l i n e s l a i d 10 to 20 c h a i n s a p a r t a c r o s s the f i e l d , a r e f a v o r e d f o r such r e c o n n a i s s a n c e s . P o i n t s a m p l i n g has a l s o been used (Anon., 1958). Re-examinations u s u a l l y cease when a minimum s t o c k i n g i s i n s u r e d , and when s e e d l i n g s have escaped from b r u s h c o m p e t i t i o n . At p r e s e n t , minimum s t o c k i n g s t a n d a r d s i n t h e P a c i f i c Northwest are a t l e a s t J l p e r c e n t by 1 - m i l a c r e , or 40 p e r c e n t by 4 - m i l a c r e q u a d r a t s (50% s u r v i v a l ) , which means t h a t 200 t o 2+00 s e e d l i n g s per a c r e a re e s t a b l i s h e d . Methods and s t a n d a r d s have been d i s c u s s e d by A l l e n e t a l . (195D, Dem-b i c k i (1955), Smith and Ker (1958), Weetman (1957), and Barn-f o r d (1968). S t a e b l e r (192+9) has shown how n o r m a l i t y and c u b i c volume o f Douglas f i r s t a n d s can be p r o j e c t e d 15 y e a r s hence, by knowing age, s i t e and 2+-milacre s t o c k i n g a t age 5, 10, o r 15. 2.3.1.2 IN YOUNG FORESTS R e g u l a r m o r t a l i t y o c c u r r i n g i n Douglas f i r s t a n d s between the age o f 20 and 160 can be a p p r e c i a t e d i n num-be r of t r e e s from normal y i e l d t a b l e s (McArdle e t a l . , 192+9). They show t h a t , b e f o r e t r e e s r e a c h 160 y e a r s , t h e i r number i s reduced 12 t o 28 t i m e s depending on s i t e , t he g r e a t e s t r e d u c -t i o n o c c u r r i n g on the p o o r e s t s i t e s . The most severe drop, e i t h e r i n a b s o l u t e v a l u e o r p e r c e n t a g e , t a k e s p l a c e between age A l i t e r a t u r e review l g 20 and AO, as i n d i c a t e d i n Table I. Thus, i n immature Douglas f i r stands, 5 to 50 percent of the l i v e trees die each decade, t h i s p r o p o r t i o n decreasing s t e a d i l y with time a f t e r age 40. TABLE I DECADAL PERCENTAGE MORTALITY IN NUMBER OF TREES AVERAGE FROM AVERAGE FROM MCAP.DLE AGE ANON. (1959) ET AL. (1949) S i t e IIO-I4O-I7O Percentage Percentage 20 30 40 50 60 70 80 35 43 48 33 39 P-7 28 22 20 19 19 15 Unless they are t r a n s l a t e d i n t o volumes, annual l o s s e s to reg u l a r m o r t a l i t y , ranging from 1.5 to 4.8 percent by number of tre e s , cannot be f u l l y appreciated. Using net y i e l d t a b l e s f o r estimating number of dead t r e e s , Staebler (1955a) b u i l t gross y i e l d t a b l e s f o r Douglas f i r . Apart from i n d i c a t i n g the f u l l productive capacity of Douglas f i r s i t e s , the gross volume-over-age curves i l l u s t r a t e , when coupled with net y i e l d curves, A l i t e r a t u r e r e v i e w 17 the p o t e n t i a l l o s t t o m o r t a l i t y . F i g u r e 1 (A and B) i s p r e -sented f o r t h a t purpose. From i t , some s t r i k i n g o b s e r v a t i o n s can be made r e l a t i v e t o m o r t a l i t y : on any s i t e ( I t o V), more than 30 p e r c e n t as much wood by c u b i c volume has d i e d as i s l e f t s t a n d i n g a t age 80, and t h r e e t i m e s more wood volume has been l o s t on s i t e I than on s i t e V. N e v e r t h e l e s s , t h e s e e s t i -mates o f m o r t a l i t y a r e b e l i e v e d t o be c o n s e r v a t i v e s i n c e f i r s t , not a l l s t a n d s support normal s t o c k ; second, h i g h e r g r o s s y i e l d e s t i m a t e s have been made ( C u r t i s , 1967); and t h i r d , o n l y r e g u l a r m o r t a l i t y has been t a k e n i n t o a c c o u n t . R e g u l a r mor-t a l i t y f i g u r e s c o u l d be i n c r e a s e d by more than 20 p e r c e n t t o i n c l u d e l o s s e s due t o i r r e g u l a r m o r t a l i t y on a r e g i o n - w i d e b a s i s ( M c t c a l f , 1968). When m o r t a l i t y i s ex p r e s s e d on an a n n u a l b a s i s , 20 t o more than 60 c u b i c f e e t o f wood per a c r e a r e l o s t on an average s i t e . I n f a c t , Johnson (1953) c a l c u l a t e d a mean a n n u a l l o s s o f 83 c u b i c f e e t on a l a r g e number o f p l o t s i n second-growth Douglas f i r s t a n d s . 2.3.1.3 IN MATURE FORESTS The average a n n u a l amount o f n a t u r a l mor-t a l i t y ( e x c l u d i n g c a t a s t r o p h i c l o s s e s ) a s e s t i m a t e d by McMahon (1961) i n o l d - g r o w t h f o r e s t s (180 y e a r s o f age and ove r ) would be 359 board f e e t (more than 60 c u b i c f e e t ) p e r a c r e p e r year (based on 342 survey p l o t s measured i n 12 c o u n t i e s throughout the Douglas f i r R egion, i n c l u d i n g t r e e s 11 i n c h e s p l u s ) ; Yield 35h Figure la- Mortality of Douglas-fir in normal stands1 (thousands of cu- ft* per acre) Sites land II 18 30 All trees 1-5 Inches and larger-From Staebler (1955 a), McArdle et al (1949)and Curtis (1967)-25h 20 YUBW 251 20 Figure lb- Mortality of Douglas-fir in normal 1 9 stands' (thousands of cu* ft- per acre) Sites HI, JSL and 3E-1AES trees 1-5 inches and larger From Staebler (l955a),McArdle etah (1949) and Curtis (1967)-A l i t e r a t u r e r e v i e w 20 moreover, t h i s l o s s would be c o n c e n t r a t e d on the Douglas f i r component. On the o t h e r hand, the an n u a l g r o s s increment p e r ac r e i n these s t a n d s ( c o v e r i n g 3 m i l l i o n a c r e s ) would range from 200 t o 800 board f e e t . T h i s s u g g e s t s t h a t the s t a n d i n g t i m b e r volume d e c r e a s e s a n n u a l l y , a t l e a s t i n some o l d e r s t a n d s . T h i s has been c o n f i r m e d by S t e e l e and W o r t h i n g t o n (1955), I s a a c (1956), and W r i g h t and L a u t e r b a c h (1958); however, blow-downs and b a r k b e e t l e s were i n v o l v e d i n most c a s e s . 2.3.2 MORTALITY IN PLANTATIONS The mere f a c t t h a t Douglas f f i r i s the main s p e c i e s used i n Northwest America f o r r e f o r e s t a t i o n i n d i c a t e s t h a t , by i t s e l f , i t does not p r e s e n t i m p o r t a n t s u r v i v a l problems. Bare-r o o t p l a n t i n g o f 2-0 s t o c k o f t e n g i v e s 70 p e r c e n t s u r v i v a l r a t e s o r b e t t e r ( P a i l l e ' , 1968). However, bad p l a n t i n g t e c h -n i q u e s , hot s l a s h burns and l o n g p e r i o d s o f drought can cause much v a r i a t i o n i n s u r v i v a l , e s p e c i a l l y on s o u t h e r l y a s p e c t s a f f e c t e d by h i g h t e m p e r a t u r e s . Repeated b r o w s i n g by a n i m a l s , almost e x c l u s i v e l y c o n c e n t r a t e d on p l a n t e d s e e d l i n g s , may not c r e a t e s e r i o u s l o s s e s , but does a f f e c t the time taken, by p l a n t e d s t o c k t o overcome b r u s h c o m p e t i t i o n which i s a s e r i o u s c o n c e r n on h i g h s i t e l a n d s . S i n c e s u r v i v a l and growth are c l o s e l y r e l a t e d t o h e i g h t a t time of p l a n t i n g ( S m i t h and A l l e n , 1962, K n i g h t , 1957), and s i n c e r e l a t i v e h e i g h t advantage o f a t r e e i s o f t e n m a i n t a i n e d o r i n c r e a s e d i n time (Warrack, 1952), b r o w s i n g can be r e g a r d e d as a f a c t o r c l o s e l y a s s o c i a t e d w i t h f u t u r e A l i t e r a t u r e r e v i e w 21 t r e e dominance, or c o m p e t i t i v e a b i l i t y . I n i t i a l s p a c i n g i s one o f the most i m p o r t a n t s i n g l e f a c -t o r s c o n t r o l l i n g amounts of j u v e n i l e m o r t a l i t y o f r e g u l a r and i r r e g u l c d r n a t u r e . F o r i n s t a n c e , f o r t y y e a r s of o b s e r v a t i o n s i n a s p a c i n g p l a n t a t i o n showed t h a t c l o s e l y spaced s t a n d s (6 x 6 f e e t or l e s s ) s u f f e r e d t w i c e as much m o r t a l i t y as w i d e l y spaced ones (8 x 8 f e e t o r l a r g e r ) . I n a 20-year p e r i o d , r e g u l a r mor-t a l i t y took 20 t i m e s more t r e e s i n a 4 x 4 f o o t - s p a c i n g than i n a 12 x 12 f o o t - s p a c i n g ; l o s s e s and damage due t o snow storms and r o o t - r o t were c o n s i d e r a b l y g r e a t e r i n the d e n s e s t s p a c i n g . The net r e s u l t was t h a t t o t a l p r o d u c t i o n was l e s s i n dense s t a n d s , and a c o n s i d e r a b l e p a r t o f the p r o d u c t i o n was and w i l l be wasted by t r e e s unable t o r e a c h merchantable s i z e s (Eeukema, 1969). S p a c i n g i n r e l a t i o n t o growth and development of c o n i -f e r s i n p l a n t a t i o n has been s t u d i e d by S j o l t e - J o r g e n s e n (1967) f o r a l a r g e number of s p e c i e s , i n c l u d i n g Douglas f i r . 2.4 DISTRIBUTION OF MORTALITY 2.i+.l FREQUENCY DISTRIBUTIONS OF TREE PARAMETERS Wh i l e r e c o g n i z i n g t h a t the b u l k of m o r t a l i t y usu-a l l y o c c u r s i n the s m a l l e r d i a m e t e r c l a s s e s ( a s i n d i c a t e d i n normal y i e l d t a b l e s by the r a p i d d e c l i n e i n number of t r e e s v/ith t i m e ) , f o r e s t e r s have seldom d e s c r i b e d d i a m e t e r f r e q u e n c y d i s t r i b u t i o n s of dead t r e e s . I n s t e a d they have attempted t o e v a l u a t e volumes o r r a t e s o f m o r t a l i t y i n o r d e r to a d j u s t growth e s t i m a t e s based on s t a n d - t a b l e p r o j e c t i o n s . Lee (1967) has A l i t e r a t u r e r e v i e w 22 shown how r a t e s of m o r t a l i t y can be r e l a t e d t o d i a m e t e r i n or d e r t o remove l o s s e s by d i a m e t e r c l a s s . An e m p i r i c a l study o f percentage d i a m e t e r d i s t r i b u t i o n of m o r t a l i t y on l o d g e p o l e p i n e was a l s o c a r r i e d out by Lee (1969). He found t h a t p e r c e n t a g e d i a m e t e r f r e q u e n c i e s were n o r m a l l y d i s t r i b u t e d , w i t h averages a t mean l i v e t r e e d i a m e t e r minus 2 i n c h e s . E x t e n s i v e s i m u l a t i o n s t u d i e s have been c a r r i e d out by Newnham (1964) and Smith e t a l . (1965) i n which they t e s t e d the i n f l u e n c e on average s t a n d d i a m e t e r , b a s a l a r e a , and number o f t r e e s o f a number o f d i f f e r e n t d i s t r i b u t i o n s o f m o r t a l i t y , o c c u r r i n g a t s p e c i f i c p e r i o d s i n tim e , w i t h a s p e c i f i c i n t e n -s i t y . They showed t h a t , f o r non-clumped d i s t r i b u t i o n s , the k i n d o f d i a m e t e r d i s t r i b u t i o n o f m o r t a l i t y i n v o l v e d was l e s s i m p o r t a n t than the amount, and t h a t the d i a m e t e r d i s t r i b u t i o n of l i v e t r e e s was not g r e a t l y a f f e c t e d by the d i s t r i b u t i o n o f dead t r e e s . Diameter d i s t r i b u t i o n s o f l i v e t r e e s have a t t r a c t e d much more a t t e n t i o n because most growth p r e d i c t i o n s a r e made by diam e t e r c l a s s e s . I t i s g e n e r a l l y r e c o g n i z e d t h a t , i n even-aged s t a n d s , these d i s t r i b u t i o n s a r e skewed, a l t h o u g h the assumption o f n o r m a l i t y i n s i d e each d i a m e t e r c l a s s i s c u r r e n t l y a c c e p t e d f o r c o n v e n i e n c e . More d e t a i l s p e r t a i n i n g t o Douglas f i r and o t h e r s p e c i e s can be found i n the f o l l o w i n g r e f e r e n c e s : Baker (1923), Meyer (1930), Munger (1945), Spurr (1952), Sammi (1961, 1969), Leak (1964, 1965), and Bennett (1964). R e c e n t l y , A l i t e r a t u r e r e v i e w 23 o p e r a t i o n s r e s e a r c h t e c h n i q u e s (Markov c h a i n s ) have been a p p l i e d to the study of changes i n di a m e t e r d i s t r i b u t i o n s (Rudra, (1968) . The g e n e r a t i o n of h y p o t h e t i c a l f o r e s t s t a n d s i n com-p u t e r s a l s o r e q u i r e d s t u d i e s o f d i a m e t e r f r e q u e n c y d i s t r i b u -t i o n s (Newnham and M a l o l e y , 1970) . S t u d i e s o f h e i g h t f r e q u e n c y d i s t r i b u t i o n s p e r se a r e l a c k i n g . I n s t e a d , crown c l a s s e s a r e sometimes t a k e n i n t o con-s i d e r a t i o n (Warrack, 1952, Ker, 1953) . However, h e i g h t d i s -t r i b u t i o n s a r e i n d i r e c t l y observed whenever h e i g h t - d i a m e t e r r e -l a t i o n s h i p s a r e e s t a b l i s h e d f o r growth p r e d i c t i o n purposes ( C r o s s l e y , 196?). 2 . 4 . 2 MORTALITY SPATIAL DISTRIBUTIONS Newnham (1964) and Smith et a l . (1965) have simu-l a t e d the e f f e c t s of v a r i o u s s p a t i a l d i s t r i b u t i o n s o f m o r t a l -i t y . They showed t h a t the amount and t i m i n g o f m o r t a l i t y a r e more i m p o r t a n t than i t s s p a t i a l arrangement. F o r example, on good Douglas f i r s i t e s , b a s a l a r e a s a t age 80 were q u i t e comparable i n s t a n d s h a v i n g s u f f e r e d up t o 70 p e r c e n t normal-, P o i s s o n - , or n e g a t i v e b i n o m i a l - m o r t a l i t y a t a j u v e n i l e stage o f development. As l o n g as the i n i t i a l number of t r e e s p e r a c r e d i d not f a l l below 300, t h e r e was no d i f f e r e n c e on y i e l d s a t age 100, p r o v i d i n g o n l y r e g u l a r m o r t a l i t y (due t o c o m p e t i t i o n ) f o l l o w e d j u v e n i l e m o r t a l i t y . I n the advent o f " f u l l y clumped" m o r t a l i t y , y i e l d a t h a r v e s t would have been reduced i n d i r e c t p r o p o r t i o n t o unoccupied a r e a s . A l i t e r a t u r e r e v i e w c% I n n a t u r a l o r p l a n t e d Douglas f i r , t h e r e i s e m p i r i c a l e v i d e n c e t o show t h a t f u l l d u m p i n e s s o f m o r t a l i t y w i l l o c c u r a f t e r c a t a s t r o p h e s ( f i r e s , wind-, snow-, or sleet-damage) and i n r o o t - r o t i n f e s t e d s t a n d s , but seldom o t h e r w i s e . The d i s -t r i b u t i o n of the l a t t e r cause of m o r t a l i t y was i n v e s t i g a t e d by F o s t e r and Johnson (1963a) i n t h r e e young p l a n t a t i o n s on Van-couver I s l a n d , B.C. 2.5 PREDICTION OF GROWTH, YIELD AND MORTALITY Methods of e v a l u a t i n g and p r e d i c t i n g growth and y i e l d can be c l a s s i f i e d i n t o t h r e e groups: 1) d i r e c t methods, based on s u c c e s s i v e measurements of permanent sample p l o t s ; 2) i n d i r e c t methods, based cn temporary sample p l o t s ; and 3) s i m u l a t i o n t e c h n i q u e s , based on the e v a l u a t i o n o f i n d i v i d u a l t r e e p e r f o r -mance and c o m p e t i t i o n i n d i c e s . S i n c e m o r t a l i t y can be exp r e s s e d as a d i f f e r e n c e between g r o s s and net measures of growth o r y i e l d , i t seems l o g i c a l t h a t methods f o r p r e d i c t i n g i t be c l a s s i f i e d i n a s i m i l a r o r d e r . A comprehensive r e v i e w o f the p r e s e n t s t a t u s o f growth and y i e l d p r e d i c t i o n s was made by C u r t i s (1969). 2.5.1 DIRECT METHODS Repeated measurements of the same p l o t s , s c a t t e r e d over l a r g e and r e p r e s e n t a t i v e a r e a s , i s the most p r e c i s e and presumably the most e x p e n s i v e means o f e s t i m a t i n g t i m b e r y i e l d s and m o r t a l i t y . I n the p a s t , l a r g e p l o t s (up t o one a c r e i n A l i t e r a t u r e r e v i e w 25 s i z e ) were e s t a b l i s h e d f o r t h a t purpose i n Douglas f i r f o r e s t s ( W i l l i a m s o n , 1963, Thomson, 1966). Nowadays, a l a r g e r number of s m a l l e r p l o t s i s used i n c o n j u n c t i o n w i t h c o n t i n u o u s f o r e s t i n v e n t o r y systems (B.C. F o r e s t S e r v i c e , 1957, Anon., 1966, Le'tourneau, 1966), The f i r s t attempt a t u t i l i z i n g the d i r e c t method t o e v a l -uate c u m u l a t i v e m o r t a l i t y and t o b u i l d g r o s s y i e l d t a b l e s was made by Hunger (194-6). K a i l (1959) d i s c u s s e d the p r e c i s i o n o f the d i r e c t method f o r e s t i m a t i n g growth by comparison t o o t h e r methods based on temporary p l o t s . 2.5.2 INDIRECT METHODS The v a r i o u s t e c h n i q u e s c l e . s s i f i e d i n t h i s group were not d e s i g n e d t o p r e d i c t m o r t a l i t y , but e i t h e r g r o s s o r net f o r e s t growth and y i e l d . 2.5.2.1 NORMAL YIELD TABLES Based on temporary p l o t s e s t a b l i s h e d i n f u l l y - s t o c k e d s t a n d s , normal y i e l d t a b l e s p r o v i d e c u b i c volume e s t i m a t e s of net y i e l d s by t e n - y e a r p e r i o d s and by s i t e ( o r average d i a m e t e r ) . The d i f f e r e n c e i n number o f t r e e s per u n i t a r e a between tv/o t e n - y e a r p e r i o d s can be re g a r d e d as p e r i o d i c m o r t a l i t y . Once volume per t r e e i s e s t i m a t e d (by means of stem a n a l y s i s o r o t h e r w i s e ) , normal g r o s s y i e l d t a b l e s can be b u i l t and used t o p r e d i c t normal m o r t a l i t y . D e t e r m i n a t i o n of nor-m a l i t y and t r e n d s towards n o r m a l i t y of a c t u a l s t a n d s w i l l A l i t e r a t u r e r e v i e w 26 p e r m i t the use o f the s e t a b l e s f o r p r e d i c t i n g m o r t a l i t y a s the d i f f e r e n c e between p r e d i c t e d g r o s s and net y i e l d s . Such t a b l e s are a v a i l a b l e f o r Douglas f i r (McArdle e t a l . , 1949, S t a e b l e r , 1955a)> and t h e i r use i n p r e d i c t i n g f u t u r e volumes was d i s -cussed by Johnson (1955)• A v a r i a b l e d e n s i t y y i e l d t a b l e has a l s o been b u i l t by McKeever (1947). 2.5.2.2 STAND-TABLE PROJECTION T h i s t e c h n i q u e s e r v e s f o r computing g r o s s growth from p a s t i n c r e m e n t i n d i a m e t e r as measured on c o r e s , and f o r p r o j e c t i n g t r e e s i z e s ( s t a n d t a b l e s ) i n t o the f u t u r e by d i a m e t e r c l a s s . Adjustments must be made f o r bark growth, and changes i n h e i g h t - d i a m e t e r r e l a t i o n s and t r e e form over t i m e . T h i s i n f o r m a t i o n i s c o n v e r t e d i n t o p r e s e n t - s t o c k t a b l e s and f u t u r e - s t o c k t a b l e s by use o f l o c a l volume t a b l e s or tree-volume t a r i f t a b l e s . S t ock t a b l e s (volume by di a m e t e r c l a s s ) can a l s o be p r o j e c t e d d i r e c t l y by means of volume growth p e r c e n t t a b l e s . However, the a c c u r a c y o f the s t a n d - t a b l e p r o j e c t i o n i n p r e d i c t i n g f u t u r e net volumes depends on how r e g e n e r a t i o n ( i n -growth) and m o r t a l i t y r a t e s a re e v a l u a t e d . The l i t e r a t u r e a v a i l a b l e on the s u b j e c t g e n e r a l l y d e s c r i b e s growth e s t i m a t e s i n much more d e t a i l t han m o r t a l i t y e s t i m a t e s (Meyer, 1942, Anon., 1947, Spurr, 1952, Lynch, 1962, T u r n b u l l e t a l . , 1963, F l o r a and Fedkiw, 1964, Pope, 1965). A l i t e r a t u r e r e v i e w 27 2.5.2.3 TWO-WAY GROWTH PREDICTION T h i s method, a l t h o u g h p a r t l y based on pe r -manent sample p l o t s , i s an i n d i r e c t method of e s t i m a t i n g mor-t a l i t y i n b a s a l a r e a and i n volume. Gross b a s a l a r e a growth per a c r e i s o b t a i n e d from i n c r e m e n t b o r i n g s and s t a n d - t a b l e p r o -j e c t i o n t e c h n i q u e s ; n e t b a s a l a r e a growth i s computed from ex-p e r i e n c e t a b l e s ; m o r t a l i t y can be exp r e s s e d as the d i f f e r e n c e , o r i g n o r e d i f o n l y the l a r g e s t t r e e s a r e ta k e n i n t o a c c o u n t . F u t u r e h e i g h t can be p r e d i c t e d from s i t e i n d e x c u r v e s and s t a n d volume growth can be e s t i m a t e d from s t a n d - o r tree-volume t a b l e s f o r f u t u r e h e i g h t s , and f u t u r e b a s a l a r e a s ; m o r t a l i t y can t h e n be ex p r e s s e d i n volume, o r i g n o r e d i n s h o r t - t e r m f o r e -c a s t s ( S p u r r , 1952). 2.5.2.A GROWTH AND YIELD FUNCTIONS D i f f e r e n t i a l e q u a t i o n s e x p r e s s i n g t h e c u r -r e n t r a t e of t r e e i n c r e m e n t have been developed by f o r e s t e r s i n s e a r c h of some m a t h e m a t i c a l e x p r e s s i o n s of the dependency be-tween growth and y i e l d o b s e r v a t i o n s . Increments i n volume were r e l a t e d t o inc r e m e n t i n h e i g h t and di a m e t e r , and y i e l d s were o b t a i n e d by i n t e g r a t i o n . Permanent and temporary p l o t d a t a were u t i l i z e d f o r t h a t purpose. Such s t u d i e s were made f o r even-aged s t a n d s by C l u t t e r (1963) f o r l o b l o l l y p i n e , and by C u r t i s (1967) f o r Douglas f i r . Moser and H a l l (1969) took the same approach f o r s t u d y i n g uneven-aged s t a n d s . R e s u l t s of C u r t i s ' study on Douglas f i r were p r e s e n t e d i n A l i t e r a t u r e r e v i e w 28 the form o f g r o s s y i e l d t a b l e s . T h e r e f o r e , they can be used f o r m o r t a l i t y e s t i m a t e s as e x p l a i n e d i n s e c t i o n 2.5.2.1 above. 2.5.3 SIMULATION TECHNIQUES 2.5.3.1 REGULAR MORTALITY Most models s i m u l a t i n g the growth o f i n -d i v i d u a l t r e e s a r e v e r y w e l l - s u i t e d f o r e s t i m a t i o n o f r e g u l a r m o r t a l i t y because they i n c l u d e a mechanism f o r e v a l u a t i n g c u r -r e n t c o m p e t i t i v e s t a t u s of each t r e e w i t h r e g a r d to i t s n e i g h -b o u rs. A number of these c o m p e t i t i o n formulae were o u t l i n e d i n s e c t i o n 2.2.1 above. Most of them, based on some measure o f a v a i l a b l e growing space, p e r m i t one t o a c c u r a t e l y " c o n t r o l " n a t u r a l growth p r o c e s s e s . M o r t a l i t y i s a s s i g n e d on the b a s i s o f p a s t growth performance; most o f the time, a t h r e s h o l d i s a r b i t r a r i l y s e t , beyond which a t r e e i s deemed to be dead. I n view o f the l a r g e v a r i a t i o n a f f e c t i n g i n d i v i d u a l t r e e b e h a v i o r , more r e f i n e m e n t i n e x p r e s s i n g the e f f e c t s of c o m p e t i t i o n on i n -d i v i d u a l t r e e m o r t a l i t y would be p o i n t l e s s , a c c o r d i n g t o Dre s s (1968). Techniques f o r p r e d i c t i n g and f a c t o r s i n f l u e n c i n g the a c c u r a c y o f e s t i m a t i o n o f i n d i v i d u a l t r e e growth have been a n a l y z e d by Smith (1964a, 1966a), Smith and W a l t e r s (1964), and Opie (1968). 2.5.3.2 IRREGULAR AND CATASTROPHIC MORTALITY Some s i m u l a t i o n models make p r o v i s i o n f o r A l i t e r a t u r e r e v i e w 29 s p a t i a l c o n t a g i o n of m o r t a l i t y which might be caused, f o r i n -s t a n c e , by l o c a l windthrows or r o o t d i s e a s e s ( D r e s s , 1968). A p a r t from t h a t , l i t t l e emphasis has been put up t o now on the f o r e c a s t i n g o f i r r e g u l a r and c a t a s t r o p h i c m o r t a l i t y . And y e t , a c o n s i d e r a b l e amount of money i s spent, and much i n f o r m a t i o n i s c o l l e c t e d i n d i s e a s e - , f i r e - , i n s e c t - and o t h e r damage-s u r v e y s . T h i s problem c o u l d b e s t be , t a c k l e d on a p r o b a b i l i s -t i c b a s i s . U n t i l t h i s i s done, however, r u l e s of thumb or o t h e r crude t e c h n i q u e s s h o u l d a t l e a s t be d e v i s e d t o make mor-t a l i t y p r e d i c t i o n s s t i l l more r e a l i s t i c . 2 . 6 CHAPTER SUMMARY T h i s l i t e r a t u r e r e v i e w was not meant t o be c r i t i c a l , but to d e s c r i b e the k i n d o f i n f o r m a t i o n c u r r e n t l y a v a i l a b l e on mor-t a l i t y o f Douglas f i r . I t has l e d 1) t o a t e n t a t i v e c l a s s i f i c a t i o n based on s p a t i a l d i s t r i b u t i o n s of dead t r e e s and causes o f m o r t a l i t y ; 2 ) t o a r e v i e w of s e v e r a l m a t h e m a t i c a l e x p r e s s i o n s o f compe t i -t i o n among t r e e s ; 3) t o a summary o f a c t u a l l o s s e s by causes; and 4) to the d e s c r i p t i o n o f s e v e r a l methods f o r growth, y i e l d and m o r t a l i t y p r e d i c t i o n s . I t i s apparent t h a t much d e s c r i p t i v e knowledge has been accumulated i n the form o f g r o s s e s t i m a t e s ; b u t , s u f f i c i e n t e m p i r i c a l d a t a a re l a c k i n g c o n c e r n i n g t i m i n g o f m o r t a l i t y i n the young age, dime n s i o n s o f dead t r e e s , r e l a t i o n s between s t a n d c h a r a c t e r i s t i c s and amounts o f m o r t a l i t y , and the A l i t e r a t u r e r e v i e w 30 o c c u r r e n c e of i r r e g u l a r and c a t a s t r o p h i c l o s s e s . Moreover, i t i s noted t h a t most c o m p e t i t i o n i n d i c e s are u s e f u l t o p r e d i c t t r e e growth, but h a r d l y h e l p i n f o r e c a s t i n g m o r t a l i t y . 31 CHAPTER 3 OBJECTIVES, DATA AND METHODS 3.1 DEFINITION OF OBJECTIVES The f i r s t o b j e c t i v e of t h i s d i s s e r t a t i o n i s t o d e s c r i b e m o r t a l i t y i n Douglas f i r s t a n d s . T h i s w i l l be done i n two s t e p s , as f o l l o w s : F i r s t l y , a n n u a l and c u m u l a t i v e amounts of m o r t a l i t y s h a l l be r e l a t e d t o s t a n d c h a r a c t e r i s t i c s by u s i n g c o r r e l a t i o n a n a l y -ses and m u l t i p l e r e g r e s s i o n t e c h n i q u e s . M o r t a l i t y s h a l l a l s o be r e l a t e d t o t r e e c h a r a c t e r i s t i c s . I n a t t e m p t i n g t o t a c k l e t h i s problem, a p r o b a b i l i s t i c approach w i l l be t a k e n . Most r e -l a t i o n s h i p s s h o u l d d i r e c t l y or i n d i r e c t l y i n c l u d e a time f a c -t o r , s i n c e b o t h amount and t i m i n g a re i m p o r t a n t . Frequency d i s t r i b u t i o n s o f t r e e parameters and t h e i r s p a t i a l arrangement are e q u a l l y i m p o r t a n t f a c t o r s t h a t s h a l l be c o n s i d e r e d . The second s t e p w i l l t h u s c o n s i s t i n d e s c r i b i n g d i s t r i b u t i o n s o f l i v e and dead t r e e s , and i n a n a l y z i n g t o which e x t e n t they a r e r e l a t e d . The o t h e r main o b j e c t i v e i s t o d e v e l o p some methods t o f a c i l i t a t e and improve p r e d i c t i o n s of m o r t a l i t y . I n t h i s r e -s p e c t , most p r e d i c t i o n e q u a t i o n s s h o u l d be u s e f u l ; b u t , beyond t h a t a stand model w i l l be b u i l t , t o s i m u l a t e f o r e s t growth and t o reproduce m o r t a l i t y . O b j e c t i v e s , d a t a and methods 32 3.2 DESCRIPTION OF DATA 3.2.1 PERMANENT SAMPLE PLOTS A l l but a few p i e c e s of i n f o r m a t i o n were not c o l -l e c t e d by the a u t h o r because l o n g - t e r m r e c o r d s were needed t o perform most of the a n a l y s e s i n c l u d e d . I n s t e a d , seven a g e n c i e s were asked t o c o n t r i b u t e t o the d a t a bank. They a r e : the B r i t i s h Columbia F o r e s t S e r v i c e , the U n i t e d S t a t e s F o r e s t Ser-v i c e , the Canada Department of F i s h e r i e s and F o r e s t r y , B.C. F o r e s t P r o d u c t s L i m i t e d , Crown Z e l l e r b a c h C o r p o r a t i o n , Weyer-haeuser Company, and the U n i v e r s i t y of B r i t i s h Columbia. The aim was to l o c a t e a number o f permanent sample p l o t s , l o n g - e s t a b l i s h e d throughout the C o a s t a l Douglas f i r Region, i n u n d i s t u r b e d n a t u r a l or p l a n t e d s t a n d s , f o r which t r e e c o o r -d i n a t e s were a v a i l a b l e t o g e t h e r v/ith as many o t h e r p i e c e s o f i n f o r m a t i o n as p o s s i b l e . C o a s t a l Douglas f i r was chosen be-cause few o t h e r s p e c i e s had been s t u d i e d w i t h the same i n t e n -s i t y i n the P a c i f i c Northwest and hence, were l i k e l y t o y i e l d the amount o f i n f o r m a t i o n needed. Furthermore, few s p e c i e s are l i k e l y t o a t t r a c t the same degree of a t t e n t i o n i n the f u t u r e . S i x t y - e i g h t p l o t s , r e p r e s e n t i n g a wide range of c o n d i t i o n s , were chosen. They were c l a s s i f i e d i n t o 5 groups, a c c o r d i n g t o t h e i r o r i g i n and c o m p o s i t i o n , as shown i n T a b l e I I . The range i n age v a r i e d i n p l a n t a t i o n s from 13 to l+S y e a r s , and i n na-t u r a l s t a n d s from 19 t o 90 y e a r s ; h e i g h t a t 100 y e a r s was be-tween 70 and 190 f e e t , as p r o j e c t e d w i t h t a b l e s from McArdle e t a l . (1949). T h i s i s i l l u s t r a t e d i n F i g u r e 2. O b j e c t i v e s , d a t a and methods 3 3 TABLE I I CLASSIFICATION OF PLOTS STUDIED BY ORIGINS-AGE AND STAND COMPOSITION AGE 1 0 2 0 3 0 LO 5 0 60 7 0 80 9 0  ORIGIN AND TOTAL SfA"ND~ PLOT NUMBER NUMBER OF PLOT MEASUREMENTS COMPOSITION1 Wind R i v e r S p a c i n g T r i a l 5 2 8 2 8 # 1 - 8 , 1 0 - 1 1 , 1 3 - 1 5 , 1 7 U.B.C. Spa c i n g T r i a l 5 # 1 4 - 1 8 U.B.C. F e r t i l i -s a t i o n P l a n t a t i o n 1 R o b e r t s o n R i v e r P l o t s 1 - 3 6 4 1 6 PLANTATIONS B.C. F o r e s t P r o d u c t s #114 B.C. F o r e s t S e r v i c e #69 Crown Z e l l e r b a c h # 5 , 8 U.S. F o r e s t S e r v i c e : 2 5 2 6 S i u s l a w # 9 , Olympic # 1 , 2 R a i n i e r # 1 , 2 , 4 , 5 , 7 , 8 , 9 V o i g h t Creek # 1 1 - 1 4 , 2 1 - 2 3 , 2 5 , 3 2 - 3 4 4 5 17 1 4 1 0 2 1 2 1 GROUP I Pure f i r ( 9 1 -1 0 0 % f i r ) B.C. F o r e s t P r o d u c t s #102 B.C. F o r e s t S e r v i c e # 8 4 , 2 0 9 , 2 8 3 U.S. F o r e s t Ser-v i c e , V o i g h t Creek 7 1 7 1 4 # 1 5 , 2 4 , 3 1 Weyerhaeuser # 1 , 4 1 1 LO GROUP I I Mixed f i r ( 7 6 -9 0 % f i r ) B.C. F o r e s t P r o d u c t s # 2 0 2 B.C. F o r e s t S e r v i c e # 6 5 , 8 5 U.B.C. P.S.P. #107 2 4 11 Weyerhaeuser # 2 , 3 7 3 1 l 2£ GROUP I I I Mixed f i r ( 5 1 -75% f i r ) B.C. F o r e s t P r o d u c t s # 2 0 3 U.B.C. P.S.P. # 2 1 2 , 2 1 4 , 2 1 6 , 2 1 7 , 2 1 8 1 1 2 5 ? 2 16 GROUP IV Mixed f i r ( l e s s t han 50% f i r ) Age from seed i n p l a n t a t i o n . P e r c e n t f i r by number of stems per a c r e . Figure 2- Classification of data by age and site quality-' Site Index-Height in 200M*** ot ag* 100 'Total stand age or age from seed;average site index evaluated at each plot measurement-190-180- x x- -x—x--x 170 *~o+*—'O P~ O -O' 1601— • o o — o — o — o 150 140 130 120 NO 90 80 -» ~~ —x— —8 ^ ». X X — X - - X X X — X — X — X — X — X o o o o X X X X x — x — x ^ j f e f i l i ^ o o o o --+---+ Legend 100- mm m + + V=- - -*«—--+ "~~ + Group I Group II o o GroupHI * x GroupIZ • + Plantation Lines connect successive 70h measurements of the same sample plot-10 20 30 40 50 60 70 80 90 Age * O b j e c t i v e s , d a t a and methods 35 I n each p l o t , a complete t a l l y o f d i a m e t e r s a t b r e a s t h e i g h t was t a k e n a t each measurement, f o r t r e e s 1.6 i n c h e s and over (except f o r V o i g h t Creek: 5 i n c h e s p l u s ) . .Some t r e e s were bored f o r age a t stump h e i g h t , and t h e i r t o t a l h e i g h t was measured t o e s t i m a t e s i t e q u a l i t y ( f o r Douglas f i r ) . I n most p l o t s , measurements of crown c l a s s were a l s o t a k e n . I n more i n t e n s i v e l y s t u d i e d a r e a s , l i k e the \7ind R i v e r s p a c i n g t r i a l and t h e U.B.C. s p a c i n g t r i a l and f e r t i l i z a t i o n p l a n t a t i o n , measurements of crown w i d t h and l i v e crown l e n g t h were a v a i l -a b l e f o r a l l o r f o r a l a r g e number of t r e e s ; Crown Z e l l e r b a c h p l o t s and some U.B.C. permanent p l o t s a l s o had l i v e crown l e n g t h measurements f o r f i r . Remeasurernent p e r i o d s v a r i e d between 3 y e a r s f o r the V o i g h t Creek p l o t s t o 20 y e a r s f o r some of the B.C. F o r e s t S e r v i c e d a t a , the average p e r i o d b e i n g about 5 y e a r s . Each p l o t was measured 4 t i m e s on the average (68 plots-282 measurements). Most of them were square or r e c t a n g u l a r i n shape; c i r c u l a r p l o t s ( V o i g h t Creek) were c u t square on stem c h a r t s . Twenty-seven a c r e s of l a n d were cove r e d by these samples, the average p l o t s i z e b e i n g O.38 a c r e . They c o n t a i n e d 13 thousand t r e e s a t time o f f i r s t measurement, of which 10 thousand were i n d i v i d u a l l y l o c a t e d i n the C a r t e s i a n system of c o o r d i n a t e s . More d e t a i l e d d e s c r i p t i o n s o f each p l o t o r group o f p l o t s o f the same o r i g i n can be o b t a i n e d from the f o l l o w i n g r e f e r -ences: 1) Wind R i v e r s p a c i n g p l o t s i n E v e r s o l e (.1955), C u r t i s and Reukema (1969), Reukema (1969); 2) U.B.C. s p a c i n g t r i a l and O b j e c t i v e s , data, and methods 36 f e r t i l i z a t i o n p l a n t a t i o n i n Osborn (1968a) and W a l t e r s e t a l . (1966); 3) R o b e r t s o n R i v e r p l o t s i n F o s t e r and Johnson (1963a); k) R a i n i e r , S i u s i a w and Olympic p l o t s , e s t a b l i s h e d i n normal s t a n d s , i n W i l l i a m s o n (1963); 5) V o i g h t Creek d a t a i n Worthing-t o n e t a l . (1962); 6) B.C. F o r e s t S e r v i c e o l d e x p e r i m e n t a l p l o t s i n Thomson (1966) and Warrack (1959, 1967); 7) no i n f o r -m ation has been p u b l i s h e d f o r the B.C. F o r e s t P r o d u c t s ' , Crown Z e l l e r b a c h ' s and Weyerhaeuser's p l o t s ; 8) i n f o r m a t i o n on the U.B.C. Rese a r c h F o r e s t permanent sample p l o t s i s c o n t a i n e d i n u n p u b l i s h e d r e p o r t s . 3.2.2 STEM MAPPED RECORDS As c o l l e c t e d , a good p a r t of the stem mapped i n f o r -m ation was unusable. Many c h a r t s had o n l y one t h i n g i n common: they showed t r e e l o c a t i o n s on a p l a n e , i n s i d e p l o t b o u n d a r i e s ; few had been drawn a t the same s c a l e , and s e v e r a l c o n t a i n e d m i s i d e n t i f i e d t r e e s . C o n sequently, s e v e r a l p l o t measurements were d i s c a r d e d f o r r e a s o n of i n c o n s i s t e n c y . I n some, azi m u t h s and d i s t a n c e s of each t r e e from the p l o t c e n t e r were g i v e n ( V o i g h t Crejek and Crown Z e l l e r b a c h ) ; these measurements were c o n v e r t e d t o r e c t a n g u l a r c o o r d i n a t e s , t a k i n g the s o u t h west c o r n e r of each p l o t as o r i g i n . The p l o t t e d p o s i t i o n o f each t r e e was determined t o the n e a r e s t f o o t by u s i n g a m a g n i f y i n g g l a s s ( i n s e v e r a l i n s t a n c e s ) and a r u l e r . I n g e n e r a l , d a t a c o n c e r n i n g a n y t h i n g but t r e e s themselves were s c a r c e and l a c k e d u n i f o r m i t y and c l a r i t y , p a r t l y because O b j e c t i v e s , d a t a and methods 37 they had been c o l l e c t e d over the y e a r s by many d i f f e r e n t p e o p l e , and p a r t l y because most o f the t i m e , these f a c t o r s were d e a l t v / i t h i n a d e s c r i p t i v e r a t h e r than an a n a l y t i c a l manner. T h e r e f o r e , we were f o r c e d t o r e l y on m e n s u r a t i o n a l i n f o r m a t i o n , and t o n e g l e c t e n v i r o n m e n t a l f a c t o r s l i k e c l i m a t e and s o i l con-d i t i o n s , even i f they have a tremendous i n f l u e n c e on f o r e s t growth, as documented by G r i f f i t h (i960) and E i s (1962). 3.3 METHODS 3.3.1 CODING AND SORTING THE INFORMATION I n o r d e r t o f a c i l i t a t e t he p r o c e s s i n g and r e t r i e v -i n g of the i n f o r m a t i o n , a l l the d a t a were punched on IBM' c a r d s (one t r e e p e r c a r d ) i n the f o l l o w i n g o r d e r : t r e e number, X and Y c o o r d i n a t e s i n f e e t (when a v a i l a b l e ) , d i a m e t e r s a t s u c c e s s i v e measurements i n i n c h e s , and c o r r e s p o n d i n g crown c l a s s . Sup-plementary i n f o r m a t i o n such a s h e i g h t , crown w i d t h and l i v e crown l e n g t h were e i t h e r e n t e r e d on the same or on s e p a r a t e c a r d s . 3.3.2 COMPUTER PROGRAMMING Four computer programs were w r i t t e n i n F o r t r a n I V f o r the IBM 360/67 d a t a p r o c e s s i n g system. D e s c r i p t i o n s of t h e i r f u n c t i o n s and s i m p l i f i e d f l o w c h a r t s a re g i v e n i n Appendix I . I n summary, the f i r s t one (YIELD) c a l c u l a t e s growth and y i e l d t a b l e s from p l o t remeasurements. The second one (STOCK) O b j e c t i v e s , d a t a and methods 38 c o m p i l e s d e t a i l e d i n f o r m a t i o n on s t o c k i n g and s t a n d d e n s i t y . The t h i r d one (SPACE) uses t r e e c o o r d i n a t e s t o e s t i m a t e s p a t i a l d i s t r i b u t i o n , and a l s o d e t e r m i n e s the f r e q u e n c y d i s t r i b u t i o n of t r e e d i a m e t e r s and t r e e h e i g h t s . The f o u r t h one (MORT) p r o -duces s i x t a b l e s , i n d i c a t i n g the annual p r o b a b i l i t y of d e a t h i n r e l a t i o n t o i n d i v i d u a l t r e e c h a r a c t e r i s t i c s ( d i a m e t e r , h e i g h t , crown w i d t h , d i a m e t e r i n c r e m e n t , h e i g h t i n c r e m e n t , and crown c l a s s ) . A l l the d a t a a n a l y z e d h e r e a f t e r were r u n t h r o u g h these f o u r programs to p r o v i d e some of the r e s u l t s p r e s e n t e d . 3.3.3 SAMPLING AND CURVE FITTING 3.3.3.1 SAMPLING FOR TREE ATTRIBUTES I n o r d e r t o e s t a b l i s h the r e l a t i o n s h i p s needed t o r u n the programs mentioned above, e i t h e r complete enumeration, s e l e c t e d or random samples were t a k e n . The method adopted was b a s i c a l l y a f u n c t i o n of the amount of d a t a a v a i l -a b l e . Douglas f i r t r e e s o n l y v/ere t a k e n i n t o account. I n the case of h e i g h t a t t r i b u t e s , the same t r e e s had been remeasured a t each p e r i o d i n many p l o t s . When t h i s number was s m a l l , o n l y one h e i g h t v a l u e was randomly chosen f o r eaich t r e e ; when the number was l a r g e ( o v e r 100), a random sample o f t r e e s was t a k e n as w e l l . I n the case o f crown w i d t h and l i v e crown l e n g t h a t t r i b u t e s , which had been e v a l u a t e d o n l y once on r e l a -t i v e l y few t r e e s , a l l measures were r e c o r d e d . I n the case of i n c r e m e n t a t t r i b u t e s , a p r e - d e t e r m i n e d p r o p o r t i o n ( u s u a l l y 10 O b j e c t i v e s , d a t a and methods 39 p e r c e n t ) of the t o t a l number of t r e e s was randomly sampled, and no more than one measurement per t r e e was r e c o r d e d . T h i s r a n -dom s e l e c t i o n was performed s o l e l y f o r the purpose of l i m i t i n g the t o t a l number o f o b s e r v a t i o n s t o b u i l d good r e g r e s s i o n e q u a t i o n s . 3.3.3.2 SAMPLING FOR SPATIAL ARRANGEMENT L i v e t r e e s have been sampled by the con-t i g u o u s quadrat method (program SPACE, Appendix I ) . Four d i f -f e r e n t quadrat s i z e s were used f o r each s a m p l i n g o p e r a t i o n of the same p l o t . The quadrat s i z e was determined by a s p e c i f i e d average number o f t r e e s p e r square, and a frequency d i s t r i b u -t i o n of q u a d r a t s c o n t a i n i n g an e q u a l number of t r e e s was b u i l t and t e s t e d f o r goodness of f i t a g a i n s t f o u r d i s c r e t e p r o b a b i l -i t y d i s t r i b u t i o n s ( b i n o m i a l , P o i s s o n , n e g a t i v e b i n o m i a l and u n i f o r m ) . I t i s known t h a t when p o i n t s a re randomly d i s p e r s e d over an a r e a , the frequency d i s t r i b u t i o n o f the number of q u a d r a t s c o n t a i n i n g the same number of p o i n t s f o l l o w s a P o i s s o n d i s t r i -b u t i o n , w i t h v a r i a n c e e q u a l t o the mean. I n f o r e s t s , t h i s a p p l i e s when the chance of a p l a n t o c c u r r i n g a t any p o i n t i n the a r e a under c o n s i d e r a t i o n i s s m a l l o r , i n o t h e r words, i f the number of i n d i v i d u a l s i n the a r e a ( q u a d r a t ) i s low r e l a -t i v e t o the p o s s i b l e number t h a t c o u l d grow,. When the P o i s s o n e x p e c t a t i o n a p p l i e s , c o n f i d e n c e l i m i t s f o r the d e p a r t u r e of the variance-mean r a t i o ( a l s o termed i n d e x of c o n t a g i o n and O b j e c t i v e s , d a t a and methods 40 c o e f f i c i e n t of d i s p e r s i o n ) from 1.0 can be c a l c u l a t e d , s i n c e the s t a n d a r d e r r o r o f t h i s d e p a r t u r e (SE) i s the square r o o t of 2 / ( n - l ) , where n i s the number o f q u a d r a t s ( G r e i g - S m i t h , 1957, Newnham, 1968). When the variance-mean r a t i o i s l a r g e r than (1+tpSE), the arrangement i s s a i d t o be more c o n t a g i o u s o r heterogeneous (clumpy) than random; and when i t i s s m a l l e r than (1-tpSE), i t i s more s y s t e m a t i c ( r e g u l a r ) t h a n random. The v a l u e t comes from " t " t a b l e s w i t h l e v e l of p r o b a b i l i t y p, and (n-1) degrees of freedom. I n the f i r s t c ase, the frequ e n c y d i s t r i b u t i o n f i t s t o a n e g a t i v e b i n o m i a l or t o a u n i f o r m d i s -t r i b u t i o n of p r o b a b i l i t i e s ; i n the second c a s e , i t f i t s t o a b i n o m i a l d i s t r i b u t i o n w i t h variance-mean r a t i o l e s s than 1.0. T h i s method, however, was not found c o m p l e t e l y s a t i s f a c -t o r y t o q u a l i f y the d i s p e r s i o n of dead t r e e s . I n some i n s t a n -ces, the number of dead t r e e s was too s m a l l t o b u i l d any f r e -quency d i s t r i b u t i o n , o r f r e q u e n c i e s were too s m a l l i n each c l a s s ( l e s s than 4) to a p p l y a C h i - s q u a r e t e s t f o r d e t e c t i n g the goodness o f f i t of observed d i s t r i b u t i o n s t o expected d i s -t r i b u t i o n s (and t h e r e was l o g i c a l l y no r e a s o n f o r combining c l a s s e s ) . Thus, the p o i n t - t o - p l a n t method was adopted, t o -get h e r w i t h the P i e l o u ' s t e s t of non-randomness ( P i e l o u , 1959)• T h i s method proved t o be the b e s t one among e i g h t s t u d i e d by Payandeh (1969) on 50-acre t r a c t s of f o r e s t s , and the t e s t was c o n s i d e r e d by F o s t e r and Johnson (1963a) as the most u s e f u l p l o t l e s s s a m p l i n g t e c h n i q u e t o study p a t t e r n s . A number of random c o o r d i n a t e s r o u g h l y e q u a l t o the number O b j e c t i v e s , d a t a and methods 41 of dead t r e e s were drawn and p l o t t e d . No p o i n t s v/ere l o c a t e d i n s i d e a b o r d e r zone of w i d t h e q u a l t o average s p a c i n g between l i v e t r e e s i n the p l o t ( t o a v o i d b o r d e r e f f e c t s ) . D i s t a n c e s from each p o i n t t o the n e a r e s t dead t r e e were measured, the p o p u l a t i o n d e n s i t y e v a l u a t e d , and the t e s t performed. Some t e s t s were made on the t o t a l number o f t r e e s dead d u r i n g the e n t i r e p e r i o d o f o b s e r v a t i o n of the same p l o t , some on t r e e s dead between two s u c c e s s i v e p l o t measurements; the procedure chosen was m a i n l y d i c t a t e d by the number of dead s u b j e c t s i n any g i v e n p e r i o d . No a n a l y s i s was performed i n p l o t s h a v i n g l e s s t h a n 20 dead t r e e s ( a r b i t r a r y l i m i t ) . 3.3.3.3 CURVE FITTING Every r e g r e s s i o n e q u a t i o n i n t h i s d i s s e r -t a t i o n was e s t a b l i s h e d by a p p l y i n g the l e a s t square procedure (Snedecor and Cochran, 196?), and by u s i n g the m u l t i p l e r e g r e s -s i o n program d e s c r i b e d by Kozak and Smith (1965), as adapted f o r the IBM 360 d a t a p r o c e s s i n g system. T h i s program was s l i g h t l y m o d i f i e d i n some i n s t a n c e s t o conform t o p a r t i c u l a r r e q u i r e m e n t s . P r e d i c t i o n e q u a t i o n s of h e i g h t , crown w i d t h , crown l e n g t h and h e i g h t t o l i v e crown were c a l c u l a t e d w i t h each s e t of d a t a . The b e s t models developed a re g i v e n i n Appendix I I . Some of them were a l s o chosen from o t h e r s o u r c e s o f i n f o r m a t i o n due t o a l a c k of adequate i n f o r m a t i o n from the p l o t s s t u d i e d . R e s i -d u a l s were not p l o t t e d i n any case to t e s t f o r the correctness O b j e c t i v e s , d a t a and methods 42 o f the r e g r e s s i o n , models. T h e i r r e l a t i v e v a l u e was judged on the magnitude of b o t h the c o e f f i c i e n t of d e t e r m i n a t i o n and the st a n d a r d e r r o r of e s t i m a t e . Smith (1966a) has suggested t h a t emphasis, i n c h o o s i n g v a r i a b l e s t o be i n c l u d e d i n r e g r e s s i o n e q u a t i o n s , s h o u l d be p l a c e d upon those which a re most l o g i c a l , s i m p l e , c o n s i s t e n t , and s t a t i s t i c a l l y e f f i c i e n t . T h i s has been the o b j e c t i v e throughout the a n a l y s i s . A c e r t a i n degree of c o r r e l a t i o n can be d e t e c t e d among the independent v a r i a b l e s i n some r e l a t i o n -s h i p s , which i s a c c e p t a b l e i n view o f the f a c t t h a t they were developed m a i n l y f o r p r e d i c t i o n purposes. Whenever t h e i r i n -d i v i d u a l c o n t r i b u t i o n t o a r e g r e s s i o n model i s not s i g n i f i c a n t a t an a c c e p t a b l e l e v e l o f p r o b a b i l i t y , i t s h a l l be i n d i c a t e d . 3.3.4 SUMMARY OF THE INFORMATION T a b l e s I I I , IV, and V c o n t a i n a summary o f the i n -f o r m a t i o n c o l l e c t e d and co m p i l e d w i t h the t h r e e computer p r o -grams (STOCK, YIELD, SPACE) d e s c r i b e d i n Appendix I . They show minimum, maximum and mean v a l u e s o f a number o f v a r i a b l e s t o -g e t h e r w i t h t h e i r s t a n d a r d d e v i a t i o n s and c o e f f i c i e n t s of v a r i a t i o n f o r a s p e c i f i e d number o f p l o t measurements. These v a l u e s a r e o u t p u t s from growth a n a l y s e s performed oh i n d i v i d u a l p l o t measurements by m u l t i p l e r e g r e s s i o n t e c h n i q u e s . Amongst the v a r i a b l e s c a l c u l a t e d a r e s e v e r a l e x p r e s s i o n s of s t o c k i n g and stand d e n s i t y . S t o c k i n g measurements a r e d i f -f e r e n t from the ones o r d i n a r i l y made on the ground. As such, Based on 121 measurements o f 2A p l o t s i n Group I . VARMEAN i s based on 65 measurements. •IDENTIFICATION OF VARIABLES MIN.: minimum MAX.: maximum ST.DEV.: s t a n d a r d d e v i a t i o n C.V. (%): c o e f f i c i e n t of v a r i a t i o n i n percentage DBH: d i a m e t e r o f the t r e e of average b a s a l a r e a TOPH: average h e i g h t of the 100 l a r g e s t t r e e s per a c r e AGE: t o t a l s t a n d age (from seed i n p l a n t a t i o n ) S I : s i t e i n d e x ( f t . a t 100 y e a r s ) BNT: t o t a l number of t r e e s per a c r e (1.6"+) BAN: t o t a l n e t b a s a l a r e a per a c r e (1.6"+) SQSP: square s p a c i n g BANORM: r a t i o of BAN t o normal BAN (McArdle e t a l . , 1949) f o r g i v e n AGE and SI , i n percentage CCF: crown c o m p e t i t i o n f a c t o r ( K r a j i c e k e t a l . , 1961) RDN: r e l a t i v e d e n s i t y .BNT/(43560/DBE~2) BASTOC: e f f e c t i v e b a s a l a r e a , c a l c u l a t e d as the r a t i o of BAN to s t o c k i n g YESTOC: e f f e c t i v e y i e l d e x p r e s s e d as the r a t i o of net y i e l d t o s t o c k i n g CCCLO: c u m u l a t i v e crown c l o s u r e o r sum t o t a l o f maximum c i r c u l a r crown a r e a of dominant, co-dominant and i n t e r m e d i a t e t r e e s p r o j e c t e d on the ground, expressed i n p e r c e n t o f p l o t a r e a CSO: p e r c e n t o f the p l o t crov/n space volume a c t u a l l y occu-p i e d by t r e e crowns STOCGR: r a t i o o f mean annual net volume increment t o g r o s s m.a.i. ( S t a e b l e r , 1955a) VARMEAN: average variance-mean r a t i o of the d i s t r i b u t i o n of number o f l i v e t r e e s p e r quadrat a f t e r 4 s u c c e s -s i v e p l o t s a m p l i n g w i t h the c o n t i n u o u s quadrat method ( G r e i g - S m i t h , 1952). O b j e c t i v e s , d a t a and methods 44 TABLE I I I STATISTICS FOR PURE NATURAL STANDS 1 p ST. C.V VARIABLES^ MIN. MAX. MEAN DEV. (%) SIZE DBH 3.2 22.0 11.6 3.7 32 TOPH 43.0 163.0 102.5 23.2 23 AGE AGE 18.0 oo n oo . U 54.0 13.8 26 SITE QUALITY SI 112.0 186.0 141.3 19.3 14 ABSOLUTE DENSITY BNT 110.0 1135.0 327.0 185.1 57 BAN 65.0 326.0 200.3 56,4 28 SQSP 6.2 19.9 12.6 3.0 24 RELATIVE DENSITY BANORM 40.0 161.0 96.4 19.6 20 CCF 92.0 377.0 229.2 48.9 21 RDN 0.26 1.37 0.8 0.2 28 EFFECTIVE DENSITY BASTOC 65.0 353.1 234.0 58.2 25 YESTOC 676.1 19383.5 9070.8 3961.8 44 STOCKING CCCLO 20.0 135.0 55.2 15.4 28 CSO 11.0 77.0 31.6 9.1 29 STOCGR 34.0 176.0 87.2 21.3 24 PATTERN VARMEAN 0.43 2.48 0.9 0.4 46 O b j e c t i v e s , d a t a and methods 45 TABLE STATISTICS FOR MIXED NATURAL STANDS 1 p ST. C.V. VARIABLES MIN. MAX. MEAN DEV. {%) SIZE DBH 2.3 • 19.5 10.9 3.3 30 TOPH 34.0 124.0 88.2 14.6 16 AGE AGE 19.0 8o.o 48.0 12.9 27 SITE QUALITY SI 87.0 196.0 139.5 28.0 20 ABSOLUTE DENSITY BNT 94.0 3456.0 376.2 551.6 147 PATTERN VARMEAN 0.6 2.8 1.3 0.7 55 Based on 85 measurements of 21 p l o t s i n Groups I I , I I I , IV. VARMEAN i s based on 29 o b s e r v a t i o n s . I n c l u d e s Douglas f i r t r e e s o n l y . I d e n t i f i c a t i o n of v a r i a b l e s i n Ta b l e I I I . O b j e c t i v e s , d a t a and methods 46 TABLE V STATISTICS FOR PLANTATIONS 1 VARIABLES 2 MIN. MAX. • MEAN ST. DEV. C.V, (%) SIZE DBH TOPH 3.3 35.0 10.2 78.0 5.5 54.9 1.7 8.9 30 16 AGE AGE 22.0 46.0 34.5 6.5 19 SITE QUALITY SI 70.0 147.0 95.3 18.2 19 ABSOLUTE DENSITY BUT BAN 335.0 37.0 2000.0 196.0 914.7 127.4 507.4 36.4 55 28 RELATIVE DENSITY BANORM CCF RDN 43.0 88.0 0.1 165.0 385.0 0.8 106.4 235.4 0.5 27.7 77.2 0.2 26 33 30 EFFECTIVE DENSITY BASTOC YESTOC 80.4 1039.1 208.3 5310.0 124.5 2443.3 26.6 927.8 21 33 STOCKING CCCLO STOCGR CSO 53.0 44.0 29.0 183.0 169.0 104.0 122.5 103.4 67.0 28.1 29.2 16.4 23 28 24 PATTERN VARMEAN 0.3 1.0 0.5 0.1 27 Based on 64 measurements of 14 p l o t s i n the Wind R i v e r S p a c i n g T r i a l ; o r i g i n a l s p a c i n g from 4 x 4 to 10 x 10 f t . VARMEAN i s based on 50 o b s e r v a t i o n s . V a r i a b l e s a re i d e n t i f i e d i n Table I I I . Objectives, data and methods 47 they represent an attempt at measuring t h i s variable i n the o f f i c e , on a tree basis, and without using a e r i a l photography. They are explained i n Appendix I I I . Also included i n the same Appendix are discussions of some measurements of r e l a t i v e and eff e c t i v e stand density made by taking stocking into account. Spatial patterns were characterized by variance-mean r a -t i o s of observed frequency d i s t r i b u t i o n s of contiguous quadrats, already explained i n s e c t i o n 3«3«3»2. Stand diameter, height, age, s i t e quality, and current expressions of density (number of trees and basal area per acre) supplement the information needed for further growth, y i e l d and mortality analyses. In. mixed stands of f i r and other species (Table IV) the number of measured variables i s reduced due to the considera-tion of Douglas f i r trees only, which renders some of them irrel e v a n t to compile. A l l variables used i n t h i s d i s s e r t a t i o n are described i n Appendix IV, and symbols are i d e n t i f i e d . 3.4 CHAPTER SUMMARY The main objective of t h i s d i s s e r t a t i o n i s to study amount, timing and d i s t r i b u t i o n of C o a s t a l Douglas f i r mortal-i t y i n r e l a t i o n to stand and in d i v i d u a l tree c h a r a c t e r i s t i c s . This i s aimed at gaining a better understanding of the process and at developing some methods for prediction of mortality and growth. O b j e c t i v e s , d a t a and methods Z+8 I t i s based on r e p e a t e d o b s e r v a t i o n s of 13 thousand t r e e s , l o c a t e d i n 68 permanent sample p l o t s l o n g e s t a b l i s h e d t h r o u g h -out the Douglas f i r R egion o f the P a c i f i c Northwest. Four computer programs have been w r i t t e n to summarize t h i s i n f o r m a t i o n c o n t r i b u t e d by seven a g e n c i e s . Data on growth and y i e l d , s t o c k i n g , s p a t i a l p a t t e r n s , amounts, t i m i n g and p r o b a b i l i t i e s of m o r t a l i t y have t h u s been generated. M u l t i p l e r e g r e s s i o n t e c h n i q u e s have been a p p l i e d t o d e s c r i b e some r e -l a t i o n s h i p s . 49 CHAPTER A AMOUNT AND TIMING OF MORTALITY Two d i f f e r e n t approaches can be t a k e n t o e s t a b l i s h r e l a -t i o n s h i p s between m o r t a l i t y and some f o r e s t c h a r a c t e r i s t i c s , the stand approach and the t r e e approach. I n t h i s c h a p t e r , the f i r s t approach i s used t o a n a l y z e amount and t i m i n g o f m o r t a l -i t y . I t i s f i r s t s t u d i e d i n normal s t a n d s , and then w i t h the d a t a d e s c r i b e d i n Chapter 3« The second approach s e r v e s i n e s t a b l i s h i n g p r o b a b i l i t i e s of i n d i v i d u a l t r e e m o r t a l i t y , which are needed t o s i m u l a t e or reproduce the n a t u r a l p r o c e s s on a stem b a s i s , and t o a p p l y management p r a c t i c e s on the ground. 4.1 MORTALITY RELATED TO STAND CHARACTERISTICS 4.1.1 PERIODIC ANNUAL MORTALITY I.N NORMAL STANDS In a f i r s t s t e p , average a n n u a l r a t e s and annual amounts ( i n numbers of i n d i v i d u a l s ) a re r e l a t e d t o n a t u r a l s t a n d c h a r a c t e r i s t i c s . Data from s e v e r a l Douglas f i r y i e l d t a b l e s have been c o m p i l e d f o r t h a t purpose. For n a t u r a l s t a n d s , d a t a from the B.C. F o r e s t S e r v i c e (1947), McArdle e t a l . (1949), and Schumacher (1930) have been combined; f o r p l a n t a -t i o n s , t a b l e s p u b l i s h e d by Hummel and C h r i s t i e (1953) (adapted by Barnes (1955) and Hoyer (1967) a n d r e v i s e d by B r a d l e y et a l . (1966)), and by D u f f (1956) were c o n s i d e r e d . The p e r i o d i c annual p e r c e n t a g e m o r t a l i t y was c a l c u l a t e d w i t h a s i m p l e i n t e r e s t formula.- The d i f f e r e n c e between low Amount and t i m i n g of m o r t a l i t y 50 r a t e s of si m p l e and compound i n t e r e s t over s h o r t p e r i o d s o f time (10 y e a r s or l e s s ) i s not a p p r e c i a b l e . As c a l c u l a t e d w i t h the f o l l o w i n g e q u a t i o n , the percentage m o r t a l i t y (APCM) be-comes the p r o p o r t i o n e x pressed i n percentage o f the i n i t i a l number o f t r e e s (BNT) t h a t w i l l d i e a n n u a l l y d u r i n g a d e f i n i t e p e r i o d o f t i m e . The annual m o r t a l i t y i n number of t r e e s (AMOR) i s s i m p l y the d i f f e r e n c e between the i n i t i a l and the f i n a l number o f t r e e s (FNT) d i v i d e d by the number o f y e a r s i n the p e r i o d c o n s i d e r e d ( N ) . Thus, APCM = ((1-(FNT/BNT))/N) 100 AMOR = (BNT-FNT)/N A c o r r e l a t i o n a n a l y s i s shows t h a t , i n n a t u r a l Douglas f i r s t a n d s , the p e r i o d i c a n n u a l p e r c e n t m o r t a l i t y (APCM) i s be s t c o r r e l a t e d w i t h age (AGE), b a s a l a r e a (BAN), number o f t r e e s (BNT, FNT), and average stand diameter (DBH), i n t h a t o r d e r . The p e r i o d i c a n n u a l m o r t a l i t y on the o t h e r hand (AMOR) i s h i g h l y c o r r e l a t e d w i t h number of t r e e s and b a s a l a r e a . I n p l a n t a t i o n s the degree o f c o r r e l a t i o n i s the same f o r APCM; however, AMOR i s b e s t c o r r e l a t e d w i t h BNT, DBH, AGE, and BAN r e s p e c t i v e l y . The b e s t p r e d i c t i o n e q u a t i o n s a re l i s t e d i n Table VI f o r n a t u r a l s t a n d s , and i n Table V I I f o r p l a n t a t i o n s . Other im-p o r t a n t r e l a t i o n s h i p s are i l l u s t r a t e d g r a p h i c a l l y ( F i g u r e s 3 - 7 ) . As r e c o g n i z e d by many a u t h o r s , s t a n d d e n s i t y , age, and s i t e q u a l i t y are the most i m p o r t a n t f a c t o r s c o n t r o l l i n g f o r e s t Amount and t i m i n g of m o r t a l i t y 51 TABLE VI PERIODIC ANNUAL PERCENT MORTALITY AND ANNUAL MORTALITY IN NUMBER OF STEMS IN NORMAL STANDS 1 SEE Independent v a r i a b l e s and -A\ U n i t o f Source I n t e r c e p t R e g r e s s i o n C o e f f i c i e n t s R ~ fr,/ PERIODIC ANNUAL PERCENT MORTALITY (APCM) McArdle e t _ _ a l . (1949) ^  +2.6L62 +7.6260 B . C . F . S . 1 1 9 4 7 ) : • BNT BAN SI +0.00062 -0.0125 +0.0116 .96 .23 10 BAN BANSq AGE -0.0374 +0,00007 -0.0160 .95 .25 15 AGE DBH -0.0234 -0 .0186 .79 .55 33 SI AGE AGESQ -0.0027 -0.0791 +0.0003 .93 .31 19 3 BAN BAN SO AGE -0.0122 +0.00002 -0.0193 .81 .48 28 PERIODIC ANNUAL MORTALITY IN NUMBER OF STEMS (AMOR) 1 McArdle e t a l . ( 1 9 4 9 ) 2 B N T BAN SI -51.1104 +0.0638 +0.0938 +0.0979 .99 4 .5 30 BAN AGE DBH +138.0980-0.8611 +0.2672 +3.8465 .36 3 6 * 4 240 McArdle e t a l . ( 1 9 4 9 ) , B.C.F.S.TT94?) , Schumacher ( 1 9 3 0 r BNT BAN -36.2750 +0.0532 +0.1034 .88 13.6 92 "^Percent of number p r e s e n t a t b e g i n n i n g o f each t e n - y e a r p e r i o d , or annual number d u r i n g each p e r i o d . Symbols a r e i d e n t i f i e d i n Appendix IV. 196 o b s e r v a t i o n s . ^272 o b s e r v a t i o n s . ^ R ^ = c o e f f i c i e n t o f d e t e r m i n a t i o n ( s i g n i f i c a n t a t the 0.05 l e v e l ) ; SEE = s t a n d a r d e r r o r o f e s t i m a t e ( u n i t of Y ); (%) = (SEE / Y ) 1 0 0 . Y = e s t i m a t e of the dependent v a r i a b l e . Amount and t i m i n g o f m o r t a l i t y 52 TABLE V I I PERIODIC ANNUAL PERCENT MORTALITY AND ANNUAL MORTALITY IN NUMBER OF STEMS IN PLANTATIONS 1 , SEE Independent V a r i a b l e s and ^2. U n i t of I n t e r c e p t R e g r e s s i o n C o e f f i c i e n t s R 1 (%) PERIODIC ANNUAL PERCENT MORTALITY (APCM) +4.7215 AGE -0.0723 .37 1.5 65 +4.5112 BANSQ 7AGE -0.00013^-0.0412 .41 1.4 61 4.8804 AGE DBH -0.0614 -0.0567 .38 1.5 65 PERIODIC ANNUAL MORTALITY IN NUMBER OF STEMS (AMOR) DBH DBHSQ +95.5571 -14.4273 +0.5576 .52 17.0 100 BAN AGE DBH +56.5457 -O.OQ73 -0.2159 -3.2948 .43 18.7 108 P e r c e n t of number p r e s e n t a t b e g i n n i n g of each t e n - y e a r p e r i o d , or an n u a l number d u r i n g each p e r i o d . From Hunnel and C h r i s t i e (1953) ,and D u f f (1956) . S i g n i f i c a n t a t the 0.05 l e v e l , w i t h 50 o b s e r v a t i o n s . U n d e r l i n e d v a r i a b l e s do not make a s i g n i f i c a n t c o n t r i b u t i o n a t t h e 0.05 l e v e l . Figure 3 * Periodic annual percent mortality related to age*1 (3)1 1 Percent of number present at beginning of ten-year periods-From McArdle eta/- (1949)-Identification of symbols in Appendix (1) A PCM = 3-88097-0-Q2625 AGE N»I96 r 2 »0-785 SE E »0-55 (33%) (2) APCM* 5-62244-0079118 AGE +0-00031097 AGE SQ-N*I96 r2 =0-927 SE E »0 -32 (19%) (3) APCM«2902 A G E " 0 ' 5 6 (0-9920 A G E j N*I96 r 2*0-98 MIN-20 0-5 MAX-150 6-1 MEAN 85 1-65 ST- DEV-40-4 1-20 C-V-4 7 % 7 2 % 150 Age (years) (2)*^ Figure 4* Periodic annual percent mortality related to basal area* ^ ' Percent of number present at beginning of ten-year periods-\ From McArdle et at- (1949)- Identification of symbols in Appendix XZ-\ (1) APCM« 50I66-O0I4I8 BAN N«196 r 2-0-77227 S E E »0-5726 (2) APCM«7-85864-0043154 BAN +0000065 BAN SO-NS 196 r 2 « 0-87371 S E E "0-427 TESTED MIN- MAX- MEAN ST- DEV- C-V-BAN 64 360 237 74 2 31% APCM 05 61 1-65 1-2 7 2 % BAN (sq-ft/acre) Figure 5- Periodic annual percent mortality related to number of trees*' % APCM 10 1 Percent of number present ot beginning of ten-year periods* From McArdle etal-K 1949) -Identification of symbols in Appendix EE-9 8 7 6 5 4 APCM « 1 - 1 0 8 5 + 0 - 0 0 1 1 7 BNT N - 1 9 6 r 2 » 0 - 6 0 S E E « 0 - 7 6 TESTED MIN- MAX- MEAN ST- DEV- CV* BNT 4 2 6 9 2 0 4 6 2 7 9 2 1 7 1 % APCM 0 - 5 6 1 1-65 1-2 7 2 % 1 ± ± 1000 2000 3000 4000 5000 6000 7000 Number of Trees per Acre Figure 6 - Periodic annual percent mortality related to stand diameter-APCM % 5 -1 Percent of number present at beginning of ten-year periods-From McArdle of al- (1949)- Identification of symbols in Appendix BE-(1) APCM* 3-18344 - 0-100925 DBH N» 196 r 2 =0-50843 SEg'0-84 (2) APCM - 4-78728 - 0-34057 DBH +0-0067466 DBH SQ-N«I96 r2»0-734 S E E »0-62 TESTED MIN* MAX- MEAN ST- DEV- CV-DBH 13 394 15 2 8 4 5 5 % APCM 0 5 61 1-65 12 7 2 % J I I I L 0 2 4 6 8 10 15 20 25 30 35 40 DBH (inches) Figure 7- Periodic annual mortality related to number of trees-AMOR N/dcre 280 260 240 220 200 180 160 140 120 100 80 60 40 20 0 Annual number dead during ten-year periods* From McArdle at al- (1949), BC- Forest Service (1947),Schumacher (1930)-+0 053168 BNT +0-10339 BAN 88 S E E » l 3 - 6 MAX* MEAN ST- OEV- CV* 6920 510 806 158% 360 231 77 3 3 % 422 14 40 272% 1000 2000 3000 4000 5000 6000 Number of 7000 Trees per Acre 2 Amount and t i m i n g of m o r t a l i t y 58 growth and y i e l d , i n n a t u r a l s t a n d s as w e l l as i n p l a n t a t i o n s . R e g u l a r m o r t a l i t y , b e i n g an i n t e g r a l p a r t o f t h i s growth p r o -c e s s i s a l s o i n f l u e n c e d by the same f a c t o r s . T h i s i s evidenced i n T a b l e s VI and V I I . That m o r t a l i t y i s a h i g h l y v a r i a b l e phenomenon i s a l s o shown i n the same t a b l e s . Every e q u a t i o n has a h i g h l y s i g n i -f i c a n t c o e f f i c i e n t of d e t e r m i n a t i o n , and y e t the s t a n d a r d e r r o r of e s t i m a t e i s f a i r l y l a r g e even i f the d a t a were t a k e n out of y i e l d t a b l e s b u i l t from harmonized c u r v e s . P a r t of t h i s v a r i a t i o n , however, c o u l d be removed i f b e t t e r r e g r e s s i o n models were t e s t e d . T h i s i s i l l u s t r a t e d i n F i g u r e s 3> k- a * i d 6, i n w h ich the i n t r o d u c t i o n of a square term (—SQ) r e d u c e s a p p r e c i a b l y the e r r o r of e s t i m a t e , and i n c r e a s e s the m u l t i p l e c o r r e l a t i o n . Some form o f e x p o n e n t i a l e q u a t i o n c o u l d g i v e a b e t t e r f i t ( F i g u r e 3)» but q u i t e o f t e n no i n f o r m a t i o n i s a v a i l -a b l e t o b a l a n c e b o t h ends of the c u r v e s (extreme v a l u e s o f the independent v a r i a b l e ) . T h i s s u g g e s t s t h a t , i n o r d e r t o de-s c r i b e a d e q u a t e l y the form o f the f u n c t i o n r e l a t i n g m o r t a l i t y and s tand c h a r a c t e r i s t i c s , more d a t a would be needed b e f o r e age 20. F i g u r e s 3 t o 6 show t h a t , i n normal Douglas f i r s t a n d s , p e r i o d i c annual p e r c e n t m o r t a l i t y (APCM) v a r i e s r o u g h l y between 1 and 5 p e r c e n t over a wide range of age, d e n s i t y and s i z e , and t h a t the number o f stems l o s t t o m o r t a l i t y i n c r e a s e s q u i t e l i n e a r l y w i t h d e n s i t y . I t i s a l s o p o s s i b l e , knowing the p e r i o d i c a n n u a l p e r c e n t Amount and t i m i n g o f m o r t a l i t y 59 m o r t a l i t y , t o f i n d out the average p r o b a b i l i t y t h a t i n d i v i d u a l t r e e s have to d i e w i t h i n a c e r t a i n p e r i o d o f t i m e . For example, a U p e r c e n t a n n u a l m o r t a l i t y means t h a t L i n d i v i d u a l s out of. 100 d i e each y e a r , or have a 0.0L p r o b a b i l i t y t o d i e every y e a r . T h i s approach was t a k e n to b u i l d T a b l e s V I I I , IX and X f o r normal Douglas f i r s t a n d s i n which m o r t a l i t y i s r e l a t e d t o d e n s i t y , s i t e and s i z e over t i m e . These t a b l e s show much more c l e a r l y than c u r v e s how c l o s e l y r e l a t e d m o r t a l i t y i s t o age and stand d e n s i t y . T able V I I I i l l u s t r a t e s t h a t m o r t a l i t y v a r i e s more when d e n s i t y goes from 100 to 300 square f e e t o f b a s a l a r e a than i t does when age i n -c r e a s e s from 20 t o 90 y e a r s . T able I X i l l u s t r a t e s t h a t the v a r i a t i o n i n m o r t a l i t y due to s i t e i n d e x i s n e g l i g i b l e as com-pared to age. Table X i n d i c a t e s t h a t , a t a g i v e n age, m o r t a l -i t y i s not i n f l u e n c e d v e r y much by changes i n average stand d i a m e t e r . At any age between 20 and 9 0 , the p r o b a b i l i t y t h a t an i n -d i v i d u a l t r e e w i l l d i e i s two t o t h r e e t i m e s g r e a t e r i n a stand c a r r y i n g 100 f e e t of b a s a l a r e a than i t i s a t 300 f e e t (due t o the number o f t r e e s p r e s e n t a t each l e v e l ) . On any s i t e , the p r o b a b i l i t y t h a t an i n d i v i d u a l t r e e w i l l d i e i s t h r e e t o f o u r t i m e s g r e a t e r a t age 20 than i t i s a t age 9 0 . The p r o b a b i l i t y t h a t a t r e e d i e s i s much more c l o s e l y r e l a t e d t o age than t o average s t a n d d i a m e t e r : a t age 9 0 , i t i s about h a l f as l a r g e as a t age 2 0 , whatever the st a n d d i a m e t e r . I n i n t e r p r e t i n g t h e s e r e s u l t s , i t must be remembered t h a t Amount and t i m i n g o f m o r t a l i t y 60 TABLE V I I I AVERAGE PROBABILITY OF INDIVIDUAL TREE DEATH BASED ON TOTAL STAND AGE AND BASAL AREA PER ACRE 1 AGE 20-30 Zf0-50 60-70 30-90 P r o b a b i l i t y of Death f o r the P e r i o d BAN/acre sq. f t . 100 .42 .39 .36 .33 150 .32 .29 .26 .23 200 .26 .23 .19 .16 250 .23 .19 .16 .13 300 .23 .20 .16 .13 Based on McArdle t a b l e s f o r Douglas f i r (McArdle e t a l . , 1949) APCM = 7.626 -0.037 BAN + 0.000069 BANSQ - 0.016 AGE N = 196 R 2 = .96 SEE = 0.25 (15.1%) P r o b a b i l i t y c a l c u l a t e d a t the b e g i n n i n g o f the p e r i o d . Amount and t i m i n g of m o r t a l i t y 61 TABLE I X AVERAGE PROBABILITY OF INDIVIDUAL TREE DEATH BASED ON TOTAL STAND AGE AND SITE QUALITY 1 AGE 20-30 40-50 60-70 80-90 S I 2 f t . age at 100 P r o b a b i l i t y ox Death f o r the P e r i o d 80 .43 .31 .22 .15 110 .43 .30 .21 .14 1 / f O .42 .30 .20 .13 170 .41 .29 .19 .12 200 .40 .28 .18 .11 Based on McArdle t a b l e s f o r Douglas f i r (McArdle et a l . , 1949) APCM = 6.0124 - 0.0027 SI - 0.0791 AGE + 0.0003 AGESQ N = 196 R 2 = 0.93 SEE = 0.31 (19%) P r o b a b i l i t y c a l c u l a t e d a t the b e g i n n i n g o f the p e r i o d . Corresponds t o s i t e c l a s s V, IV, I I I , I I , I r e s p e c t i v e l y . Amount and t i m i n g of m o r t a l i t y 62 TABLE X AVERAGE PROBABILITY OF INDIVIDUAL TREE DEATH BASED ON STAND AGE AND AVERAGE S I Z E 1 AGE 20-30 40-50 60-70 80-90 P r o b a b i l i t y of dea t h f o r the P e r i o d DBH i n . 2 .34 4 .33 .29 6 .33 .29 .24 8 .33 .28 .24 .19 10 .33 .27 .23 .19 12 .27 .23 .18 14 .27 .22 .18 16 .22 .17 18 .22 .17 20 .21 .17 "^Based on McArdle t a b l e s f o r Douglas f i r (McArdle et a l . , 1949) APCM = 3.920 - 0.023 AGE - 0.019 DBH N = 196 R 2 = 0.79 SEE = 0.55 (33%) P r o b a b i l i t y c a l c u l a t e d a t the b e g i n n i n g of the p e r i o d . Amount and t i m i n g of m o r t a l i t y 63 they were o b t a i n e d from s t a n d s of normal ( i . e . h i g h ) d e n s i t y from age 20 to 90, which i s not the case i n most n a t u r a l s t a n d s . T h e r e f o r e , the a c t u a l p r o b a b i l i t y of t r e e m o r t a l i t y i n any par-t i c u l a r s t a n d w i l l d i f f e r from t h a t i n d i c a t e d i n T a b l e s V I I I to X, depending on the d i f f e r e n c e between a c t u a l and normal changes i n stand d e n s i t y w i t h t i m e . 4.1.2 PERIODIC ANNUAL MORTALITY IN SOME NATURAL AND PLANTED STANDS I n t h i s s e c t i o n , the p e r i o d i c a n n u a l p e r c e n t mor-t a l i t y (APCM) has been c a l c u l a t e d the same way as i n s e c t i o n 4.1.1 above. The p e r i o d i c annual m o r t a l i t y , however, i s ex-p r e s s e d i n number of t r e e s per a c r e (AMOR), i n b a s a l a r e a per a c r e (BAMOR), and i n c u b i c volume per a c r e (VOLMOR), as shown i n Table X I . Three t y p e s of Douglas f i r s t a n d s are c o n s i d e r e d : 1) pure n a t u r a l s t a n d s , 2) mixed n a t u r a l s t a n d s , and 3) p l a n -t a t i o n s . Pure f i r s t a n d s are i n c l u d e d i n Group I : mixed s t a n d s p o f f i r , hemlock, ced a r , or f i r and a l d e r are r e p r e s e n t e d by Groups I I , I I I , and IV; p l a n t a t i o n Douglas f i r c o m p r i s e s o n l y the Wind R i v e r S p a c i n g T r i a l ( T a b l e I I ) . The h y p o t h e s i s i n d i r e c t l y t e s t e d here i s t h a t m o r t a l i t y i s a f u n c t i o n o f , o r can be p r e d i c t e d by t a k i n g i n t o account a l l o r p a r t o f the f o l l o w i n g s t a n d and s i t e c h a r a c t e r i s t i c s : 'Western hemlock, Tsuga h e t e r o p h y l l a ( R a f . ) Sarg. Western r e d cedar, Thuja p l i c a t a Donn. Red a l d e r , A l n u s r u b r a Bong. Amount and t i m i n g of m o r t a l i t y TABLE XI MORTALITY CHARACTERISTICS MEASURED IN SAMPLE PLOTS ^ Standard C o e f f i c i e n t M o r t a l i t y Minimum Maximum Mean D e v i a t i o n o f V a r i a t i o n U n i t of -, GROUP I Mo r t a l i t y " 1 " (%) APCM 0.0 7.0 1.67 1.36 81 AMOR 0.0 45.0 5.42 6.00 110 BAMOR 0.0 6.8 1.41 1.16 82 VOLMOR 0.0 260.0 48.03 43.10 90 GROUP I I , I I I , IV APCM 0.0 7.8 2.51 1.92 76 AMOR 0.0 219.0 12.16 30.50 251 BAMOR 0.0 5.8 1.34 1.20 90 MOLMOR 0.0 185.0 35.90 35.98 100 PLANTATION APCM 0.0 6.5 1.10 1.16 106 AMOR 0.0 71.0 12.58 14.99 119 BAMOR 0.0 6.4 0.90 1.19 132 VOLMOR 0.0 113.0 14.22 20.16 142 APCM = p e r i o d i c a n n u a l p e r c e n t m o r t a l i t y i n number o f sterns per a c r e . AMOR = p e r i o d i c a n n u a l m o r t a l i t y i n number of stems per a c r e . BAMOR = p e r i o d i c a n n u a l m o r t a l i t y i n square f e e t of b a s a l a r e a p e r a c r e . VOLMOR = p e r i o d i c a n n u a l m o r t a l i t y i n c u b i c f e e t per a c r e . Amount and t i m i n g of m o r t a l i t y 65 MORTALITY = f (TIME, DENSITY, STOCKING, PATTERN, SIZE, SITE) The i n d i v i d u a l c o n t r i b u t i o n o f each c h a r a c t e r i s t i c i s e v a l u a t e d by means of c o r r e l a t i o n a n a l y s e s , and t h e i r composite i n f l u e n c e by m u l t i p l e r e g r e s s i o n t e c h n i q u e s . 4 . 1 . 2 . 1 CORRELATIONS BETWEEN PERIODIC ANNUAL MORTALITY AND STAND PARAMETERS Simple c o r r e l a t i o n c o e f f i c i e n t s e m p i r i c a l l y i n d i c a t e the degree of a s s o c i a t i o n between two v a r i a b l e s (Dra-per and Smith, 1966). I n T a b l e s X I I , X I I I , and XIV, s tand c h a r a c t e r i s t i c s have been i n d i v i d u a l l y r e l a t e d t o f o u r e x p r e s -s i o n s o f m o r t a l i t y . Except i n p l a n t a t i o n s , f o r which the d a t a were c o l l e c t e d on a l i m i t e d range of c o n d i t i o n s by comparison t o n a t u r a l s t a n d s ( F i g u r e 2 ) , the degree of c o r r e l a t i o n i s g e n e r a l l y low, and i n many c a s e s , not s i g n i f i c a n t . T h i s i n d i c a t e s t h a t n a t u r a l m o r t a l i t y was h i g h l y v a r i a b l e i n the sample p l o t s s t u d i e d . T h i s i s shown by the l a r g e c o e f f i c i e n t of v a r i a t i o n , and by the wide range and the extreme skewness (towards the l e f t ) o f the d i s t r i b u t i o n o f i n d i v i d u a l o b s e r v a t i o n s ( T a b l e X I ) . For example, AMOR v a r i e d i n Group I between 0 and 45 t r e e s per a c r e per y e a r , w i t h a mean of 5«4> and a s t a n d a r d d e v i a t i o n o f 6 . 0 . W i t h most independent v a r i a b l e s l i s t e d , maximum and minimum v a l u e s are b o t h l o c a t e d w i t h i n the 95 p e r c e n t l i m i t s of the normal d i s t r i b u t i o n , i . e . the mean p l u s or minus two s t a n d a r d d e v i a t i o n s . I n the case of AMOR, however, the l o w e r l i m i t Based on 97 p l o t measurements i n Group I , M o r t a l i t y was r e -p o r t e d a t the b e g i n n i n g of each p e r i o d ; t h u s , l a s t p l o t measurement i s not i n c l u d e d . I d e n t i f i c a t i o n of independent v a r i a b l e s i n Appendix IV. (":") c o r r e l a t i o n between dependent and independent v a r i a b l e not s i g n i f i c a n t a t the 0 . 0 5 l e v e l . APCM = p e r i o d i c a n n u a l p e r c e n t m o r t a l i t y i n number of t r e e s . AMOR = p e r i o d i c annual m o r t a l i t y i n number o f t r e e s . BAMOR = p e r i o d i c a n n u a l m o r t a l i t y i n square f e e t of b a s a l a r e a per a c r e . VOLMOR = p e r i o d i c a n n u a l m o r t a l i t y i n c u b i c f e e t . Based on 65 p l o t measurements. Amount and t i m i n g o f m o r t a l i t y 67 TABLE X I I SIMPLE CORRELATION BETWEEN SOME EXPRESSIONS OF MORTALITY AND STAND CHARACTERISTICS (PURE NATURAL DOUGLAS FIR STANDS) X Independent V a r i a b l e Simple C o r r e l a t i o n C o e f f i c i e n t s Dependent V a r i a b l e TIME ABSOLUTE DENSITY RELATIVE DENSITY EFFECTIVE DENSITY STOCKING PATTERN*1" SIZE SITE APCM AMOR BAMOR VOLHOR A G E -.38 .34 B N T » .57 -.24 -.35 B A H * * .27 .43 YlELDN * -.24 .27 .48 S O S P -.41 . i—L .37 B A N O R M * •* .23 S T O C G R .43 •* C C F * * .23 R D N .26 .43 B A S T O C -.38 .27 .42 Y E S T O C * -.34 .26 .47 C S O * .46 * C C C L O * .44 V A R M E A N * * « D B H * -.36 .29 .50 T O P H " .30 .50 si •"- .27 .46 Amount and t i m i n g of m o r t a l i t y 68 TABLE X I I I SIMPLE CORRELATION BETWEEN SOME EXPRESSIONS OF MORTALITY AND STAND CHARACTERISTICS (MIXED DOUGLAS FIR STANDS) 1 2 Simple C o r r e l a t i o n C o e f f i c i e n t s Independent V a r i a b l e ^ " Dependent V a r i a b l e 3 P o r t i o n - ^ APCM AMOR BAMOR VOLMOR TIME AGE DF •"- -.37 ABSOLUTE DENSITY BNT DF .26 .89 * PATTERN 5 BAN YIELDN VARMEAN DF DF -><• •tt ***• .28 .29 * .27 .34 •ir SIZE DBH DF -/<r -.52 •ir TOPH DF •>{. -.61 .26 SITE QUALITY SI DF Based on 64 measurements of 21 p l o t s i n Groups I I , I I I , IV. (") c o r r e l a t i o n between dependent and independent v a r i a b l e not s i g n i f i c a n t a t the O.05 l e v e l . DF i n d i c a t e s t h a t the independent v a r i a b l e has been measured on Douglas f i r o n l y ; o t h e r w i s e , the whole stand was con-s i d e r e d . V a r i a b l e s a r e i d e n t i f i e d i n Table X I I and i n Appendix IV. 'Based on 21 p l o t measurements. Amount and t i m i n g of m o r t a l i t y 6 9 TABLE XIV SIMPLE CORRELATION BETWEEN SOME EXPRESSIONS OF MORTALITY AND STAND CHARACTERISTICS (PLANTATIONS) 1 2 Simple C o r r e l a t i o n C o e f f i c i e n t s Independent V a r i a b l e - ^ Dependent V a r i a b l e APCM AMOR BAMOR VOLMOR TIME AGE .52 .38 .52 .52 ABSOLUTE DENSITY BNT .30 .61 .35 .30 BAN .55 .56 .59 .58 YIELDN .46 .35 .47 .49 SQSP -.33 -.62 - . 3 9 -.33 RELATIVE DENSITY BANORM .51 .66 .57 .54 STOCGR .44 .64 .52 .48 CCF .48 .70 .55 .51 RDN .55 .56 .59 .58 STOCKING CSO .47 .64 .54 .52 CCCLO .47 .63 .54 .51 PATTERN VARMEAN .50 .61 .50 .40 SIZE DBH * -.32 * * TOPH SITE SI - . 3 9 -.53 -.42 -.37 Based on 50 measurements o f 14 p l o t s i n the WIND RIVER s p a c i n g t r i a l . '(*) c o r r e l a t i o n betv/een dependent and independent v a r i a b l e not s i g n i f i c a n t a t the 0.05 l e v e l . V a r i a b l e s a r e i d e n t i f i e d i n Table X I I and i n Appendix I V . Amount and t i m i n g of m o r t a l i t y 70 (5*4 ~ 12.0) extends f a r below the minimum (0.0), and the upper l i m i t (5«4 + 12.0) f a l l s w e l l beyond the maximum observed (i+5.0). T h i s v a r i a t i o n i n m o r t a l i t y can be a t t r i b u t e d e i t h e r t o an i n f i l t r a t i o n of i r r e g u l a r m o r t a l i t y i n our d a t a , t o the i r r e g u l a r i t y i n remeasurement p e r i o d s o f each p l o t ( v a r y i n g be-tween 3 and 20 y e a r s ) , o r t o the m o r t a l i t y p r o c e s s i t s e l f . M o r t a l i t y i s b e t t e r e x p r e s s e d i n a b s o l u t e terms than i n percentage e s p e c i a l l y v/hen d a t a a r e not grouped by age or den-s i t y c l a s s e s . At the l o w e r end of the s c a l e , s m a l l v a r i a t i o n s i n a b s o l u t e number r e s u l t i n l a r g e v a r i a t i o n s i n p e r c e n t a g e s , and v i c e v e r s a a t the upper end. T h i s showed c l e a r l y , f o r i n s t a n c e , i n a s c a t t e r g r a m o f the r e l a t i o n s h i p between number of t r e e s per a c r e (BNT) and annual p e r c e n t a g e m o r t a l i t y f o r Group I . Among the t h r e e v a r i a b l e s e x p r e s s i n g m o r t a l i t y i n a b s o l u t e terms, a n n u a l m o r t a l i t y i n number o f t r e e s per ac r e (AMOR) i s be s t r e l a t e d t o s t a n d c h a r a c t e r i s t i c s . I n n a t u r a l s t a n d s (Table X I I ) , AMOR i s h i g h l y c o r r e l a t e d w i t h s tand d e n s i t y , s t o c k i n g , age, d i a m e t e r , and h e i g h t ; i n mixed s t a n d s (Table X I I I ) , number of t r e e s p e r a c r e , h e i g h t , d i a m e t e r and age y i e l d good c o r r e l a t i o n s ; i n p l a n t a t i o n s ( T a b l e X I V ) , s tand d e n s i t y , p a t t e r n and age are c l o s e l y r e l a t e d t o AMOR. I n gen-e r a l , m o r t a l i t y i n c r e a s e s w i t h an i n c r e a s e i n d e n s i t y and s t o c k i n g , but d e c r e a s e s w i t h an i n c r e a s e i n age ( o v e r l o n g p e r i o d s ) , d i a m e t e r , and h e i g h t . S i t e q u a l i t y does not a f f e c t m o r t a l i t y s i g n i f i c a n t l y . Amount and t i m i n g o f m o r t a l i t y 71 Some c o n t r a d i c t i o n s between T a b l e X I I and XIV can be i n -t e r p r e t e d by k e e p i n g i n mind t h a t the d a t a i n T a b l e XIV are from a p l a n t a t i o n measured over a r e l a t i v e l y s h o r t p e r i o d o f time, i n which t r e e s e v o l v e d a t l a r g e l y d i f f e r e n t s p a c i n g s from time o f p l a n t i n g . Thus, they show t r e n d s t h a t are o n l y appar-e n t l y o p p o s i t e t o those e x p e r i e n c e d i n n a t u r a l s t a n d s ; f o r i n s t a n c e , m o r t a l i t y i n c r e a s e d between age 22 and 46, whereas i t d e c r e a s e s when observed over l o n g e r p e r i o d s o f t i m e ; i t was g r e a t e r i n c l o s e r s p a c i n g s , g i v i n g h i g h e r c u b i c volume y i e l d s which i n c r e a s e m a i n l y w i t h age i n n a t u r a l s t a n d s , and on l o w e r q u a l i t y s i t e s measured by average h e i g h t o f dominant and c o d i m i n a n t t r e e s (because h e i g h t s were s m a l l e r i n c l o s e r s p a c i n g s , and m o r t a l i t y h e a v i e r ) . Annual m o r t a l i t y i n square f e e t o f b a s a l a r e a (BAMOR) and c u b i c volume (VOLMOR) i s b e s t c o r r e l a t e d w i t h h e i g h t , d i a m e t e r , s i t e q u a l i t y and d e n s i t y i n n a t u r a l s t a n d s ( T a b l e X I I ) . I n p l a n t a t i o n s ( T a b l e X I V ) , d e n s i t y , s t o c k i n g , age, p a t t e r n and s i t e i n d e x are r e s p e c t i v e l y the most i m p o r t a n t f a c t o r s . i+,1.2.2 MULTIPLE REGRESSIONS OF PERIODIC ANNUAL MORTALITY ON STAND PARAMETERS F i r s t - and second- o r d e r l i n e a r r e g r e s -s i o n models o n l y were developed t o d e s c r i b e some r e l a t i o n s h i p s between m o r t a l i t y and stand c h a r a c t e r i s t i c s . Even i f the f i t t o the d a t a i s not p e r f e c t i n a l l c a s e s , l i n e a r models a r e s i m p l e a p p r o x i m a t i o n s q u i t e c o n v e n i e n t t o a n a l y z e a p r o c e s s Amount and t i m i n g of m o r t a l i t y 72 such as m o r t a l i t y . However, due t o t h a t s u s p e c t e d l a c k of f i t , one must be c a u t i o u s i n i n t e r p r e t i n g b o t h m u l t i p l e c o r r e l a t i o n c o e f f i c i e n t s and s t a n d a r d e r r o r o f e s t i m a t e s , f o r r e a s o n s enumerated i n Draper and Smith (1966). A number o f the most l o g i c a l c o m b i n a t i o n s of independent v a r i a b l e s was s t u d i e d . However, i t was not p o s s i b l e t o b r i n g the p e r c e n t a g e s t a n d a r d e r r o r o f e s t i m a t e t o a l e v e l l o w e r than 7k p e r c e n t o f the mean response ( a n n u a l m o r t a l i t y ) o r t o push the c o e f f i c i e n t o f d e t e r m i n a t i o n over 0,6k. i . FOR PURE NATURAL STANDS In pure n a t u r a l s t a n d s , no r e l a t i o n -s h i p was e s t a b l i s h e d betv/een p e r i o d i c a n n u a l p e r c e n t m o r t a l i t 3 r (APCM) and s e v e r a l c o m b i n a t i o n s o f s t a n d and/or s i t e c h a r a c t e r -i s t i c s . The b e s t e q u a t i o n t o p r e d i c t p e r i o d i c a n n u a l m o r t a l i t y i n number o f stems p e r a c r e (AMOR) i n c l u d e s c u m u l a t i v e crown c l o s u r e (CCCLO) and age. I t a c c o u n t s f o r k3 p e r c e n t o f the v a r i a t i o n i n AMOR. I n the second b e s t e q u a t i o n , crown space o c c u p a t i o n (CSO) o r crown c l o s u r e and average s t a n d d i a m e t e r (DBH) account f o r kl p e r c e n t o f the v a r i a t i o n i n AMOR. The t h i r d b e s t i n c o r p o r a t e s square s p a c i n g (SQSP) o r n o r m a l i t y i n b a s a l a r e a (BANORM) and age. T h i r t y p e r c e n t o f the v a r i a t i o n i n AMOR i s t a k e n c a r e o f by a l i n e a r c o m b i n a t i o n o f these v a r i -a b l e s . Each v a r i a b l e makes a s i g n i f i c a n t c o n t r i b u t i o n (0.05 l e v e l ) t o the r e g r e s s i o n s ; n e v e r t h e l e s s , the s t a n d a r d e r r o r o f Amount and t i m i n g of m o r t a l i t y 73 e s t i m a t e i s l a r g e r than 83 p e r c e n t o f the mean response i n a l l c a s e s . These and some o t h e r e q u a t i o n s a r e l i s t e d i n Table XV. Annual m o r t a l i t y i n b a s a l a r e a (BAMOR) can be p r e d i c t e d by knowing s t a n d age and s i t e i n d e x or h e i g h t a t age 100 ( S I ) . I t i s of i n t e r e s t t o note t h a t t h i s e q u a t i o n groups the same v a r i -a b l e s C u r t i s (1967) used t o p r e d i c t b a s a l a r e a growth r a t e . A l l o t h e r c o m b i n a t i o n s o f two o r more stand o r s i t e c h a r -a c t e r i s t i c s c o n t a i n e d terms t h a t d i d n o t , when added l i n e a r l y , c o n t r i b u t e s i g n i f i c a n t l y t o i m p r o v i n g the r e l a t i o n s . From a p r a c t i c a l p o i n t o f view, the s i m p l e s t and b e s t equa-t i o n t o p r e d i c t AMOR i s the one i n c l u d i n g square s p a c i n g (square r o o t of a r e a i n square f e e t d i v i d e d by number o f t r e e s ) and s t a n d age. i i . FOR MIXED NATURAL STANDS In mixed s t a n d s o f Douglas f i r and o t h e r c o n i f e r s , a n n u a l m o r t a l i t y (AMOR) can b e s t be p r e d i c t e d by knowing s t a n d d e n s i t y (BNT) and age, or average s t a n d d i a -meter. However, the s t a n d a r d e r r o r o f e s t i m a t e i s v e r y l a r g e due t o extreme v a r i a t i o n s i n the o r i g i n a l d a t a ( T a b l e X I ) . Some p r e d i c t i o n e q u a t i o n s a r e g i v e n i n Table XVI. i i i . FOR PLANTATIONS I n p l a n t a t i o n s , the b e s t e q u a t i o n s t o p r e d i c t p e r i o d i c a n n u a l p e r c e n t m o r t a l i t y (APCM) from stand and/or s i t e c h a r a c t e r i s t i c s a l l i n c l u d e age and e i t h e r s i t e Amount and t i m i n g of m o r t a l i t y 74 TABLE XV MULTIPLE REGRESSION EQUATIONS OF PERIODIC ANNUAL MORTALITY ON STAND CHARACTERISTICS (PURE STANDS - GROUP I ) 1 SEE Dependent Independent v a r i a b l e s and ~2 U n i t of V a r i a b l e I n t e r c e p t R e g r e s s i o n C o e f f i c i e n t s R Y (%) AMOR SQSP AGE AGESQ 39.9776 -0.4847 -0.9860 +0.0030 .28 5.2 96 CCCLO AGE 3.0025 +0.2923 -0.2488 .43 4.5 83 CSO DBH 0.9256 +0.4444 -0.7937 .41 4.6 85 CCCLO DBH 0.0686 +0.2644 -0.7619 .38 4.3 39 BANORM AGE AGESQ 29.9945 +0.0783 -1.0257 +0.0075 .30 5.1 94 BNT BNTSQ AGE 87 0.6562 -0.0092 +0.0003 -O.O327 .39 4.7 BAN SI BNT -3.4402 -0.0124 +0.0665 +0.0209 .35 4.9 91 AGE BAN DBH 13.9352 -0.1938 +0.0419 -0.5304 .21 5.4 100 BAMOR AGE AGESQ SI 73 -4.4945 +0.1340 -0.0011 +0.0155 .13 1.1 I d e n t i f i c a t i o n o f v a r i a b l e s i n Appendix IV. S i g n i f i c a n t a t the 0.05 l e v e l w i t h 97 o b s e r v a t i o n s ; under-l i n e d v a r i a b l e s do not make a s i g n i f i c a n t c o n t r i b u t i o n a t the 0.05 l e v e l . Amount and t i m i n g of m o r t a l i t y 75 TABLE XVI MULTIPLE REGRESSION EQUATIONS OF PERIODIC ANNUAL MORTALITY ON STAND CHARACTERISTICS (MIXED STANDS - GROUPS I I , I I I , IV) SEE Dependent Independent V a r i a b l e s and _2 U n i t of V a r i a b l e I n t e r c e p t R e g r e s s i o n C o e f f i c i e n t s R Y ( 0 / AMOR BNT AGE -6.0696 +0.0492 -0.0067 .80 lk 115 DBH 66.6188 -5.2397 .28 26 214 DBH DBHSQ 176.6500 -29.0250 +1.1703 .56 20 166 —r AGE BAN DBH^ 83.1928 -0.9582 +0.1345 -5.0252 .38 24 198 AGE DBH 92.3763 -0.7117 -4.6356 .34 25 207 I d e n t i f i c a t i o n of symbols i n Appendix IV. S i g n i f i c a n t a t the 0.05 l e v e l w i t h 63 o b s e r v a t i o n s . Under-l i n e d v a r i a b l e s do not make a s i g n i f i c a n t c o n t r i b u t i o n a t the 0.05 l e v e l . 'BAN measured on e n t i r e s t a n d ; AGE and DBH on Douglas f i r o n l y . Amount and t i m i n g of m o r t a l i t y 76 i n d e x , s p a t i a l p a t t e r n , or s t a n d d e n s i t y . Age, s i t e i n d e x and stand d i a m e t e r a re the v a r i a b l e s t a k i n g c a r e of the l a r g e s t p r o p o r t i o n of the t o t a l sum o f squares (49 p e r c e n t ) when l i n e a r l y r e l a t e d t o APCM. An i n d i c a t o r o f t r e e d i s p e r s i o n (variance-mean r a t i o of the fr e q u e n c y d i s t r i b u t i o n of q u a d r a t s h a v i n g the same number of t r e e s , VARMEAN) removes up t o 48 p e r c e n t of the v a r i a t i o n i n APCM when a s s o c i a t e d w i t h stand age. S i t e i n d e x and age or a b s o l u t e s t a n d d e n s i t y and age a l s o account f o r 39 t o 46 p e r -cent of the v a r i a t i o n i n APCM (T a b l e X V I I ) . A l l th e s e equa-t i o n s , however, have h i g h s t a n d a r d e r r o r s of e s t i m a t e (74 "to 83 p e r c e n t o f the mean r e s p o n s e ) . Except f o r age, a l l independent v a r i a b l e s show a b e t t e r c o r r e l a t i o n w i t h p e r i o d i c a n n u a l m o r t a l i t y (AMOR) than w i t h APCM. Some e x p r e s s i o n s of d e n s i t y , e s p e c i a l l y the crown compe-t i t i o n f a c t o r (CCF), n o r m a l i t y i n b a s a l a r e a (BANORM), and square s p a c i n g (SQSP) are s t r o n g l y c o r r e l a t e d w i t h AMOR (Ta b l e X I V ) . T h e r e f o r e , a l a r g e r p r o p o r t i o n of t h e v a r i a t i o n i n AMOR i s accounted f o r by some measures of d e n s i t y a l o n e t h a n by com-b i n a t i o n s o f s e v e r a l stand v a r i a b l e s i n the case of APCM. A maximum of 63 p e r c e n t of the v a r i a t i o n i n AMOR i s accounted f o r by a l i n e a r c o m b i n a t i o n of s t a n d age, average d i a m e t e r , s i t e i n d e x and r e l a t i v e or a b s o l u t e s t a n d d e n s i t y (RDN, YIELDN). Crown c l o s u r e , s p a c i n g and n o r m a l i t y a re a g a i n , as observed i n n a t u r a l s t a n d s , good m o r t a l i t y p r e d i c t o r s when cou p l e d w i t h age (T a b l e X V I I I ) . Amount and t i m i n g of m o r t a l i t y 77 TABLE X V I I MULTIPLE REGRESSION EQUATIONS OF PERIODIC ANNUAL PERCENT MORTALITY ON STAND CHARACTERISTICS (PLANTATIONS) , SEE Dependent Independent V a r i a b l e s and" U n i t of V a r i a b l e I n t e r c e p t R e g r e s s i o n C o e f f i c i e n t s R Y (%) APCM -2.8312 AGE +0.1241 .30 1.01 87 -10.0648 AGE +0.3346 DBH SI -0.8622 +0.0517 .49 .87 75 -3.3723 AGE +0.0882 VARMEAN +3.0716 .40 .92 79 6.4021 AGE -0.5589 AGESQ VARMEAN +0.0102 +3.4072 .48 .86 74 13.0100 AGE -0.7178 AGESQ SI -0.0127 -0.0232 .44 .92 79 -1.3930 AGE +0.1181 SQSP -0.16.11 .39 .95 82 9.3344 AGE -0.7090 AGESO BNT +0.0130 +0.0009 .46 .88 76 -I.804O VARMEAN +2.5770 RDN +3.0660 .33 .93 80 AGE SI BAN CCCLO -3.%9 +O.162+0.023 +0.034 VARMEAN DBH +1.464 -0.668 .50 .88 76 I d e n t i f i c a t i o n of symbols i n Appendix I V . S i g n i f i c a n t a t the 0.05 l e v e l w i t h 50 o b s e r v a t i o n s ; under-l i n e d v a r i a b l e s do not make a s i g n i f i c a n t c o n t r i b u t i o n a t the 0.05 l e v e l . A m o u n t a n d t i m i n g o f m o r t a l i t y 78 T A B L E X V I I I M U L T I P L E R E G R E S S I O N E Q U A T I O N S O F P E R I O D I C A N N U A L M O R T A L I T Y O N S T A N D C H A R A C T E R I S T I C S ( P L A N T A T I O N S ) S E E D e p e n d e n t I n d e p e n d e n t V a r i a b l e s a n d 2 2 U n i t o f V a r i a b l e I n t e r c e p t R e g r e s s i o n C o e f f i c i e n t s R Y (%) A M O R A G E D B H S I R D N -101.4100 +2.9780 -12.8340 +0.6640 +43.0500 .63 9.5 75 A G E D B H S I -124.2600 +4.3984 -14.5879 +0.7471 .57 10.0 80 Y I E L D N A G E D B H S I 885.0467. +0.0082 +2.9576 -14.2915 +0.6121 .63 9.5 75 Y I E L D N D B H 23^2215 +0.0137 -7.8865 .55 10.3 82 C C C L O D B H S Q A G E A G E S Q .53 105.9400 +0.2395 -0.3524 - 8 . 1 7 9 9 +0.1425 10.1 80 S Q S P A G E A G E S Q 151.9800 -4.7499 -7.6836 +0.1359 .57 10.2 81 B A N O R M A G E A G E S Q 139.2400 +0.4290 -10.8129 +0.1653 .55 10.4 82 S T O C G R A G E A G E S Q 121.2039 +0.3519 -9.5493 +0.1525 .51 10.8 86 C C F -17.0180 +0.1292 .49 10.8 86 C C C L O 11.3 -23.9378 +0.3007 .39 94 A G E S I B A N C C C L O V A R M E A N D B H -64.165+2.173+0.476+0.357-0.218+12.207-11.765 .64 9.5 75 B A M O R A G E A G E S Q S I B A N 15.7349 -0.9705 +0.0149 -0.0192 +0.0201 .50 .87 98 V O L M O R A G E A G E S Q S I Y I E L D N 273.2830 -14.8447 +0.2253 -0.4992 +0.0120 .43 15.9 112 I d e n t i f i c a t i o n o f s y m b o l s i n A p p e n d i x I V . S i g n i f i c a n t a t t h e 0.05 l e v e l w i t h 50 o b s e r v a t i o n s ; u n d e r -l i n e d v a r i a b l e s d o n o t m a k e a s i g n i f i c a n t c o n t r i b u t i o n a t t h e 0.05 l e v e l . Amount and t i m i n g of m o r t a l i t y 79 The l i n e a r i n c l u s i o n of age, d e n s i t y , s t o c k i n g , p a t t e r n , s i z e and s i t e e x p r e s s i o n s i n the same model does not improve the r e l a t i o n s h i p , nor low e r the s t a n d a r d e r r o r o f e s t i m a t e be-low 75 p e r c e n t of the mean response i n APCM o r i n AMOR because of i n t e r - c o r r e l a t i o n s amongst many of these v a r i a b l e s . Annual m o r t a l i t y i n b a s a l a r e a (BAMOR) or i n c u b i c volume (VOLMOR) i s b e s t p r e d i c t e d by t a k i n g age, s i t e i n d e x and stand d e n s i t y i n t o account ( T a b l e X V I I I ) , I t must be p o i n t e d out t h a t Spurr (1952) and C l u t t e r (1963) r e l a t e d the same v a r i a b l e s ( i n a d i f f e r e n t form o f e q u a t i o n ) t o e x p l a i n growth i n b a s a l a r e a and c u b i c volume, t h u s s u g g e s t i n g t h a t growth and m o r t a l -i t y a r e c o n t r o l l e d and c o u l d be p r e d i c t e d by the same st a n d and s i t e p a rameters. Two common c h a r a c t e r i s t i c s of the p r e d i c t i o n e q u a t i o n s p r e s e n t e d i n T a b l e s XV t o X V I I I i s t h a t they account f o r a r e l a t i v e l y s m a l l p r o p o r t i o n of the v a r i a t i o n i n m o r t a l i t y , and they have v e r y l a r g e s t a n d a r d e r r o r s of e s t i m a t e . On t h e one hand, the c o n s i d e r a t i o n of p h y s i c a l parameters ( s u c h as s l o p e o f t e r r a i n , s o i l t h i c k n e s s , h u m i d i t y r e g i m e s , and wind p a t -t e r n s ) , and the development o f more a c c u r a t e m a t h e m a t i c a l f u n c -t i o n s c o u l d perhaps improve the r e l a t i o n s h i p s based on stand c h a r a c t e r i s t i c s . On the o t h e r hand, i t must be remembered t h a t annual m o r t a l i t y r e p r e s e n t s o n l y a m i n i m a l f r a c t i o n o f t h e t o t a l y i e l d (0.5 t o 5 p e r c e n t of the number of l i v e t r e e s , or r o u g h l y 0.5 t o 1.0 p e r c e n t o f the s t a n d i n g volume); t h e r e f o r e , l a r g e v a r i a t i o n s i n m o r t a l i t y e s t i m a t e s do not c o r r e s p o n d t o Amount and t i m i n g of m o r t a l i t y 8 0 l a r g e v a r i a t i o n s i n y i e l d e s t i m a t e s , towards which most f o r e s -t r y s t u d i e s a r e d i r e c t e d . At the o u t s e t , a more r i g o r o u s c l a s s i f i c a t i o n might perhaps reduce a p p r e c i a b l y the observed v a r i a t i o n i n a n n u a l m o r t a l i t y . 4.1.3 GROSS Al© NET GROWTH AND YIELD So f a r i n t h i s c h a p t e r , m o r t a l i t y has been d e a l t w i t h on a p e r i o d i c a n n u a l b a s i s , w i t h o u t d i r e c t r e f e r e n c e t o stand growth. The amount of t i m b e r l o s t t h r ough time by com-p a r i s o n t o the amount growing must a l s o be c o n s i d e r e d . Table XIX c o n t a i n s a summary of the growth and y i e l d c h a r a c t e r i s t i c s measured i n pure s t a n d s and p l a n t a t i o n s . T h i s i n f o r m a t i o n was c o m p i l e d w i t h the program YIELD (Appendix I ) . Minimum, maximum, mean and s t a n d a r d d e v i a t i o n of y i e l d s i n b a s a l a r e a and c u b i c volume per a c r e are g i v e n t o g e t h e r w i t h p e r i o d i c annual and mean annual i n c r e m e n t s . V a r i a b i l i t y of the o b s e r v a t i o n s i s measured by a c o e f f i c i e n t of v a r i a t i o n expressed i n p e r c e n t a g e . C o r r e l a t i o n a n a l y s e s show t h a t y i e l d s i n c u b i c volume and square f e e t o f b a s a l a r e a are r e l a t e d t o s t a n d h e i g h t , d i a m e t e r , age and s i t e i n d e x , i n d e c r e a s i n g o r d e r o f i m p o r t a n c e . I n c r e -ments i n volume a r e f u n c t i o n s of s i t e i n d e x , a b s o l u t e and r e -l a t i v e s t a nd d e n s i t y , h e i g h t and average d i a m e t e r ; whether they are e x p r e s s e d i n p e r i o d i c or mean an n u a l u n i t s does not make much d i f f e r e n c e . P e r i o d i c a n n u a l increment i n b a s a l a r e a i s b e s t c o r r e l a t e d w i t h number of t r e e s per a c r e , s t o c k i n g , age, Amount and t i m i n g of m o r t a l i t y 81 TABLE XIX GROWTH AND YIELD CHARACTERISTICS MEASURED IN SAMPLE PLOTS V a r i a b l e 1 Min. Max. Mean St.dev. C.V.(%) N GROUP I YIELDN 933.0 18802.0 7778.6 3712.0 48 121 MAIN 34.6 256.8 140.0 45.7 33 121 GVOINC 28.0 413.2 326.0 204.9 85.0 97 BAN 65.0 200.3 56.4 28 121 BAIN 1.6 8.0 3.8 0.9 25 121 GBAINC 1.4 9.7 4.2 1.4 33 97 PLANTATION' YIELDN 476.0 5310.0 2485.3 956.1 38 64 MAIN 21.7 116.2 70.0 19.4 28 64 GVOINC 51.2 247.6 119.4 42.3 35 50 BAN 37.0 196.0 127.4 36.4 29 64 BAIN 1.7 5.3 3.6 0.7 20 64 GBAINC 2.2 8.0 4.6 1.5 33 50 "YIELDN = net c u b i c volume y i e l d per ac r e ( c u b i c f e e t ) . MAIN = mean an n u a l n e t inc r e m e n t i n c u b i c f e e t p a r a c r e . GVOINC = p e r i o d i c a n n u a l g r o s s i n c r e m e n t i n c u b i c f e e t per a c r e . BAN = net b a s a l a r e a i n square f e e t p e r a c r e . BAIN = mean annual net b a s a l a r e a i n c r e m e n t i n square f e e t per a c r e . GBAINC = p e r i o d i c annual g r o s s b a s a l a r e a i n c r e m e n t i n square f e e t p e r a c r e . Comprises the Wind R i v e r s p a c i n g t r i a l o n l y . Amount and t i m i n g of m o r t a l i t y 82 s p a c i n g and n o r m a l i t y ; mean annual i n c r e m e n t s i n b a s a l a r e a a re s t r o n g l y dependent on a b s o l u t e and r e l a t i v e d e n s i t y , s t o c k i n g and s i t e i n d e x . The degree of c o r r e l a t i o n between s t a n d and s i t e p a r a -meters and mean annual i n c r e m e n t i s g e n e r a l l y h i g h e r than i t i s v/ith p e r i o d i c a n n u a l i n c r e m e n t s . 4.1.3.1 GROSS GROWTH The b e s t e q u a t i o n s developed t o p r e d i c t p e r i o d i c annual g r o s s b a s a l a r e a i n c r e m e n t (GBAIITG) i n c l u d e e i t h e r age and b a s a l a r e a , o r age and s i t e i n d e x . Up t o 80 p e r -cent o f the v a r i a - t i o n i n GBAING i s removed by l i n e a r combina-t i o n s o f t h e s e v a r i a b l e s . Some models are p r e s e n t e d i n Table XX, f o r pure n a t u r a l s t a n d s and f o r p l a n t a t i o n Douglas f i r . I n o r d e r t o o b t a i n homogeneity of v a r i a n c e , and t o e x p r e s s the r e l a t i o n s h i p between GBAING and age, s i t e i n d e x and r e l a -t i v e d e n s i t y (RD) i n a l i n e a r form, C u r t i s (196?) developed the f o l l o w i n g e q u a t i o n f o r Douglas f i r , i n which a l l v a r i a b l e s were s i g n i f i c a n t a t the 0.01 l e v e l : log(dGB/dAGE)= 1.5306-0.48?0 (logAGE)+0.0781(SI/AGE)-Q.2174(1/RD) R 2= 0.79 SEE=0.83 F=257 N=208 where, dGB/dAGE: g r o s s b a s a l a r e a growth r a t e RD: r a t i o of a c t u a l b a s a l a r e a t o p r e d i c t e d b a s a l a r e a , termed r e l a t i v e d e n s i t y . Amount and t i m i n g of m o r t a l i t y 83 Up t o 51 p e r c e n t of the v a r i a t i o n i n p e r i o d i c a n n u a l g r o s s i n c r e m e n t i n c u b i c volume (GVOINC) was accounted f o r by age, s i t e i n d e x and b a s a l a r e a i n n a t u r a l s t a n d s ; i n p l a n t a t i o n s , where GVOINC showed l e s s v a r i a b i l i t y ( T a b l e X I X ) , the m u l t i p l e p c o r r e l a t i o n c o e f f i c i e n t was l a r g e r (max. R = 0.86). E q u a t i o n s are g i v e n i n T a b l e XX. C u r t i s (1967) e x p r e s s e d t h i s r e l a t i o n s h i p by u s i n g essen-t i a l l y the same v a r i a b l e s i n a l o g a r i t h m i c form as f o l l o w s : l o g (dGV/dAGE)= -0.2878 -0.0051 (AGE-^) +0.224-0 (logAGE b h) -1-1.3743 ( l o g S I ) -0.2566 (1/RD) R 2 = 0.69 SEE =0.09 F = 115 N = 208 where, dGV/dAGE: g r o s s c u b i c volume growth r a t e A G E ^ : age a t b r e a s t h e i g h t Based on t h i s a n a l y s i s o f g r o s s i n c r e m e n t , a v a r i a b l e den-s i t y y i e l d t a b l e was b u i l t by p o o l i n g Group I and Group I I d a t a which r e p r e s e n t s t a n d s h a v i n g more th a n 90 p e r c e n t Douglas f i r by number o f stems. Observed v a l u e s o f p e r i o d i c annual g r o s s b a s a l a r e a i n c r e m e n t were c o m p i l e d i n 33 p l o t s (128 measure-ments) and p r e s e n t e d by age and net b a s a l a r e a c l a s s e s . P e r i o d i c annual m o r t a l i t y i n b a s a l a r e a was a l s o computed i n each b a s a l a r e a c l a s s . A m u l t i p l e r e g r e s s i o n e q u a t i o n was then f i t t e d t o a l l p l o t measurements, t o p r e d i c t average i n c r e m e n t s , and f i l l the v o i d s i n the t a b l e ; i t i s : GBAINC = 11.5610 -0.3099 AGE +0.0019 AGESQ +0.0173 BAN R 2 = 0.58 SEE = 0.89 (20%) N = 128 Amount and timing of m o r t a l i t y 84 TABLE XX MULTIPLE REGRESSION EQUATIONS OF PERIODIC ANNUAL GROSS INCREMENT IN BASAL AREA AND CUBIC VOLUME ON STAND AND SITE CHARACTER!STICS Dependent Independent V a r i a b l e s and 1 V a r i a b l e Intercept Regression C o e f f i c i e n t s R 2 SEE 2 Unit of GROUP I (N = 97) GBAINC AGE AGESQ BAN SI 12.8081 -0.3538 +0.0022 +0.0196 -0.0018 .61 0.88 21 BAN AGE 6.5000 +0.0203 -0.1195 .49 1.00 24 AGE AGESQ SI 10.5276 -O.3068+O.OO22+O.O229 .45 1.04 25 GVOINC BA AGE SI -164.1960 +0.5658 :T71277 +2.2482 AGE AGESQ, SI -288.3350 +2.3577 -0.0183 +2.9896 AGE SI BAN BANSQ -291.7080 -0.6435 +2.6477 +1.2875 -0.0023 .51 60.5 29 PLANTATION^ (N = 50T .51 60.6 29 .48 62.5 30 GBAINC BAN AGE DBH 12.2709 +0.0153 -O .38OO +0.5070 .82 0.65 14 AGE AGESQ SI -2.0512 +0.3340 -0.0077 +0.0415 .80 0.69 15 GVOINC BAN AGE SI -20.6208 +0.4345 -3.0456 +1.9051 .76 21.4 17 AGE DBH SI 82.9825 -3.5950 +10.2391+1.0020 .73 22.6 19 AGE SI BAN BANSQ -127.9450 -3.8295 +2.1173 +2.4749 -0.0035 .36 16.5 8 I d e n t i f i c a t i o n of symbols i n Appendix IV. p S i g n i f i c a n t at the 0.05 l e v e l . Underlined v a r i a b l e s do not make a s i g n i f i c a n t c o n t r i b u t i o n at the 0.05 l e v e l , ^Includes the Wind River spacing t r i a l only. Amount and t i m i n g o f m o r t a l i t y 85 Observed and p r e d i c t e d v a l u e s a r e g i v e n i n Table XXI f o r stan d s s u p p o r t i n g 70 t o 326 square f e e t o f b a s a l a r e a , and r a n g i n g i n age from 30 to SO y e a r s . I t shows t h a t GBAINC de-c r e a s e s as st a n d s get c i d e r , but i n c r e a s e s w i t h i n c r e a s i n g den-s i t y . Average p e r i o d i c a n n u a l m o r t a l i t y a l s o i n c r e a s e s w i t h an i n c r e a s e i n s t a n d d e n s i t y . As computed, annual increment i s s m a l l e r and annual mor-t a l i t y h e a v i e r than c a l c u l a t e d by E l l i o t (1969) f o r Douglas f i r s t a n d s i n New Zea l a n d . T h i s d i f f e r e n c e i s p r o b a b l y due i n l a r g e p a r t t o a d i f f e r e n c e i n average s i t e q u a l i t y (141 f e e t a t 100 y e a r s , compared t o 30 f e e t a t 30 y e a r s f o r New Zeal a n d ' s d a t a ) . U.l.3.2 NET GROWTH AND YIELD Mean a n n u a l i n c r e m e n t i n b a s a l a r e a p e r acr e (BAIN) can b e s t be p r e d i c t e d from age and b a s a l a r e a . These two v a r i a b l e s account f o r more than 94 p e r c e n t o f the v a r i a t i o n i n BAIN, and show a s t a n d a r d e r r o r of e s t i m a t e o f 3 t o 6 p e r c e n t of the mean response. S i t e i n d e x , when added t o t h i s r e g r e s s i o n , does not improve the m u l t i p l e c o r r e l a t i o n . S i t e i n d e x and age, or top h e i g h t and age g i v e poor r e s u l t s by comparison t o b a s a l a r e a and age (Ta b l e X X I I ) . Age, b a s a l a r e a , top h e i g h t and s i t e i n d e x a r e i m p o r t a n t v a r i a b l e s i n r e l a t i o n w i t h mean annual i n c r e m e n t i n c u b i c v o l -ume per a c r e (MAIN). L i n e a r c o m b i n a t i o n s o f age, s i t e i n d e x and b a s a l a r e a remove more than 95 p e r c e n t o f the v a r i a t i o n i n Amount and t i m i n g of m o r t a l i t y 86 TABLE XXI VARIABLE DENSITY YIELD TABLE GROUPS I AND I I Net B a s a l Area M o r t a l i t y i n (BAN) Number of ., B a s a l A r e a ( S q . f t . P l o t Gross B a s a l Area Increment ( S q . f t . / / a c r e ) Measurements ( s q . f t . / a c r e / y e a r ) a c r e / y e a r ) 30 40 A G E 50 60 70 80 290 8.9 7.2 5.8 4.8 4.2 3.9 18' (4.8) (4.9) (3.7) (3.4) (1.83) 235 8.0 6.3 4.9 4.4 3.2 3.0 17 (7.5) (4.3) (5.2) (3.9) (3.5) (1.53) 200 7.4 5.7 4.3 3.3 2.6 2.4 33 (6.5) (5.6) (4.3) ( 3 - D (4.0) (1.81) 165 6.8 5.0 3.7 2.7 33 (6.9) (4.4) ( 4 . D (1.73) 140 6.4 4.6 3.2 2.2 13 (4.7) (4.3) (3.0) (0.71) 120 6.0 4.3 2.9 6 (7.3) (3.9) (3.2) (0.45) 105 5.8 4.0 2.6 2 (3.9) (3.0) (0.30) 95 5.6 3.8 2.4 1 (3.7) (0.50) 85 5.4 3.7 2.3 4 (1.7) (2.0) (1.30) 75 5.3 3-5 (0.40) 1 ( 3 . D Predicted from: GBAINC = 11.561 - 0.309 AGE + 0.002 AGESQ + 0.017 BAN. Observed v a l u e in b r a c k e t s . Amount and t i m i n g of m o r t a l i t y 87 TABLE X X I I MULTIPLE REGRESSION EQUATIONS OF MEAN ANNUAL INCREMENT ON STAND AND SITE CHARACTERISTICS SEE Dependent Independent V a r i a b l e s and" ^2 U n i t o f V a r i a b l e I n t e r c e p t R e g r e s s i o n C o e f f i c i e n t s R" Y {%) GROUP I (N = 121) BAIN 5.4537 AGE -0.1421 AGESQ BAN +0.0005 +0.0221 .94 .23 6 6.8891 AGE -0.1938 AGESO TOPK +0.0010 +0.0408 .41 .74 19 3.8646 AGE -0.1145 AGESO SI +0.0007 +0.0271 .41 .75 20 MAIN -92.5605 AGE -0.2521 AGESQ SI -0.0103 +0.9044 BAN +0.7511 .95 10.0 7 6.4296 AGE -0.7611 AGESQ BAN -0.0093 +0.0158 .87 16.4 12 55.2609 AGE -4.8727 AGESQ TOPK +0.0172 +2.8735 .72 24.4 17 -157.1820 AGE +0.7086 AGESQ SI -0.0032 +1.9018 .70 25.2 18 PLANTATION 3 (N = 64) BAIN 1.0673 AGE" +0.0325 AGESQ BAN -0.0019 +0.0299 .97 .13 3 -5.9752 AGE +0.5689 AGESQ SI -0.0079 -0.0027 .35 .62 17 -6.8889 AGE +0.5856 AGESQ TOPH -0.0082 +0.0085 .35 .62 17 Amount and t i m i n g of m o r t a l i t y TABLE X X I I ( c o n t ' d ) 8 8 1 , S E E Dependent independent V a r i a b l e s and" ^2. U n i t of V a r i a b l e I n t e r c e p t R e g r e s s i o n C o e f f i c i e n t s R Y PLANTATION 5 (N = 6 4 ) MAIN AGE AGESQ SI BAN - 1 4 6 . 6 8 0 0 + 5 . 2 3 7 3 - 0 . 0 8 1 2 + 0 . 6 1 0 4 + 0 . 6 1 1 0 . 9 7 3 . 5 5 AGE AGESQ BAN 3 1 . 3 6 5 3 - 1 . 1 5 4 0 + 0 . 0 0 2 6 + 0 . 5 9 0 6 . 7 6 9 . 8 1 4 AGE AGESQ TOPH - 1 4 2 . 4 5 2 0 + 7 . 3 5 0 4 - 0 . 0 1 7 9 + 1 . 6 7 2 1 . 6 8 1 1 . 2 16 AGE AGESQ SI - 2 9 5 . 9 2 4 0 + 1 6 . 3 8 4 7 - 0 . 2 0 6 1 + 0 . 5 7 3 4 . 5 8 1 2 . 9 1 8 I d e n t i f i c a t i o n of symbols i n Appendix IV. S i g n i f i c a n t a t the 0 . 0 5 l e v e l . U n d e r l i n e d v a r i a b l e s do not make a s i g n i f i c a n t c o n t r i b u t i o n a t the 0 . 0 5 l e v e l . 'Includes the Wind R i v e r s p a c i n g t r i a l o n l y . Amount and t i m i n g of m o r t a l i t y 89 MAIN and b r i n g the s t a n d a r d e r r o r of e s t i m a t e down t o 5 p e r -c e n t . Age and BAN, age and TOPH, or age and SI g i v e much l e s s p r e c i s e r e l a t i o n s h i p s w i t h M I N , a l t h o u g h they a re u s e f u l f o r r e p r e s e n t i n g changes i n MAIN over time (Table X X I I ) . Net b a s a l a r e a (BAN) can be p r e d i c t e d by knowing number o f t r e e s (BNT) and average h e i g h t of the 100 l a r g e s t p e r a c r e (TOPH). A more u s e f u l but l e s s p r e c i s e r e l a t i o n i s e s t a b l i s h e d by r e l a t i n g BAN t o age and s i t e i n d e x , o r t o age and top h e i g h t . Between 70 and 90 p e r c e n t of the v a r i a t i o n i n BAN i s accounted f o r by the s e parameters ( T a b l e X X I I I ) . Net c u b i c volume y i e l d (YIELDN) i s a c c u r a t e l y p r e d i c t e d by knowing b a s a l a r e a and average s t a n d d i a m e t e r , or b a s a l a r e a , s i t e i n d e x and age. The l a t t e r r e g r e s s i o n i s a l s o u s e f u l i n p r e d i c t i n g MAIN. A l t h o u g h l e s s p r e c i s e , r e g r e s s i o n s o f YIELDN on age and s i t e i n d e x , o r on age and top h e i g h t a re more u s e f u l i f y i e l d i s t o be r e p r e s e n t e d by s i t e c l a s s e s over age. Top h e i g h t and b a s a l a r e a account i n d i v i d u a l l y f o r more than 75 p e r c e n t o f the v a r i a t i o n i n y i e l d ( T a b l e X X I I I ) . A.1.3.3 C U M U L A T I V E M O R T A L I T Y P r e d i c t e d v a l u e s of g r o s s a n n u a l increment (T a b l e XX) can be c o n s i d e r e d as average ann u a l g r o s s growth r a t e s f o r 5-year p e r i o d s , s i n c e they were c o m p i l e d from p l o t s remeasured a t 5-year i n t e r v a l s on the average (Chapter 3). I n o r d e r t o o b t a i n g r o s s y i e l d s , one has to " i n t e g r a t e " such increment f u n c t i o n s from age 0 onward ( o r from the l o w e s t A m o u n t a n d t i m i n g o f m o r t a l i t y 90 T A B L E X X I I I M U L T I P L E R E G R E S S I O N E Q U A T I O N S O F N E T Y I E L D O N S T A N D A N D S I T E - C H A R A C T E R I S T I C S D e p e n d e n t V a r i a b l e I n t e r c e p t 1 I n d e p e n d e n t V a r i a b l e s a n d R e g r e s s i o n C o e f f i c i e n t s 2 2 R S E E G R O U P I ( N = 121) B A N -I49.472O T O P H B N T +2.9101 +0.1575 .88 S q . f t . 19". 4 % 9 59.7792 A G E A G E S Q T O P H -2.8718 +0.0241 +2.1524 .74 28.9 14 -86.0341 A G E A G E S Q S I +1.2791 +0.0095 +1.3279 .71 30.8 15 Y I E L D N -12218.0 A G E S I - B A N +89.8551+60.1760+33.1509 .97 C u . f t . 624.4 8 -5478.0 B A N D B H +41.6012+425.1560 .97 631.6 8 —2246.9 A G E A G E S Q T O P H -238.0270+1.8907+165.9320 .94 903.0 12 - I 5 6 5 9 . O A G E A G E S Q +O2.3501+0.7556 +103.6150 .90 1182.0 15 -7993.6 T O P H +153.8940 .92 1033.1 13 -4662.7 B i ^ + 6 2 . 1 1 7 1 .89 1226.8 16 A m o u n t a n d t i m i n g o f m o r t a l i t y 91 T A B L E X X I I I ( c o n t ' d ) D e p e n d e n t V a r i a b l e I n t e r c e p t 1 I n d e p e n d e n t V a r i a b l e s a n d R e g r e s s i o n C o e f f i c i e n t s 2 2 R • S E E P L A N T A T I O N 3 ( N - = 64) • B A N -145.4310 T O P H • B N T +4.OO43 +0.0579 .90 S q . f t . 11.86 % 9 -266.1880 A G E A G E S Q T O P H +18.3902-0.2102 +0.3304 .71 20.21 16 -244.2670 A G E A G E S Q S I +18.2449-0.2046 -0.0606 .70 20.30 16 Y I E L D N -4117.3600 A G E S I B A N +49.9812+21.8337+21.9530 .97 C u . f t . 169.6 % 7 -1467.3500 B A N D B H +20.4697+242.742 .98 136.9 5 -4515.0100 A G E A G E S Q T O P H +153.3920-1.3972+62.4476 O -r .OJ? 397.3 16 -10156.40 A G E A G E S Q S I +487.4620-5.0233+21.1125 .77 467.1 19 -2626.7900 T O P H +93.0792 • 75 478.4 19 -543.3970 B A +23.7778 .82 4U.2 17 • ^ I d e n t i f i c a t i o n o f s y m b o l s i n A p p e n d i x I V . 2 S i g n i f i c a n t a t t h e m a k e a s i g n i f i c a n t 0.05 l e v e l . U n d e r l i n e d v a r i a b l e s c o n t r i b u t i o n a t t h e 0.05 l e v e l . d o n o t ^ I n c l u d e s t h e W i n d R i v e r s p a c i n g t r i a l o n l y . Amount and t i m i n g of m o r t a l i t y 92 l i m i t of a v a i l a b l e d a t a , i . e . age 20). However, s i n c e the e q u a t i o n s developed here have not been e x p r e s s e d i n a d i f f e r -e n t i a l form, the i n t e g r a l must be r e p l a c e d by a summation over s u c c e s s i v e 5-year p e r i o d s . The c u m u l a t i v e increment d u r i n g one p e r i o d i s t h u s assumed t o be 5 t i m e s the annua l amount c a l c u l a t e d a t the b e g i n n i n g of the p e r i o d . On the o t h e r hand, net y i e l d s can be p r e d i c t e d d i r e c t l y from e q u a t i o n s g i v e n i n Table X X I I I . I f i n g r o w t h i s n e g l e c t e d , c u m u l a t i v e m o r t a l i t y a t any g i v e n time can be c a l c u l a t e d as the d i f f e r e n c e between g r o s s and net y i e l d a t t h a t time. Table XXIV i s g i v e n as an example; i t has been b u i l t by f o l l o w i n g t h i s method, f o r n a t u r a l s t a n d s (Group I ) . Cumulative m o r t a l i t y i s e x p r e s s e d i n a b s o l u t e terms and as a p r o p o r t i o n o f g r o s s and net y i e l d i n c u b i c volume per a c r e . T h i s t a b l e c l e a r l y i n d i c a t e s the amount and t i m i n g o f m o r t a l i t y on good Douglas f i r s i t e s ( S I = 1L0). B e f o r e age 80, n e a r l y 20 p e r c e n t o f the g r o s s p r o d u c t i o n and more than 20 p e r -cent o f the amount a c t u a l l y growing has been l o s t t o m o r t a l i t y . The a c c u r a c y o f t h i s method of a c c u m u l a t i n g g r o s s growth r a t e s over y i e l d a t age 20 depends h e a v i l y on the p r e c i s i o n w i t h which t h i s e a r l y y i e l d can be p r e d i c t e d . I n Table XXIV, i t has p r o b a b l y been o v e r e s t i m a t e d due t o a l i m i t e d amount of d a t a f o r n a t u r a l s t a n d s i n the range of 20 to LO y e a r s . Com-p a r i s o n s w i t h normal y i e l d t a b l e s , a l s o l a c k i n g i n f o r m a t i o n c o n c e r n i n g e a r l y s t and development (e.g. s i x p l o t s i n McArdle Amount and ti m i n g of m o r t a l i t y 93 TABLE XXIV PREDICTED CUMULATIVE MORTALITY ON SITE CLASS I I I DOUGLAS FIR (GROUP I) AGE YIELD CUMULATIVE MORTALITY' GROSS NET PROPORTION OF GROSS YIELD NET YIELD cu. ft77acre cu. f t . / a c r e 20 2796.3 2796.3 0.00 0.00 0.00 30 4534.9 3997.6 535.3 0.12 0.13 40 6408.7 5350.1 1058.6 0.16 0.20 50 8381.1 6853.6 1527.5 0.18 0.22 60 10413.5 8508.3 1905.2 0.18 0.22 70 12473.1 10314.5 2158.6 0.17 0.21 80 14521.4 12270.9 2250.5 0.15 0.18 From GVOINC =-288.3350 +2.3577 AGE -0.0185 AGESQ +2.9396 SI R 2 = 0.48 SEE = 62.5 (30%) N = 97 where SI = 140 GROSS YIELD = NET Y I E L D ? n + 5 ( p r e d i c t e d increment at 5 ° ^ u age 20) + 5 ( p r e d i c t e d increment at age 25) 'From YIELDN = -13659.0 +82.3501 AGE +0.7556 AGESQ +103.6150 SI R 2 = 0.90 SEE = 1182.0 (15%) N = 121 Underlined v a r i a b l e s do not make a s i g n i f i c a n t c o n t r i b u t i o n a t the 0.05 l e v e l . I d e n t i f i c a t i o n of symbols i n Appendix IV. Amount and t i m i n g of m o r t a l i t y 94 et a l . (1949) f o r age 20-29), must be made c a u t i o u s l y . The use of age and s i t e index only f o r y i e l d p r e d i c t i o n s i s a second source of i n a c c u r a c i e s . As shown i n Table XXIII, the i n c l u s i o n of stand d e n s i t y i n the model g r e a t l y improves the r e l a t i o n s h i p , but makes i t i m p r a c t i c a l to use, u n l e s s den-s i t y i s expressed i n r e l a t i v e terms (e.g. C u r t i s , 1967). 4.2 MORTALITY RELATED TO TREE CHARACTERISTICS L i f e insurance companies c u r r e n t l y prepare v/hat i s c a l l e d "experience" t a b l e s of m o r t a l i t y , i n which the yearly, proba-b i l i t y of dying or l i v i n g i s given f o r i n d i v i d u a l s of a c e r t a i n age. These t a b l e s r e f l e c t the accumulated experience of many companies i n sampling the process of m o r t a l i t y among human po p u l a t i o n s . They are r e l i a b l e because the number of observed cases i s tremendously l a r g e . L i f e insurance a n n u i t i e s are c a l c u l a t e d from them. N a t i o n a l v i t a l s t a t i s t i c s have a l s o been compiled by the Canadian Government since 1921, g i v i n g the r a t e of m o r t a l i t y by 100,000 people, by age group, sex group, e t c . (Dominion Bureau of S t a t i s t i c s , 1969). In f o r e s t r y , an attempt was made with i n s e c t s (Ives, 1964). To our knowledge, no such t a b l e s e x i s t f o r f o r e s t s , even i f f o r humans, they have been c a l c u l a t e d f o r a l o n g time (Hart, 1924). T h i s approach to m o r t a l i t y seems to be one of the simplest, because a l a r g e p a r t of the data compiled to charac-t e r i z e f o r e s t s comes from r e p r e s e n t a t i v e samples of some popu-l a t i o n s measured r e p e a t e d l y over the years, i . e . by means of Amount and t i m i n g of m o r t a l i t y 95 permanent p l o t s . I f o f f e r s , the tremendous p o s s i b i l i t y of a l l o w i n g f o r accumulation of i n f o r m a t i o n c o l l e c t e d by v a r i o u s agencies, to improve the r e l i a b i l i t y and broaden the range of m o r t a l i t y t a b l e s . I t could p o s s i b l y help i n d e v e l o p i n g f o r e s t insurance (Walters, 1951). Two s e t s of t a b l e s are developed i n t h i s study, one f o r n a t u r a l Douglas f i r stands, and one f o r p l a n t a t i o n s . They are b a s i c a l l y s i m i l a r , but the time span i s longer f o r n a t u r a l stands than f o r p l a n t a t i o n s , and the d i v e r s i t y of d a t a c o l -l e c t e d i s l e s s . In f a c t , s i x groups of m o r t a l i t y t a b l e s were b u i l t 9 one f o r each s p e c i f i c c l a s s of i n d i v i d u a l s c h a r a c t e r -i z e d by t h e i r s i z e and v i g o r . Trees were a l l o c a t e d to a p a r t i c u l a r c l a s s a f t e r compari-son with an average c h a r a c t e r i s t i c of the p o p u l a t i o n i n which they are bound to l i v e a f t e r they get e s t a b l i s h e d . This pro-cedure would not be a p p l i c a b l e to mobile i n d i v i d u a l s which are, to a c e r t a i n extent, independent of t h e i r neighbourhood as f a r as m o r t a l i t y causes are i n v o l v e d . But, i n p l a n t communities, i t i s r e a d i l y accepted that the l a r g e r and t a l l e r i n d i v i d u a l s ( i . e . f a s t e s t growing) do suppress surrounding s m a l l e r s u b j e c t s . Thus, based on t h i s assumption, 13 thousand i n d i v i d u a l s were grouped i n t o c l a s s e s and observed d u r i n g v a r i a b l e p e r i o d s of time. C l a s s e s of r e l a t i v e s i z e , r e l a t i v e height, r e l a t i v e increment i n s i z e and height, r e l a t i v e crown w i d t h - s i z e r a t i o , and p o s i t i o n i n the canopy were considered. Q u a r t e r - c l a s s e s were a r b i t r a r i l y considered at any given Amount and t i m i n g of m o r t a l i t y 96 moment i n time to s i m p l i f y c o m p i l a t i o n and t a b u l a t i o n . Proba-b i l i t i e s of d y i n g were c a l c u l a t e d as f o l l o w s : n = number of t r e e s observed i n one p l o t , i n one c l a s s k r k nd = number of t r e e s dead at the end of the p e r i o d among r k the n„ observed t r e e s r k N = number of p l o t s observed i n any age and r e l a t i v e c l a s s r^. r ^ = r e l a t i v e c l a s s k, where k equals a c t u a l t r e e value d i v i d e d by average stand value at the beginning of the o b s e r v a t i o n p e r i o d a^ = number of years of o b s e r v a t i o n between two measurements of p l o t ' P. , = annual p r o b a b i l i t y of dy i n g 3 • K TD . , = t o t a l number of dead t r e e s J >K TO. , = t o t a l number of t r e e s observed TY . , = number of t r e e - y e a r s of o b s e r v a t i o n 3 ? J£ A. = ten-year age p e r i o d , i n c l u d i n g A-4 to A+5. Thus, when A j , j = 20, 30, 40 90 r k , k = 1, 16 ( r x = 0.00; r 2 = 0.25, r l 6 = 3.75) then, N TO. , = 5T n i Z-1 r k N TD = ^ nd J ' K i =1 r k N 3 >k i j,k 1 Amount and t i m i n g of m o r t a l i t y 97 = (TD . , / TO J > ^  )/10 When no r e l a t i v e c l a s s e s are considered (e.g. crown c l a s s ) , the same formulae apply; r must be dropped, and k i s given a crown c l a s s value of 1,2, 3» or 4, i n s t e a d of 0.00 to 3.75* For example, a number of twenty-year-old t r e e s (age 16 to 25 at the beginning of the p e r i o d of o b s e r v a t i o n ) , one-fourth the s i z e of the average t r e e ( r ^ = 0.25) were observed d u r i n g a ten-year p e r i o d , at the end of which the number of dead and the number of l i v e t r e e s were recorded. The p r o p o r t i o n of dead t r e e s to the t o t a l number, d i v i d e d by 10 ( l e n g t h of the p e r i o d ) was then entered i n t o the t a b l e showing the y e a r l y p r o b a b i l i t y of dying based on r e l a t i v e s i z e , i n the case l a b e l e d : year 20, r a t i o 0.25 (Appendix V). The number of t r e e - y e a r s of observa-t i o n was obtained by m u l t i p l y i n g the number of t r e e s observed by the number of y e a r s of o b s e r v a t i o n . The annual p r o b a b i l i t y of death was c a l c u l a t e d because most p e r i o d s of o b s e r v a t i o n were d i f f e r e n t from 10 y e a r s i n each i n d i v i d u a l •plot. 4.2.1 MORTALITY AND RELATIVE TREE SIZE F i g u r e s 8 and 9 i l l u s t r a t e how the p r o b a b i l i t y of death v a r i e s with time i n r e l a t i o n to the r a t i o of a c t u a l t r e e diameter to average stand diameter ( r e l a t i v e s i z e ) 3 . They Percentage dead t r e e s k i l l e d by bark b e e t l e s have been l i s t e d by diameter c l a s s (Balch, 1942), but no attempt to use t h i s technique as a p r e d i c t i o n t o o l was made. Figure 8* Periodic probability of individual tree mortality by relative size classes- Natural stands,Groups I , H , IE , EC-9 9 Figure 8 (continued)- Periodic probability of individual tree mortality by relative size classes* Natural stands;Groups I t H f m , 1ST* 'Number of tree-years of observation-774 0-3 O-Ql—V 60-70 years O'QI 4097 70-80 years 0-3 o-o»—V 0 5 80-90 years h5 dbh/OBH Figure 9- Periodic probability of individual tree mortality in the Wind 1 0 0 River spacing plantation-1 Number of tree-years of observation* 2399 0-2 dbhin/OBHin* OI L I367 O-Ol-H' 0-3 -507 OO1—4 635 0 - 0 * — | r 3375 3 0 - 4 0 years dbh in-/DBH in h/TOPH 3 0 - 4 0 years years Amount and t i m i n g of m o r t a l i t y 101 show t h a t : 1) t r e e s are not l i k e l y to d i e between the age of 20 and 90 years when they are at l e a s t 1.50 times the average s i z e a l l the time; 2) the smaller the r e l a t i v e s i z e ever a c e r t a i n minimum, the l a r g e r the p r o b a b i l i t y to d i e i s ; 3) Douglas f i r t r e e s are most v u l n e r a b l e between the age of LO and 50 years; L) the p r o b a b i l i t y of dying becomes more or l e s s n e g l i g i b l e at any time f o r t r e e s of average s i z e or bigger; and 5) other t h i n g s being equal, t r e e s growing i n n a t u r a l stands (Figure 8) have more chances to d i e than i n p l a n t a t i o n s (Figure 9). These c o n c l u s i o n s , based on 168,589 t r e e - y e a r s of observa-t i o n , could p r o f i t a b l y be used i n p l a n n i n g i n t e n s i v e management p r a c t i c e s f o r young Douglas f i r f o r e s t s (e.g. salvage and t h i n -n i n g o p e r a t i o n s aimed at the r e c u p e r a t i o n or r e d u c t i o n of l o s -ses by m o r t a l i t y ) . Appendix V shows' the same t h i n g as F i g u r e 8 and F i g u r e 9 i n more d e t a i l , f o r 10-year p e r i o d s . 4.2.2 MORTALITY AND RELATIVE HEIGHT F i g u r e s 9 and 10 i l l u s t r a t e how m o r t a l i t y v a r i e s v/ith time, i n r e l a t i o n to the r a t i o of a c t u a l t r e e height to stand top height (or average height of the 100 l a r g e s t t r e e s per a c r e ) . They show t h a t : 1) t r e e s of average height (top height) or l a r g e r are not l i k e l y to d i e between the age of 20 and 90 years; 2) g e n e r a l l y , the s h o r t e r the t r e e i s , the higher i t s p r o b a b i l i t y to d i e ; however, a f t e r the age of 60, i t seems that t r e e s one-half top height have more chances to d i e than e i t h e r s h o r t e r or t a l l e r t r e e s ; 3) t r e e s are most v u l n e r a b l e 1 0 2 Figure 10- Periodic probability of individual tree mortality by relative height classes- Natural stands-, Groups I,II,IE,ET_-103 Figure 10 (continued)- Periodic probability of individual tree mortality by relative height classes- Natural stands-,Groups I t H,IH t I3Z-1 Number of tree-years of observation-Amount and t i m i n g of m o r t a l i t y 104 between the age of 40 and 50 years; and A) t r e e s growing i n n a t u r a l stands have more chances to d i e than i n p l a n t a t i o n s , up to age 50. These c o n c l u s i o n s are based on 153>942 t r e e - y e a r s of ob-s e r v a t i o n . They are more or l e s s the same as f o r those drawn a f t e r o b s e r v a t i o n of the m o r t a l i t y - r e l a t i v e s i z e r e l a t i o n s h i p , except f o r two t h i n g s . F i r s t , the range i n height around the mean i s much s m a l l e r than the range i n diameter; t h i s i s p a r t l y e x p l a i n e d by stand composition and p a r t l y by the form of the height-diameter r e l a t i o n s h i p used to p r e d i c t h e i g h t . Stands observed were even-aged, and the curve used was a second degree polynomial which f l a t t e n s out over a c e r t a i n diameter, as shown by p l o t t i n g any height r e l a t i o n s h i p presented i n Appendix I I . T h i s was e x p l a i n e d i n more d e t a i l by C r o s s l e y (1967). Secondly, t r e e s much smaller than the average have a b e t t e r chance to l i v e than t r e e s s l i g h t l y s h o r t e r than the average, at l e a s t i n the l a t e r stage of stand development. During the p e r i o d of r a p i d l y i n c r e a s i n g d e n s i t y (before age 50), the d i f f e r e n t i a t i o n be-tween height c l a s s e s i s not yet pronounced and most t r e e s are l o c a t e d i n the main canopy. A f t e r t h a t stage i n development, however, t r e e s f a l l i n g o f f the main canopy or d e v e l o p i n g under i t would have a s l i g h t l y i n c r e a s e d chance of l i v i n g over the weakest s u b j e c t s , remaining r i g h t under the main cover. T h i s i s evidenced by the o v e r a l l lower p r o b a b i l i t y of death observed i n p l a n t a t i o n s where the v a r i a n c e of t r e e height i s s m a l l e r than i n n a t u r a l stands. Appendix V c o n t a i n s more d e t a i l s than Amount and t i m i n g of mortality-F i g u r e 9 and Fi g u r e 10 on these r e l a t i o n s h i p s . 105 4.2.3 MORTALITY AND CROWN CLASS The process j u s t explained i s w e l l e x e m p l i f i e d by the r e l a t i o n s h i p s m o r t a l i t y - crown c l a s s ( F igure 11) i n which the p r o b a b i l i t y of m o r t a l i t y i s r e l a t e d to f o u r crown c l a s s e s (4* suppressed; 3- i n t e r m e d i a t e ; 2: co-dominant; 1:dominant t r e e s ) . C o n c l u s i o n s are roughly the same as f o r h e i g h t - m o r t a l -i t y r e l a t i o n s h i p s . Suppressed t r e e s always have a higher pro-b a b i l i t y of dying than others, except i n young p l a n t a t i o n s where the d i f f e r e n t i a t i o n process i s not r e a l l y completed. Co-dominant and dominant t r e e s always have a f a i r chance of l i v -i n g , whereas the p r o b a b i l i t y of dyi n g f l u c t u a t e s a p p r e c i a b l y with i n t e r m e d i a t e t r e e s . Appendix V g i v e s annual f l u c t u a t i o n s i n number, and p r o b a b i l i t i e s f o r 101,852 t r e e - y e a r s of observa-t i o n on t r e e s c l a s s i f i e d by crown c l a s s i n the o r i g i n a l data. 4.2.4 MORTALITY AND RELATIVE INCREMENT IN SIZE AND HEIGHT The measured range i n s i z e increment v a r i e d between 0.00 (or even negative) to 3*25 times the average stand diame-t e r increment. The f o l l o w i n g o b s e r v a t i o n s are based on data l i s t e d i n Appendix V: 1) i n any one r e l a t i v e increment c l a s s ( r a t i o of t r e e increment to average stand increment), the pro-b a b i l i t y of dyi n g i n c r e a s e s up to age 40 a f t e r which i t de-creases; 2) the p r o b a b i l i t y of dyi n g s h a r p l y decreases when the Figure Ii- Periodic probability of individual tree mortality by crown classes- Natural standsjGroups 1,31, Figure Ii (continued I)- Periodic probability of individual tree mortality by crown classes* Natural stands; Groups i tn,m vnL-Figure II (continued 2) Periodic probability of individual tree mortality by crown classes-Wind River Spacing Plantation-Crown Class 3 0 - 4 0 years Crown Class 4 » suppressed tree 3 • intermediate tree 2 • co-dominant tree I • dominant tree Amount and t i m i n g of m o r t a l i t y 109 annual diameter increment goes from c l a s s 0 . 0 0 to 0 . 2 5 , and from c l a s s 0 . 2 5 to 0 . 5 0 ; 5) t r e e s growing at the average r a t e or f a s t e r are u n l i k e l y to d i e from r e g u l a r m o r t a l i t y . These c o n c l u s i o n s are based on 1 0 4 , 0 8 7 t r e e - y e a r s of o b s e r v a t i o n . F i g u r e 12 shows the r e l a t i o n s h i p m o r t a l i t y - r e l a t i v e d i a -meter increment between age 40 and 5 0 . Both curves were drawn from a negative e x p o n e n t i a l model f i t t e d to the data i n c l u d e d i n Appendix V. The form of the equation i s : b y Y = aX c % where a, b, and c are the r e g r e s s i o n c o e f f i c i e n t s . The slope of t h i s curve i s very sharp up to X = 0 . 5 0 , a f t e r which l e v e l i t g r a d u a l l y becomes asymptotic to the a x i s . Re-g r e s s i o n c o e f f i c i e n t s are given i n Appendix V f o r every curve shown i n F i g u r e s 8 , 9 and 1 0 , as w e l l as f o r r e l a t i v e diameter increment f u n c t i o n s . Height increment r e l a t i o n s h i p s are not i l l u s t r a t e d because h e i g h t s were p r e d i c t e d by means of height-diameter equations, and consequently, they show e s s e n t i a l l y the same trend as d i a -meter increment curves, except that the range i n r e l a t i v e h e ight increment i s much smaller ( 0 . 0 0 to 2 . 0 0 times the aver-age as compared to 0 . 0 0 to 3 . 2 5 f o r d i a m e t e r s ) . 4 . 2 . 5 MORTALITY AND RELATIVE CROWN WIDTH/DIAMETER RATIO Observations on m o r t a l i t y and crown width were made i n the Wind R i v e r p l a n t a t i o n only, because no d i r e c t measure-ment of crown widths had been made i n n a t u r a l stands (Appendix I I ) . T h i s a n a l y s i s does not l e a d to i n t e r e s t i n g c o n c l u s i o n s 0 5 -Y«P Figure 12 Periodic probability of individual tree mortality on relative increment in size* Wind River Spacing Plantation 4 0 - 5 0 years y« 0-4679 x0-056l (o-|466x ) N» 35,992 r 2 » 0 - 7 8 0 0 J I I I I i - 0 -Nafural stands; Groups I, H , I H , HE-4 0 - 5 0 years 0-5 y* I 903 xO-0908 (o-0025x) N« 68,095 r 2 = 0-75 ° ? 0 JL I I J L 01 0-2 0-3 0-4 0-5 X-dbh in/DBH In-1-0 1-5 2 0 Amount and t i m i n g of m o r t a l i t y 111 on m o r t a l i t y perhaps because r a t i o s of r a t i o s are d i f f i c u l t to i n t e r p r e t (the i n d i v i d u a l crown width/diameter r a t i o v/as com-pared to the average crown width/diameter r a t i o ) . B e t t e r r e l a -t i o n s h i p s could probably be developed by r e l a t i n g m o r t a l i t y to i n d i v i d u a l crown width-diameter r a t i o s ( l i k e i t was done f o r crown c l a s s ) r a t h e r than to r e l a t i v e crown width r a t i o s . Appendix V c o n t a i n s our f i n d i n g s f o r 53,539 t r e e - y e a r s of ob-s e r v a t i o n i n the Wind R i v e r spacing p l a n t a t i o n . 4.3 CHAPTER SUMMARY Annual m o r t a l i t y (expressed both i n absolute terms and as a r a t e ) was measured i n s e v e r a l p u b l i s h e d y i e l d t a b l e s , and analyzed by m u l t i p l e r e g r e s s i o n techniques. C o r r e l a t i o n ma-t r i c e s show th a t stand age and absolute d e n s i t y are s i g n i f i -) c a n t l y r e l a t e d to annual m o r t a l i t y , both i n n a t u r a l stands and p l a n t a t i o n s . However, the p r e c i s i o n of a l l l i n e a r r e g r e s s i o n models developed i s f a i r l y low. Data from Table I I were a l s o analyzed i n a s i m i l a r manner. I t appears that m o r t a l i t y i s best expressed i n number of stems per a c r e . S i g n i f i c a n t but weak c o r r e l a t i o n s are noted with stand and s i t e c h a r a c t e r i s t i c s . For i n s t a n c e , the percentage v a r i a t i o n i n annual m o r t a l i t y taken care of by the best com-b i n a t i o n of c h a r a c t e r i s t i c s t e s t e d i n n a t u r a l stands i s i+3, and the standard e r r o r of estimate i s l a r g e r than ?8 percent i n every equation. Simultaneous c o n s i d e r a t i o n of age, d e n s i t y , s t o c k i n g , p a t t e r n , s i z e , and s i t e c h a r a c t e r i s t i c s does not Amount and t i m i n g of m o r t a l i t y H 2 improve the r e l a t i o n s h i p w i t h m o r t a l i t y , nor the p r e c i s i o n o f the p r e d i c t i o n . I t i s observed i n s e v e r a l i n s t a n c e s t h a t mor-t a l i t y can be p r e d i c t e d by making use of the same s t a n d p a r a -meters e n t e r i n g growth f u n c t i o n s . L i n e a r c o m b i n a t i o n s of age, s i t e i n d e x and st a n d d e n s i t y form q u i t e a c c u r a t e growth and y i e l d p r e d i c t i o n e q u a t i o n s . A v a r i a b l e d e n s i t y y i e l d t a b l e i s b u i l t by making use o f the r e -l a t i o n s h i p between g r o s s b a s a l a r e a growth r a t e , age and b a s a l a r e a . An example o f how amount and t i m i n g o f m o r t a l i t y can be computed by a c c u m u l a t i n g g r o s s growth r a t e s over y i e l d a t age 20 i s g i v e n . The t r e e approach t o m o r t a l i t y i s t a k e n i n b u i l d i n g s i x groups of m o r t a l i t y t a b l e s f o r i n d i v i d u a l t r e e s , 20 t o 90 y e a r s of age i n n a t u r a l s t a n d s , and 20 t o 50 y e a r s of age i n p l a n t a -t i o n s . I n s i d e each group, t h e r e i s one t a b l e f o r each p a r t i -c u l a r t r e e s i z e or v i g o r c l a s s . A p a r t from g i v i n g p r o b a b i l -i t i e s n e c e s s a r y t o b u i l d s t o c h a s t i c s t a n d models, these t a b l e s i n d i c a t e w hich t r e e i s most l i k e l y t o d i e and when, based e i t h e r on i t s s i z e , i t s r a t e o f growth o r i t s p o s i t i o n i n s i d e a g i v e n f o r e s t community. CHAPTER 5 113 DISTRIBUTION OF MORTALITY The i n i t i a l p a t t e r n of p l a n t p o p u l a t i o n s i s determined when seeds germinate. L o c a t i o n of seed-trees, d i r e c t i o n of p r e v a i l i n g winds, and topography f i r s t i n f l u e n c e seed d i s p e r -s i o n . Then, genetic p o t e n t i a l , m i c r o - c l i m a t e and e x t e r n a l agents ( b i r d s and rodents, f u n g i and i n s e c t s ) i n f l u e n c e germin-a t i o n p a t t e r n s . T h e r e a f t e r , morphological, s o c i o l o g i c a l and environmental f a c t o r s i n t e r a c t i n c r e a t i n g p a t t e r n m o d i f i c a -t i o n s and va r i a n c e i n frequency d i s t r i b u t i o n s of t r e e para-motors. The e c o l o g i c a l aspects of p l a n t establishment ( i n c l u d i n g Douglas f i r ) have been e x t e n s i v e l y i n v e s t i g a t e d (Aaltonen, 1926, A l l e n , 1942, Isaac, 1943, G r i f f i t h , i960, Zinke, 1962, Kershaw, 1963, 1964, F o s t e r and Johnson, 1963b, Smith et a l . , 1966). B i o m e t r i c i a n s have p e r f e c t e d numerous methods of e s t i -mating p r o b a b i l i t y d i s t r i b u t i o n s from experimental designs and of e v a l u a t i n g a s s o c i a t i o n between s p e c i e s . M o r t a l i t y spreading from e p i c e n t e r s i n t o contagious d i s t r i b u t i o n s has a l s o been documented from a t h e o r e t i c a l viewpoint (Cole, 194-6, Thompson, 1952, Evans, 1953, Barton and David, 1956, P i e l o u , 1961). But, apart from s t u d i e s of p a t t e r n s and d i s t r i b u t i o n s of d i s e a s e s i n p l a n t a t i o n s ( F o s t e r and Johnson, 1963a), there has been r e l a t i v e l y l i t t l e e m p i r i c a l knowledge accumulated from Douglas f i r stands concerning a c t u a l s p a t i a l arrangements and frequency D i s t r i b u t i o n of m o r t a l i t y 11A d i s t r i b u t i o n s of dead and l i v e t r e e parameters. In t h i s chapter, diameter d i s t r i b u t i o n s and s p a t i a l d i s -t r i b u t i o n s of m o r t a l i t y are analyzed i n some n a t u r a l and planted stands. The h y p othesis being t e s t e d i s that dead t r e e s should have the same d i s t r i b u t i o n as l i v e t r e e s . Thus, l i v e t r e e d i s t r i b u t i o n s a l s o have to be s t u d i e d . F i n a l l y , s i t e occupancy i s d i s c u s s e d i n the l i g h t of v a r i a t i o n s i n s t o c k i n g , stand den-s i t y and s p a t i a l p a t t e r n s . 5.1 DISTRIBUTION OF DEAD TREES 5.1.1 DIAMETER DISTRIBUTIONS In pure n a t u r a l stands observed over l o n g p e r i o d s of time (between the age of 46 and 85), a l l percentage diameter d i s t r i b u t i o n s of dead t r e e s t e s t e d (except one) correspond to a negative b i n o m i a l d i s t r i b u t i o n . T h e i r c o e f f i c i e n t of v a r i -a b i l i t y decreases with an i n c r e a s e i n average diameter up to 20.5 i n c h e s , and a s i g n i f i c a n t p r o p o r t i o n of the b i g g e s t dead t r e e s are l a r g e r than the average stand diameter. About 75 per-cent of them are l a r g e r than 5 i n c h e s up to age 75> and l a r g e r than 8 i n c h e s t h e r e a f t e r . The variance-mean r a t i o i n d i c a t e s a wide v a r i a t i o n around the mean (1.18 to 1.67) which can p a r t l y be explained by the 4-inch range i n average diameter ( F i g u r e 15 A,B). Percentage diameter d i s t r i b u t i o n s of Douglas f i r t r e e s a l s o f i t to the negative b i n o m i a l d i s t r i b u t i o n i n young mixed n a t u r a l stands; as stands grow o l d e r , however, d i s t r i b u t i o n s Figure 13 A- Percentage diameter distributions of dead frees related to 1 1 5 average stand diameter In pure natural stands-Group I* AVDBH-8-5" (6*6- IO-5) AVDBH«I2-5B (106- 14-5) Negative Binomial Chi2 « 7-6 df »I0 *5-5 w Var«-7-5 K»l-3 Varmean = 1*36 C- V-• 49% AVDBH«l6-5n (14^-18-5) Negative Binomial Chi2=43-4 df = 20 Mean •KM)* Var-s|3-l K»7-9 Varmean «1*31 C V- • 36 % Negative Binomial Chi 2 «15-3 df « 14 Mean 8 7 2 ° Var-°8-5 K«4-2 Varmean = 118 C-V- =40% AVDBH s20-5M (18-6-22-5) Negative Binomial Chi 2 =229 df « 21 Meon»l45° Var-824-l K*3-4 Varmean • 1-67 C- V** 34% Figure 13 B- Percentage diameter distributions of dead trees related "6 to stand age-In pure natural stands-Group I-Age 46-55 Negative Binomial C h i 2 * 13-6 d f * ! 3 Mean» 7-3" Vara 7-9 K*5I Varmean =108 C-V- = 38% Age 66-75 Negative Binomial Chi2 =8-7 df-18 Mean-* 9 0 " Var =174 K»3-2 Varmean = 193 C-V««45% Age 56-65 Negative Binomial C h i 2 « 7 0 d f * l 3 Mean*»6-9M Var*9-5 K*2-7 Varmean • I -38 C-V- • 45 % Age 76-85 „ Normal Chi2 =32 8 df = 24 Mean «14 5M Var =474 Varmean = 3-26 C-V-=48% D i s t r i b u t i o n of m o r t a l i t y 117 tend toward normal. T h e i r c o e f f i c i e n t of v a r i a t i o n decreases r a p i d l y as age and s i z e i n c r e a s e ( F i g u r e 13 C,D). Stu d i e s of the Wind R i v e r spacing p l a n t a t i o n show that, when c o r r e l a t e d e i t h e r with average stand diameter (range 2.6 to 10.5 i n c h e s ) , or stand age (26 to /+5 y e a r s ) , percentage diameter d i s t r i b u t i o n of dead t r e e s does not f i t any of f i v e standard p r o b a b i l i t y d i s t r i b u t i o n s t e s t e d , i . e . normal, bino-m i a l , Poisson, negative binomial and uniform. In tho four average diameter c l a s s e s t e s t e d , the p r o b a b i l i t y of g e t t i n g a l a r g e r Chi-square than the one obtained i n a t e s t of goodness of f i t was always l e s s than 0.05 (which r e j e c t e d the n u l l hypo-t h e s i s t h a t the data came from one of the aforementioned d i s -t r i b u t i o n s ) . Within the range of diameter and age analyzed, the c o e f f i c i e n t of v a r i a b i l i t y stayed at about 35 percent. However, the variance-mean r a t i o of the observed d i s t r i b u t i o n s i n c r e a s e s with an i n c r e a s e i n diameter and age. F i g u r e 13 (E,F) i l l u s t r a t e s the percentage m o r t a l i t y by diameter c l a s s f o r four average stand diameters, and two age c l a s s e s . Wide c l a s s e s of average diameter have been c a l c u l a t e d i n order to o b t a i n enough data to b u i l d a s i g n i f i c a n t d i s t r i b u t i o n . These r e s u l t s do not correspond with those of unpublished s t u d i e s on lodgepole pine (Pinus c o n t o r t a Dougl.), i n which Lee (1969) found that the percentage diameter d i s t r i b u t i o n of dead t r e e s was normally d i s t r i b u t e d , with mean at mean diameter minus 2 i n c h e s . Figure 13 C- Percentage diameter distributions of dead trees related to average stand diameter-In mixed natural stands - Groups H , 111,131 AVDBH=4-5 (2-6 - 6-5) AVDBH*8-5* (6-6-10-5) 4 Negative Binomial Negative Binomial C h i 2 * 0-3 d f *3 Chi 2 * 5-7 d f *7 Mean = 21" Var* 1-2 K = 9-3 Mean = 5-5" Var-*3-2 K*9-7 Varmean * 0-57 C V * 5 2 % Varmean * 0-58 C V - 3 3 % AVDBH*I2-5W (10-6-14-5) , Normal •'NS*Probability of C h i 2 * 11-3 df*4 NS1 a larger chi- Mean = 6-3" Var-= 2-6 K*l4-8 square is less Varmean *0-4l C V - « 2 5 % than «05-Figure 13 D- Percentage diameter distributions of dead trees related to 1 1 9 stand age-In mixed natural stands - Groups H,m,EC-Age 25 (16-35) Age 45 (36-55) Negative Binomial Poisson Chi 2 "1-8 d f » 5 Chi z »5-4 d f « 9 Mean• 2-1a V a r « 2 0 Ka|-3 Mean«5-7 B Var*3-7 K«0 0003 Varmean = 0 95 C-V««67% Varmean»0-65 O V - « 3 3 % Age 65 (56-75) '"NS* Probability of a larger chi-square is less than 05-Poisson Chi 2 »16-7 df»6 NS' Mean = 5-5 " Var» 1-9 K = 000l Varmean=0-35 CV » 2 5 % Figure 13 E- Percentage diameter distributions of dead trees related to average stand diameter In the Wind River spacing plantation* 120 AVDBH=3-5* (2-6 - 4-5) A V D B H s 5 - 5 B (4-6-6-5) Chi2 «150 Binomial df « 3 Meon « 2-8 Var-8 0-8 NS1 P«0-37 Varmean • 0-29 C V ° 3 2 % AVDBH8 7-5" (6-6-8-5) Poisson C h i 2 * 20-4 d f « 8 NS1 Mean" 3-7 a Var- • 1-4 Varmean « 0-38 C V = 32 % AV D B H 8 95" (8-6-10-5) Negative Binomial Chi2 =23-2 d f « 5 NS1 Mean* 5-2" Vor-«4-3 K«8*6 Varmean = 0-33 C V - ° 4 0 % Normal Chi 2 « 46-3 df « 4 NS1 Mean * 5-5" Var- 8 2-7 Varmean 8 Q - 4 9 C-V-829% NS« Probability of a larger chi-square is less than -05-Figure 13 F- Percentage diameter distributions of dead trees related to stand age* In the Wind River spacing plantation* 121 Age 26-35 Binomial Chi 2 *»24l d f « 3 NS1 Mean«2-8M Var « 0*9 P«0-36 Varmean-<0*32 C V » 3 5 % Age 36-45 , Poisson ' N S « Probability of Chi2 «16-1 d f » 8 NS1 a larger chi- Meon»3*6n V a r « 1-8 ??Uari-8 Varmean-0-50 C V « 3 7 % than 05-D i s t r i b u t i o n of m o r t a l i t y 122 5*1.2 SPATIAL DISTRIBUTIONS Three separate t e s t s v/ere performed to analyse s p a t i a l d i s t r i b u t i o n s of dead t r e e s . The f i r s t one i n c l u d e d 10 p l o t s from pure Douglas f i r stands (Group I, Table I I ) , i n which a l a r g e number of dead t r e e s were recorded, due to l a r g e p l o t s i z e s (one acre) and l o n g p e r i o d s of o b s e r v a t i o n (about 30 y e a r s ) . Program SPACE (Appendix I) was used to perform the t o s t on cumulative number of dead t r e e s at time of l a s t p l o t remeasurement. The second one was made i n the l ^ - y e a r - o l d U.B.C. spacing t r i a l (Table I I ) where t r e e s were pla n t e d at 3, 6, 9, 12, and 15 f e e t apart i n ci square l a t t i c e , on a high q u a l i t y s i t e (160 f e e t at 100 y e a r s ) . In 1969, a m o r t a l i t y survey was c a r r i e d out i n which every dead tr e e was l o c a t e d . Data were a l s o run through program SPACE to i n v e s t i g a t e s p a t i a l arrangement of m o r t a l i t y . The l a s t t e s t was performed i n the Wind River spacing t r i a l . In t hat case, the p o i n t - t o - p l a n t method ( P i e l o u , 1959) was used r a t h e r than the quadrat method to c h a r a c t e r i z e dead t r e e p a t t e r n s . The main reason f o r t h i s change i n technique was the l i m i t e d number of dead t r e e s present i n the l / A - a c r e p l o t s observed. Even with the d i s t a n c e method, the cumulative number of dead t r e e s s i n c e the f i r s t p l a n t a t i o n measurement (age 22) had to be considered i n order to o b t a i n enough t r e e s to form a d i s t r i b u t i o n ; one p l o t (No. 13), showing l i g h t mor-t a l i t y , s t i l l had to be l e f t out. D i s t r i b u t i o n of m o r t a l i t y 123 R e s u l t s presented i n Table XXV show that, i n n a t u r a l stands t r e e s dead d u r i n g p e r i o d s of 10 to 30 years were randomly d i s p e r s e d when sampled v/ith quadrats v a r y i n g i n s i z e from 6 to 62 m i l a c r e s . In most cases, quadra.t frequency d i s t r i b u t i o n s conformed to the negative binomial d i s t r i b u t i o n of p r o b a b i l i t y , and no b a s i c d i f f e r e n c e was detected between Chi-square values or variance-mean r a t i o s obtained with d i f f e r e n t quadrat s i z e s . A s i g n i f i c a n t degree of dumpiness of d i s p e r s i o n v/as detected only i n two p l o t s out of ten. Fourteen years a f t e r p l a n t a t i o n , dead t r e e s occur i n clumps only i n the c l o s e s t spacing t e s t e d at the U n i v e r s i t y Research F o r e s t (Table XXVI). V i s u a l i n t e r p r e t a t i o n confirms t h i s r e -s u l t . The r e l a t i v e l y small number of dead t r e e s (and small number of quadrats) i n wider spacing, however, d i d not f a c i l i -t a t e p a t t e r n i d e n t i f i c a t i o n ; and the t e s t of goodness of f i t , having l i m i t e d number of degrees of freedom, can not be c o n s i d -ered f u l l proof. Nevertheless, i n a l l four t e s t s performed i n each s p a c i n g v/ith v a r i o u s quadrat s i z e s (of which only one i s shown i n Table XXVI), there was a c o n s i s t e n t conformity e i t h e r to the negative b i n o m i a l or the uniform d i s t r i b u t i o n of pro-b a b i l i t i e s . Table XXVII summarizes the a n a l y s i s performed with the Wind R i v e r p l a n t a t i o n data. The p o i n t - t o - p l a n t method ( d i s -tances between random p o i n t s and dead t r e e s ) i n d i c a t e s that some dumpiness or aggregation of s p a t i a l d i s t r i b u t i o n i s detected i n almost a l l spacings.. However, i n each spacing, D i s t r i b u t i o n of m o r t a l i t y 1 2 A TABLE XXV SPATIAL DISTRIBUTION OF DEAD TREES IN NATURAL STANDS1 GROUP I Dead Av. Quad- Num-t r e e s No. r a t ber per t r e e s s i z e of Var x— Age acre per ( n i l - quad- ance-PLOTS (years) (2) quadrat acre) r a t s mean F i t to RAINIER 1 ••• 86 1 11.6 81 1.07 Neg.binomial 50-81 Chi: 2.5 df 2 2 23.2 36 1.25 Neg.binomial C h i : 3.1 df 4 3 35.7 25 1.05 Neg.binomial C h i : 5.2 df 5 4 47.6 16 0.93 Neg.binomial C h i : 2.7 df 6 RAINIER 2 132 1 7.6 121 1.03 Poisson 50-80 Ch i : 6.4 df 4 2 15.2 64 1.31 Neg. bi n o m i a l C h i : 8.1 df 6 3 22.7 36 0.87 Poisson C h i : 2.4 df 6 4 30.3 25 1.26 Neg.binomial C h i : 9.7 df 8 RAINIER 4 - 86 1 11.5 54 1.01 Poisson 56-86 C h i : 1.7 df 2 2 23.4 24 0.75 Poisson C h i : 4.9 df 4 3 35.7 15 0.61 Poisson C h i : 2.0 df 3 4 46.9 12 1.09 Uniform C h i : 2.0 df 5 RAINIER 5 49 1 19.2 49 0.66 Binomial 53-83 Ch i : 0.2 df 1 2 47.6 16 1.18 Uniform D i s t r i b u t i o n of m o r t a l i t y 125 TABLE XXV (cont • •d) Dead Av. Quad- Num-t r e e s no. r a t ber per t r e e s s i z e of Vari--Age acre per ( m i l - quad- ance-PLOTS (years) (2) quadrat acre) r a t s mean F i t to RAINIER 7 84 1 11.9 81 1.22 Neg.binomial 52-82 C h i : 4.9 df 3 2 23.8 36 1.73* * Ne g. b i n om i a l C hi: 8.7 df 4 3 35.7 25 1.93**Uniform C h i : 7.1 df 7 4 47.6 16 1.70 Neg.binomial C h i : 6.8 df 8 RAINIER 8 79 1 12.6 64 1.18 Neg.binomial 56-62 Ch i : 0.7 df 2 2 25.6 36 1.25 Neg.binomial C h i : 1.0 df 4 3 38.5 25 0.97 Poisson C h i : 4.9 df 4 4 52.6 16 1.34 Neg.binomial C h i : 2.3 df 6 RAINIER 9 156 1 6.4 144 1.49* *Neg.binomial 58-88 Ch i : 6.2 df 2NS 2 12.3 64 1.92* *Neg.binomi a l C h i : 6.9 df 6 3 19.2 49 2.02* *Neg.binomial C h i : 4.7 df 7 4 25.6 36 2.24**Neg.binomial C h i : 9.4 df 10 OLYMPIC 1 156 1 6.4 144 0.92 Poisson 51-32 Chi: 2.1 df 4 2 12.8 64 1.27 Neg.binomial C h i : 5.2 df 4 3 19.2 49 1.33 Neg.binomial C h i : 7.6 df 6 4 25.6 36 1.13 Neg.binomial D i s t r i b u t i o n of m o r t a l i t y 1 2 6 TABLE XXV (cont'd) Dead Av. Quad- Num-t r e e s no. r a t ber per t r e e s s i z e of • V a r i -Age • acre per ( m i l - quad- ance-PLOTS (years) (2) quadrat acre) r a t s mean F i t to Olympic 2 146 1 6.8 144 1.07 Neg.binomial 51-•8l C h i : 2.5 df 3 2 13.7 64 l . l l Neg.binomial C h i : 5.4 df 4 3 20.8 36 O.89 Poisson C h i : 7.4 df 6 L 27.8 36 1.14 Neg.binomial C h i : 5.2 df 6 SIUSLAW 9 67 1 14.9 64 1.22 Neg.binomial 67-•92 C h i : 1.9 df 2 2 3 C 3 25 1.09 Neg.binomial Chi : 3.5 df 5 3 45.4 16 1.21 Neg.binomial Chi : 10.4 df 1 4 62.5 16 1 . 6 0 Neg,binomial Chi : 8.8 df 8 l A s determined with program SPACE (Appendix I ) . 2 T o t a l number of t r e e s dead d u r i n g the p e r i o d s p e c i f i e d . 3 ( * * ) above the 95% upper confidence l i m i t of the t h e o r e t i c a l variance-mean r a t i o c o n s i s t e n t with randomness; i n d i c a t i v e of aggregated p a t t e r n . Otherwise, w i t h i n the l i m i t s . D i s t r i b u t i o n of m o r t a l i t y 127 TABLE XXVI SPATIAL DISTRIBUTION OF DEAD TREES 14 YEARS AFTER PLANTATION IN A SQUARE LATTICE 1 (U.B.C. SPACING TRIAL) Spacing Quadrat Number T r i a l Spacing S i z e of 7 No. ( f e e t ) ( milacre)Quadrats Varmean^ F i t to 15 3x3 3.09 154 1.92* Neg.pbinomial (Chi =5.4, df=12) 17 6x6 49.00 6 2.24 Uniform (Chi2=4 . o , df=8) 14 9x9 83.68 4 1.57 Uniform ( C h i 2 r , 6 . 5 , df=5) 16 12x12 79.34 4 1.88 Uniform 18 (Chi 2=5.0 3 df=4) 15x15 41.30 12 1.64 Neg.pbinomial (Chi =3.3, cif=2) In c l u d e s only one of four t e s t s performed i n each spacing with a d i f f e r e n t number of quadrats. Quadrat s i z e determined by a s p e c i f i e d average number of t r e e s expected per quadrat, and the t o t a l number of dead t r e e s i n the p l o t s t u d i e d . Variance-mean r a t i o of the observed d i s t r i b u t i o n . (*): Above the 95% upper confidence l i m i t of the v a r i a n c e -mean r a t i o c o n s i s t e n t with randomness; i n d i c a t i v e of aggre-gated p a t t e r n . • C a l c u l a t e d as 1 + t \ /~""2/N-l; where N i s the number of quadrats, and t i s from Student's t a b l e s f o r p = .05. D i s t r i b u t i o n of m o r t a l i t y TABLE XXVII INDEX OF NON-RANDOMNESS IN POPULATION OF DEAD TREES (WIND RIVER SPACING PLANTATION) 128 I n i t i a l Time Since P l o t Spacing P l a n t a t i o n No. ( f e e t ) (years) Alpha Confidence-^ P a t t e r n 4 x 4 33 37 42 1.830 1.477 1.333 0.74-1.30 0.74-1.30 0.74-1.30 Aggregated Aggregated Aggregated 4 x 4 33 37 42 1.233 1.227 0.985 0.74-1.30 0.74-1.30 0.74-1.30 Random Random Random 4 x 4 33 37 42 46 1.012 1.1.63 1.086 1.219 0.74-1.30 0.74-1.30 0.74-1.30 0.74-1.30 Random Random Random Random 4 5 x 5 37 42 1.171 1.391 0.74-1.30 0.74-1.30 Random Aggregated 5 x 5 33 37 42 1.192 1.492 1.787 0.74-1.30 0.74-1.30 0.74-1.30 Random Aggregated Aggregated 5 x 5 33 37 42 O.903 1.303 1.537 0.74-1.30 0.74-1.30 0.74-1.30 Random Aggregated Aggregated 7 6 x 6 33 42 46 2.783 1.135 0.840 0.74-1.30 O.74-I.30 0.74-1.30 Aggregated Random Random P l o t No. I n i t i a l Spacing ( f e e t ) D i s t r i b u t i o n of m o r t a l i t y TABLE XXVII (cont'd) Time Since P l a n t a t i o n ^ (years) Alpha Confidence^ 129 P a t t e r n 8 6 x 6 42 1.762 0.74-1.30 Aggregated 10 8 x 8 42 1.278 0.67-1.39 Random 11 8 x 8 42 1.433 0.67-1.39 Aggregated 17 8 x 8 28-46 1.233 0.67-1.39 Random 14 10 x .10 46 1.700 O.6I-I.48 Aggregated 15 10 x 10 42 1.516 0.61-1.48 Aggregated Cumulative number of dead t r e e s s i n c e f i r s t measurement (age 22) up to 1 i n d i c a t e d age, unless s p e c i f i e d otherwise. From: P i e l o u (1959); d=I"I DW, where D i s the number of t r e e s per square f e e t of area, and W i s the sum of squares of p o i n t - t o - p l a n t d i s t a n c e s d i v i d e d by the number of d i s t a n c e s measured. 95% confidence i n t e r v a l s for<sK i n a random p o p u l a t i o n . D i s t r i b u t i o n of m o r t a l i t y 130 every p l o t does not i n d i c a t e the same p a t t e r n . In the smaller spacing (A x 4 f e e t ) , one sample shows a high degree of aggre-g a t i o n i n m o r t a l i t y p a t t e r n s , whereas the other two r e f l e c t randomness. In the 5 x 5-f°°t spacing, m o r t a l i t y was randomly d i s t r i b u t e d at an e a r l y stage of development; with time, dum-piness showed up i n a l l three p l o t s , p o s s i b l y i n d i c a t i n g con-t a g i o n . At wider spacings, there i s no trend, f o r m o r t a l i t y d i s t r i b u t i o n was evaluated only once. A b e t t e r i n t e r p r e t a t i o n of these r e s u l t s can be made i n the l i g h t of h i s t o r i c a l i n f o r m a t i o n , which has been analyzed by Reukema (1969). I t appears that i r r e g u l a r m o r t a l i t y (as de f i n e d i n Chapter 2) played a r o l e i n the development of t h i s p l a n t a t i o n . Root r o t and snow breakages have v i s i b l y a f f e c t e d r e g u l a r m o r t a l i t y d i s p e r s i o n i n the densest stands; while not being c l e a r l y d i s c e r n i b l e , these f a c t o r s may a l s o have i n -fluen c e d to a l e s s e r extent normal p a t t e r n s at wider spacings. F o s t e r and Johnson (1963a) made an e x c e l l e n t mathematical a n a l y s i s to demonstrate that r o o t - r o t a f f e c t e d t r e e s had con-t a g i o u s d i s t r i b u t i o n s i n Douglas f i r p l a n t a t i o n s . That snow breakages behave i n a s i m i l a r manner does not need demonstra-t i o n . Smith et a l . (1961, 1965) have shown by s i m u l a t i o n s t u d i e s that up to 70 percent of the t r e e s present at time of stand establishment can be l o s t through m o r t a l i t y without a f f e c t i n g y i e l d a t harvest, provided m o r t a l i t y d i s t r i b u t i o n , i s f u l l y sys-tematic. I f i t were f u l l y clumped, the l o s s would be d i r e c t l y D i s t r i b u t i o n of m o r t a l i t y 131 p r o p o r t i o n a l to the area a f f e c t e d . T h i s study i n d i c a t e s that, most of the time, m o r t a l i t y d i s t r i b u t i o n i s somewhere between the two extremes j u s t mentioned. Trends i n d i s p e r s i o n of m o r t a l i t y cannot be e a s i l y detec-ted w i t h i n short p e r i o d s of time. For i n s t a n c e , i n s i x 1/10-acre p l o t s randomly l o c a t e d through the Robertson R i v e r p l a n -t a t i o n (Table I I ) , only one t r e e out of 376 died between the age of 20 and 25 years, thus p r e v e n t i n g any a n a l y s i s of mor-t a l i t y p a t t e r n . 5.2 DISTRIBUTION OF LIVING TREES 5.2.1 DIAMETER DISTRIBUTIONS In pure n a t u r a l stands (Group I ) , 6l p l o t measure-ments were analyzed. The Voight Creek p l o t s (Table I I ) , con-t a i n i n g t r e e s l a r g e r than 5 i n c h e s only, were t r e a t e d separ-a t e l y . Almost a l l of the Voight Creek p l o t s t e s t e d (9 p l o t s , 26 remeasurements) had skewed d i s t r i b u t i o n s of diameters. In the other group.of p l o t s (12 p l o t s , 35 remeasurements), .normal diameter d i s t r i b u t i o n s were i d e n t i f i e d i n 1L remeasurements, negative b i n o m i a l d i s t r i b u t i o n s i n 13 remeasurements, and no s i g n i f i c a n t f i t was detected i n 8 remeasurements. Negative binomial diameter d i s t r i b u t i o n s were found i n the densest stands. In most cases, the nature of the diameter d i s t r i b u -t i o n of each p l o t d i d not change ever p e r i o d s of 30 years of o b s e r v a t i o n , i . e . a negative binomial d i s t r i b u t i o n remained D i s t r i b u t i o n of m o r t a l i t y 1^2 negative b i n o b i a l d u r i n g 30 years. In Groups I I , I I I , and IV (mixed stands i n which only Douglas f i r t r e e s were taken i n t o account), 27 negative bino-m i a l or uniform diameter d i s t r i b u t i o n s were observed among 27 p l o t remeasuroments s t u d i e d . In the Wind R i v e r spacing p l a n t a t i o n (14 p l o t s ) , 64 diame-t e r d i s t r i b u t i o n s were analyzed i n stands remeasured between 22 and 46 years of age. I t appears t h a t : 1) i n each p l o t , . t h o nature of the diameter d i s t r i b u t i o n d i d not change over obser-v a t i o n p e r i o d s averaging more than 20 years; 2) Poisson and normal diameter d i s t r i b u t i o n s were i d e n t i f i e d i n 33 cases (out of 6k) mainly i n the wider spacings ( 8 x 8 and 10 x 10 f e e t ) , whereas negative binomial diameter d i s t r i b u t i o n s were observed i n 23 cases, mostly i n k x k~> 3 x 5-> and 6 x 6-foct spacings; i n 8 cases, there was no s i g n i f i c a n t f i t to any of the f i v e d i s t r i b u t i o n s t e s t e d by the program SPACE (Poisson, normal, binomial, negative binomial and uniform). Inasmuch as variance-mean r a t i o s can c h a r a c t e r i z e tho p o p u l a t i o n s s t u d i e d , i t i s of i n t e r e s t to note that i n pure f i r stands, the average variance-mean r a t i o of diameter d i s -t r i b u t i o n s was 1.24 (0.39 - 2.60); i t was 1.73 (0.26 - 2.74) i n mixed stands, and 0.47 (0.29 - 0.85) i n p l a n t a t i o n . T h i s r e l a t i v e measurement of d i s p e r s i o n (Blackman, 1942) was p o s i -t i v e l y c o r r e l a t e d w i t h stand age and average stand diameter, and n e g a t i v e l y c o r r e l a t e d with number of t r e e s per aero. In pure stands, the average diameter of a l l p l o t s was D i s t r i b u t i o n of m o r t a l i t y ]_^ 3 11.6 i n c h e s (3.3 - 22.0) and i n p l a n t a t i o n s 5*5 in c h e s (3.3 -10.2). R a t i o s of s m a l l e s t t r e e to average were 0.35 (0.18 -0.64) i n n a t u r a l stands, and 0.33 (0.19 - 0.50) i n p l a n t a t i o n s . R a t i o s of l a r g e s t t r e e to average were 1.95 (1.41 - 2.74) i n pure n a t u r a l stands, and 1.75 (1.44 ~ 2.34) i n p l a n t a t i o n s . R a t i o s of l a r g e s t to average correspond with p r e v i o u s observa-t i o n s by Smith et a l . (1961), but r a t i o s of sm a l l e s t t r e e to average seem lower than the 0.50 r e p o r t e d by the same authors. 5.2.2 HEIGHT DISTRIBUTIONS Height i s a tr e e parameter c l o s e l y c o r r e l a t e d with diameter at bre a s t h e i g h t . Therefore, height d i s t r i b u t i o n s should be c l o s e l y a s s o c i a t e d with diameter d i s t r i b u t i o n s . T h i s hypothesis can be accepted i n the l i g h t of the data s t u d i e d here. I t has been found t h a t , most of the time, a normal d i s t r i b u t i o n of h e i g h t s corresponds to a normal d i s t r i -b u t i o n of diameters.. However, due p o s s i b l y to the p a r a b o l i c nature of the height-diameter r e l a t i o n s h i p and to some other f a c t o r s a f f e c t i n g stand composition and height growth of i n d i -v i d u a l t r e e s , height was found to e x h i b i t a much l a r g e r number of i r r e g u l a r d i s t r i b u t i o n s ( h i g h l y skewed, with s e v e r a l modes, etc.) that d i d not match any of the f i v e d i s t r i b u t i o n s t e s t e d by the program SPACE. T h i s g r e a t e r v a r i a t i o n i n d i s t r i b u t i o n s i s r e f l e c t e d i n the f o l l o w i n g variance-mean r a t i o s of height d i s t r i b u t i o n s : D i s t r i b u t i o n of m o r t a l i t y 134 Min. Max. Mean St.dev. C V . N Pure f i r 1.00 8.30 3.05 1.70 56 61 Mixed f i r 0.90 7.50 4.69 1.65 35 27 P l a n t a t i o n 1.50 4.30 2.63 0.59 22 64 Not s u r p r i s i n g l y , t h i s parameter was p o s i t i v e l y c o r r e l a t e d with stand age, height, diameter, variance-mean of the diameter d i s t r i b u t i o n , and n e g a t i v e l y c o r r e l a t e d with number of t r e e s per a c r e . 5.2.3 SPATIAL DISTRIBUTIONS S p a t i a l p a t t e r n analyses were performed with sample p l o t s f o r which r e c t a n g u l a r t r e e c o o r d i n a t e s were a v a i l a b l e . The p a t t e r n was evaluated only a t p e r i o d s corresponding to a p l o t remeasurement made at l e a s t 4 or 5 years l a t e r than the preceeding one. Thus, between one and f i v e p a t t e r n e v a l u a t i o n s were made i n each p l o t , with program SPACE (Appendix I ) . The i n i t i a l average expected number of t r e e s per quadrat was a r b i t r a r i l y sot at two, and i n c r e a s e d up to f i v e i n each run. Therefore, the goodness of f i t of the observed d i s t r i b u -t i o n s of t r e e s per square to four standard p r o b a b i l i t y d i s t r i -b u t i o n s was obtained by four Chi-square t e s t s performed i n each run. The mean and v a r i a n c e of the observed quadrat 'Two p l o t s had no t r e e c o o r d i n a t e s i n Group I, and 8 p l o t s i n Groups I I , I I I , IV. D i s t r i b u t i o n of m o r t a l i t y 135 frequency d i s t r i b u t i o n s were a l s o c a l c u l a t e d i n the program. Quadrat s i z e s ranged from 0.50 to 100.0 m i l a c r e s . The main purpose of t h i s study was not. to d e f i n e p r e c i s e l y tho nature of tho p a t t e r n s , but to d e t e c t trends i n d i s p e r s i o n with time. In F i g u r e s lk, 15, and IS, each dot r e p r e s e n t s the average of four variance-moan r a t i o s c a l c u l a t e d at time of each p l o t measurement.. Each l i n o j o i n s tho v a l u e s obtained a t s u c c e s s i v e measurements of the same p l o t (from l e f t to r i g h t ) , . The general trend of tho 22 curves i n F i g u r e li+ i s down-ward, and t h e i r c o n c e n t r a t i o n i s s l i g h t l y under the v a r i a n c e -mean r a t i o of 1.0, P l a i n l i n o s i n d i c a t e that, between age 55 and 85} the s p a t i a l d i s p e r s i o n of pure n a t u r a l stands of Douglas f i r d i d not change very much; and even without drawing the confidence l i m i t s , i t i s evident that there i s randomness. Dotted l i n o s show a more i r r e g u l a r arrangement f o r 39- to 56-y e a r - o l d stands; t h i s would i n d i c a t e that the s p a t i a l arrange-ment of l i v i n g t r e e s has not yet reached i t s r e l a t i v e s t a b i l i t y . A pronounced heterogeneous (clumpy) d i s p e r s i o n was d i s c e r n a b l e only i n 3 p l o t s out of 22 s t u d i e d . F i g u r e 15 i l l u s t r a t e s how Douglas f i r t r e e s wore d i s t r i -buted i n mixed stands. Except f o r two p l o t s , tho trend i s again downward i n most cases, but the r e l a t i o n s h i p between arrange-ment and age, i . e . more s t a b i l i t y with i n c r e a s i n g age, does not seem to hold any l o n g e r . I t i s b e l i e v e d that tho p a t t e r n of other s p e c i e s mixed with Douglas f i r d i d i n f l u e n c e the Figure 14- Trend in spatial pattern of living trees in Group I (pure natural stands)-Average Variance-Mean Ratio 2*6-2-4-2 2 -2 0 -1-8-1-6-1-4-1 2 -10 -0 - 8 -0-6 -0-4 -0 - 2 -ooi- JL Legend Each dot represents the average of 4 measurements* Each line represents one plot-Each dot on the same line (from left to right) indicates trend in pattern with time-Voight Creek Plots (39-56 years of age) Others (55-85 years of age) J. Oi 8 10 20 30 Average Quadrat Size in Milacres 40 Figure 15* Trend in spatial pattern of living trees in Groups H , H I , I 2 (mixed natural stands) Average Variance-Mean Re" lata 2-e|-2-6 -2*4-2-2 20 1-8 i-6 1-4 -12 1.0-76) 0*6 0 4 0 2 0.0 E(34)0 © S(4I) o •°v(55-66) '•(30-39) 0 8 | ^19-29) (64-71) ° (53=64)° Legend Distribution of Douglas-fir only GroupE ( 9 0 % ftr+) • » Group UK (75% fir +) e i »e- G r o u p ( 5 0 % fir -i-) ( ) Stand age See legend in Figure 14* L ± 8 ID 12 14 16 18 20 30 40 Average Quadrat Size in miiscrss Figure 16* Trend in spatial pattern of living trees in plantations* Average Variance-Mean Ratio l-2h Legend Wind River Spacing Trial • o 4x4 feet • • 5x5 feet o o 6x6 feet 0 - - - 0 8 x 8 feet x x 10 x 10 feet See legend in Figure 14* V / -o J L 4 5 6 7 8 Average Quadrat Size in Milacres D i s t r i b u t i o n of m o r t a l i t y 139 i n i t i a l p a t t e r n and subsequent development of those stands. Only a small number of p l o t s (3 out of 13) showed a t r u l y aggregated s p a t i a l d i s t r i b u t i o n of t r e e s . F i g u r e 16 was drawn wit h data gathered between age 22 and 46 i n the Wind R i v e r spacing p l a n t a t i o n . A l l 14 p l o t s show an upward trend i n variance-mean r a t i o , which s i g n i f i e s t h a t the s p a t i a l arrangement of l i v i n g t r e e s i s c u r r e n t l y going away from the u n i f o r m i t y sot at time of p l a n t i n g . The r a t e of t r a n s -formation would be f a s t e r i n c l o s e r spacings, as i n d i c a t e d by the slope of the curves. Moreover, i t appears that, at time of l a s t measurement, the p a t t e r n has not yet s e t t l e d . Data from Robertson R i v e r (Table I I ) were analyzed i n de-t a i l 14 years a f t e r p l a n t a t i o n by F o s t e r and Johnson (1963a). Ey v a r y i n g quadrat s i z e from 5 to 300 m i l a c r e s and quadrat shape from square to l o n g narrow s t r i p s , they obtained l a r g e v a r i a t i o n s i n t h e i r c o e f f i c i e n t of d i s p e r s i o n (variance-mean r a t i o ) . They concluded that t h i s c o e f f i c i e n t i n c r e a s e s with an i n c r e a s e i n quadrat s i z e , and that l o n g r e c t a n g u l a r quadrats c o n t r i b u t e to smaller i n t e r - q u a d r a t d i f f e r e n c e s . Quadrat f r e -quency d i s t r i b u t i o n s obtained with l a r g e quadrats ( c o n t a i n i n g up to 60 t r e e s ) d i d not resemble any known model. With quadrat shapes and s i z e s comparable to those that would have been a p p l i e d by program SPACE, they obtained variance-mean r a t i o s i n d i c a t i v e of i r r e g u l a r d i s p e r s i o n s , i . e . no r e g u l a r i t y as set at time of p l a n t i n g , 14 years e a r l i e r . D i s t r i b u t i o n of m o r t a l i t y ]_^Q 5.3 RELATIONSHIPS BETWEEN DISTRIBUTIONS OF DEAD TREES AND LIVING TREES I t appears from the a n a l y s i s j u s t completed that the hypo-t h e s i s of dead t r e e s being d i s t r i b u t e d l i k e l i v i n g t r e e s can be accepted, with r e s e r v a t i o n s . In b r i e f , data showed t h a t : 1 . M o r t a l i t y would be d i s p e r s e d randomly i n stands showing a random s p a t i a l arrangement, un l e s s some i r r e g u l a r causes are i d e n t i f i e d . When i t so happens, c h a r a c t e r i s t i c s of the c a u s a l agent would c o n t r o l the d i s t r i b u t i o n . Species other than f i r i n mixed stands would play the r o l e of these c a u s a l agents. Clumpy stands have a clumpy d i s t r i b u t i o n of r e g u l a r m o r t a l i t y . 2. At low stand d e n s i t y , diameter and height d i s t r i b u t i o n s of l i v e t r e e s f i t t e d to normal or Poisson s e r i e s ; otherwise, they were best d e s c r i b e d by a negative b i n o m i a l or other forms not t e s t e d ( p o s s i b l y the Neyman Type A s e r i e s with s e v e r a l modes f o r h e i g h t s , and some others f o r diameters (Meyer, 1930, F o s t e r and Johnson, 1963a, B l i s s and Reinkor, 1964, Leak*, 1965)). Almost a l l diameter d i s t r i b u t i o n s of dead t r e e s (ex-pressed i n percentage) had skewed p r o b a b i l i t y d i s t r i b u t i o n s . F o l l o w i n g t h i s review of d i s p e r s i o n s and frequency d i s t r i -b u t i o n s , second-degree l i n e a r r e g r e s s i o n models were computed to r e l a t e average m o r t a l i t y diameter and stand diameter. They are i l l u s t r a t e d i n F i g u r e s 17 and 18. In pure n a t u r a l stands, t h i s study i n d i c a t e s that the r a t e ,6t~ metes* F , 9 u r e 17- Mortality DBH related to stand DBH in natural stands - 1 1) MDBH*0-5940 SOBH +0*0058 SDBH2 ( N » i 4 3 , r 2 * » 8 l , S E E '1-45 (19%)) 2) MDBH = 0-5912 SOBH + 00065 SOBH2 (N«9l,r 2 -**83,SE E *H*60 (19%)) 3) MDBH• 0*8557S DBH + 0*0203 SDBH2 (N»52,r 2 »*69 % SE E »0*99 (16%)) 14-12 10-Stand characteristics * Min- Max Mean St* Dev CV-(1) MDBH 1-2 22*3 7*5 3*4 45 SDBH 20 265 11*2 4*2 37 (2) MDBH 14 22*3 8 4 3*7 45 SDBH 32 26-5 12-2 4-4 36 (3) MDBH 1-2 101 60 18 29 SDBH 20 16*6 94 29 31 e (3) GroupsII,HI,BE (2) Group I ** (I) Groups i,E,m,iz • ' Every natural stand described in Table 2 has been taken into account,and every period during which some mortality was reported'Symbols are identified in Appendix EE* ± SDBH, inches I I 22 24 Figure 18* Mortality DBH related to stand DBH in plantations*1 MDBH, inches 6 -.o Wind River Spacing Trial MDBH»0-8385 SDBH-00218 SOBH2 N * 4 4 f r 2 « . 7 3 , S E E a - 5 6 (15%) Stand characteristics' Min- Max* Mean St-Dev- C V-MDBH 2 22 6-97 3 76 103 28 SDBH 3-30 9-70 5-27 1-54 29 1 Every plot described in Table 2 has been taken into account,and every period during which some mortality was reported* Symbols are identified in Appendix HZ-i - i ' i ' I J — . 4 5 SDBH (inches 6 8 D i s t r i b u t i o n of m o r t a l i t y 1^3 of i n c r e a s e i n m o r t a l i t y diameter (MDBH) i s about 1/2 the r a t e of i n c r e a s e of the stand diameter (SDBH). Thus, when SDBH equals U inches, MDBH i s 3 inches; and f o r every two-inch i n -crease i n SDBH (up to 26 i n c h e s ) , the d i f f e r e n c e between SDBH and MDBH i n c r e a s e s by one i n c h . The square term, making a s i g n i f i c a n t but small c o n t r i b u t i o n to the r e g r e s s i o n , does not have much e f f e c t on the shape of the curve (Figure 17). In mixed stands of Douglas f i r and other c o n i f e r s , the r a t e of i n c r e a s e i n MDBH i s f a s t e r than f o r pure stands i n the lower diameter c l a s s e s , but decreases r a p i d l y when stand diame-t e r goes over 10 i n c h e s . The same trend has been measured by Hoyor y f o r young stands of western hemlock up to 18 inc h e s i n DBH. His r e g r e s s i o n was: MDBH = -3.1009 + 0.5366 SDBH + 0.0206 SDBH 2 N = 3 0 R 2 = 0.23 T h i s equation f o r hemlock f o l l o w s the mixed f i r curve q u i t e c l o s e l y ( F i g u r e 17). • In young f i r p l a n t a t i o n s (Figure 18), m o r t a l i t y diameter a l s o f o l l o w s e x a c t l y the same trend as i n the mixed n a t u r a l stands. However, the l a c k of d i v e r s i t y i n p l a n t a t i o n data pre-vents one from d e r i v i n g any c o n c l u s i o n about t h i s s i m i l a r i t y . Nevertheless, the Wind R i v e r Spacing T r i a l i l l u s t r a t e s the 'Personal communication, 1969. Hoyer, G.C. Fo r e s t Land Manage-ment Center, Dept. of Nat. Res., State of Washington. D i s t r i b u t i o n of m o r t a l i t y IL^U m o r t a l i t y process over a wide range of stand d e n s i t y on poor to medium s i t e , which could w e l l be r e p r e s e n t a t i v e of a l a r g e r number of f i r p l a n t a t i o n s . Observation of F i g u r e s 17 and 18 i n d i c a t e s that, even i f n o r t a l i t y occurs mainly on smaller t r e e s , t h e i r average s i z e becomes s i g n i f i c a n t l y l a r g e as stand diameter i n c r e a s e s and t h e i r volume comes to represent an i n c r e a s i n g p r o p o r t i o n of the stand volume, as shown by S t a c b l e r (1955^) • .Stacbler r e l a t e d the average volume of f i r t r e e s that d i e (Y) to the average volume of l i v e t r e e s (X) i n the f o l l o w i n g two equations: (1) f o r X l e s s than 299 board f e e t (approx. 60 cubic f e e t ) , Y = O.li+8 X (2) f o r X g r e a t e r than 299 board f e e t (approx. 60 cubic f e e t ) , Y = -196 + 0.80/+ X T h i s l i n e a r r e l a t i o n s h i p i s q u i t e comparable to the one drawn i n Fi g u r e 17 except f o r the u n i t s of measurements; s i m i -l a r c o n c l u s i o n s can be der i v e d from both. 5.4 EFFECTS OF MORTALITY DISTRIBUTIONS ON SITE OCCUPANCY AND USE OF SITE CAPACITY S i t e occupancy can be measured simply by the p r o p o r t i o n of a f o r e s t e d area covered by the sum of crown p r o j e c t i o n a l areas. Expressed as a percentage, t h i s measurement determines what might be c a l l e d the l e v e l of s t o c k i n g . Within the area stocked, the i n t e n s i t y of s i t e occupancy or degree of crowding can be evaluated by measuring stand d e n s i t y . D e t a i l e d i n f o r m a t i o n D i s t r i b u t i o n of m o r t a l i t y . 145 concerning these measurements i s i n c l u d e d i n Appendix I I I . Since l i v e t r e e s are d i s t r i b u t e d on f o r e s t s i t e s more or l e s s l i k e dead t r e e s , a c t u a l l i v e t r e e p a t t e r n s and changes i n these p a t t e r n s ( c r e a t e d mainly by m o r t a l i t y and ingrowth) c h a r a c t e r i z e s i t e occupancy. Crown c l o s u r e i s one measure of o v e r a l l s i t e occupancy. The p r o p o r t i o n of empty quadrats i n a sample of contiguous quadrats i s i n d i c a t i v e of the d i s t r i b u -t i o n and s i z e of gaps a s s o c i a t e d with each p a t t e r n . And the p r o p o r t i o n of the p o t e n t i a l mean annual increment used by i n d i -v i d u a l t r e e s or groups of t r e e s of the same s i z e , r e v e a l s how much of the s i t e c a p a c i t y has been, and how much of i t could be u t i l i z e d depending on t r e e e f f i c i e n c y , and m o r t a l i t y . These measurements are d i s c u s s e d h e r e a f t e r . 5.4.1 ANALYSIS OF SITS OCCUPANCY INDICATORS In Tables I I I and V (Chapter 3), a summary of s t a -t i s t i c s from pure n a t u r a l stands and p l a n t a t i o n s was presented. These t a b l e s i n d i c a t e d t;h'at young-growth Douglas f i r stands were found to occupy f o r e s t s i t e s of d i f f e r e n t q u a l i t i e s i n many d i f f e r e n t ways. S p a t i a l p a t t e r n s v a r i e d between u n i f o r m i t y (or r e g u l a r i t y ) and severe dumpiness (VARMEAN : 0.43 to 2.48);. the percentage of area covered by crown p r o j e c t i o n s went from 20 to 135; a n d r e l a t i v e d e n s i t y f l u c t u a t e d between a minimum of 0.26 to a maximum of 1.37. In the spacing p l a n t a t i o n , a somewhat wider v a r i a t i o n i n degrees of s i t e occupancy was a r -t i f i c i a l l y b u i l t i n , by spacing; i n t e n s i t y of s i t e occupancy D i s t r i b u t i o n of m o r t a l i t y 1^5 f l u c t u a t e d a c c o r d i n g l y . R e s u l t s of c o r r e l a t i o n analyses performed on measurements of s i t e occupancy are summarized i n Tables XXVIII and XXIX f o r n a t u r a l stands and. p l a n t a t i o n s , r e s p e c t i v e l y . They show that i n t e n s i t y of s i t e occupancy i n c r e a s e s as crown c l o s u r e c l o s e s , or as a l a r g e r p r o p o r t i o n of the crown space i s occupied. However, t r e e s occupy the s i t e d i f f e r e n t l y as s t o c k i n g and d e n s i t y i n c r e a s e : i n n a t u r a l stands, t r e e d i s -p e r s i o n showed a trend towards more u n i f o r m i t y , whereas i n p l a n t a t i o n the i n v e r s e s i t u a t i o n was observed. T h i s d i v e r -gence, i n d i c a t e d by the variance-mean r a t i o of the s p a t i a l d i s -t r i b u t i o n ( F i g u r e s XIV and XVI), can be explained by the f a c t that planted t r e e s had, at time of p l a n t i n g , a p e r f e c t l y sys-tematic d i s t r i b u t i o n , whereas n a t u r a l t r e e s o r i g i n a t e d i n a somewhat more heterogeneous p a t t e r n , from which they both evolved towards random d i s p e r s i o n as d e n s i t y b u i l t up with time. T h i s e x p l a i n s why variance-mean of the p a t t e r n and den-s i t y are n e g a t i v e l y c o r r e l a t e d i n n a t u r a l stands, and p o s i t i v e l y c o r r e l a t e d i n p l a n t a t i o n s . In n a t u r a l stands, i n t e n s i t y of s i t e occupancy ( d e n s i t y ) was p o s i t i v e l y c o r r e l a t e d with s i t e q u a l i t y , but fewer and l a r g e r t r e e s occupied the area as age i n c r e a s e d . In spacing p l a n t a t i o n , t r e e s p l a n t e d at wider spacings grew f a s t e r and i n d i c a t e d , when measured f o r height at a given age (average of dominants and codominants), a higher s i t e q u a l i t y than t r e e s planted at c l o s e r spacings. Thus, a negative c o r r e l a t i o n was TABLE X X V I I I RELATIONSHIPS BETWEEN SITE OCCUPANCY MEASUREMENTS IN PURE NATURAL STANDS TIME SITE DENSITY STOCKING PATTERN SIZE AGE SI BNT BAN BANORM CCF RDN BASTOC CCCLO CSO VARMEAN DBH SIMPLE CORRELATION COEFFICIENT SIZE DBH .83 .70 -.74 .74 * .28 .74 .89 * PATTERN VARMEAN * * * -.43 -.55 -.50 -.43 * -.50 -.43 STOCKING CSO .38 .40 .61 .55 .40 * .98 1.00 CCCLO * «• .39 .37 .61 .54 .37 * 1.00 DENSITY BASTOC .94 .49 -.73 .74 * .31 .74 1.00 RDN .72 .64 -.27 .99 .73 .84 1.00 CCF .34 .39 .27 .84 .96 1.00 BANORM * .39 .40 .73 1.00 BAN .72 .64 -.27 1.00 BNT -.65 -.35 1.00 SITE QUALITY SI .31 1.00 TIME AGE 1.00 Not s i g n i f i c a n t a t 0.05 l e v e l w i t h 121 o b s e r v a t i o n s i n Group I . V a r i a b l e s are i d e n t i f i e d i n Table I I I and Appendix IV.--P--N3 TABLE XXIX RELATIONSHIPS BETWEEN SITE OCCUPANCY MEASUREMENTS IN PLANTATIONS TIME SITE DENSITY STOCKING PATTERN SIZE AGE SI BNT BAN BANORM CCF RDN BASTOC CCCLO CSO VARMEAN DBH SIMPLE CORRELATION COEFFICIENT SIZE DBH PATTERN VARMEAN STOCKING CSO CCCLO DENSITY BASTOC . RDN CCF BANORM BAN BNT SITE QUALITY SI TIME AGE .50 .45 -.74 .31 -.55 .62 .43 -.68 .74 .47 -.68 .73 .77 * -.51 .79 -.48 .33 .40 -.74 .88 .48 -.71 .76 .80 -.49 .36 * -.68 1.00 -.47 1.00 1.00 .30 * -.38 .32 .46 .58 .65 .45 .81 .96 .92 .79 .83 .97 .93 .81 .51 * * .51 .98 .82 .731.00 .76 .96 1.00 .85 1.00 1.00 .86 * * .57 * .98 * 1.00 1.00 * -.36 l.oo .60 1.00 1.00 Not s i g n i f i c a n t at 0.05 l e v e l with 64 observations i n the Wind River Spacing t r i a l . Variables are i d e n t i f i e d i n Table III and Appendix IV. t—1 -p-Co D i s t r i b u t i o n of m o r t a l i t y ]_^< observed between d e n s i t y and s i t e index, even i f , w i t h i n each spacing, the process was e s s e n t i a l l y s i m i l a r to the one j u s t d e s c r i b e d f o r n a t u r a l stands. 5.4.2 ANALYSIS OF EMPTY QUADRATS In t h i s d i s s e r t a t i o n , s i t e occupancy was evaluated by a crown summation procedure or "cramming method (Pope, I960)," which could not i n d i c a t e how the t r e e s were d i s t r i b u t e d over the area (Greig-Smith's rooted frequency). Some of t h i s i n f o r m a t i o n was computed by the program SPACE (Appendix I ) . With t h i s program, each p o p u l a t i o n was sampled with a g r i d of quadrats, the s i z e of which was determined by the sample p l o t s i z e , and an expected average number of stems per quadrat. By u s i n g t h i s procedure, i t i s p o s s i b l e to look up i n a t a b l e of Poisson p r o b a b i l i t i e s what i s the maximum t h e o r e t i c a l pro-p o r t i o n of empty quadrats f o r the Poisson e x p e c t a t i o n to apply. T h i s maximum i s l i s t e d here, under the mean numbers of t r e e s used i n program SPACE: Expected mean number of t r e e s per quadrat 1 2 3 4 5 Maximum percentage of empty quadrats c o n s i s t e n t with randomness 37 13 5 ?- 1 Apart from p r o v i d i n g a way of a s s e s s i n g randomness of t r e e s i n the f i e l d , t h i s method s e t s some l i m i t s as to the maximum s i z e of openings i n the canopy or the maximum sur f a c e without t r e e s c o n s i s t e n t with random d i s p e r s i o n . Some areas D i s t r i b u t i o n of m o r t a l i t y 150 are l i s t e d f o r r e f e r e n c e i n Table XXX. The t a b l e can be read as f o l l o w s : f o r example, an acre s u p p o r t i n g 1,000 t r e e s .should not present more than 5 empty spaces l a r g e r than U m i l a c r e out of 250 ( s u b j e c t to other c o n d i t i o n s enumerated at the bottom of the t a b l e ) i f t r e e s are to be considered randomly d i s p e r s e d . L i v e and dead t r e e s p a t i a l d i s t r i b u t i o n s analyzed i n here conformed to the r u l e , i . e . when sampled with four s u c c e s s i v e mean number of t r e e s per quadrat, the p r o p o r t i o n s of empty quadrats, were above the maximum p e r m i s s i b l e where the s p a t i a l p a t t e r n was i d e n t i f i e d as clumpy, and roughly equal or i n f e r i o r when i d e n t i f i e d as random or r e g u l a r . In most cases, except i n a few clumpy p o p u l a t i o n s , p r o p o r t i o n s were s l i g h t l y s u p e r i o r to 0.13, p e r m i s s i b l e with samples of 2 t r e e s per square, and i n f e r i o r to .05, .02, or ,01 allowed with 3j 4> a n d 5 t r e e s per square. Thus, i n most p l o t s , not more than 20 percent of the area was f r e e of t r e e s , when broken down as i n d i c a t e d i n Table XXX. 5.4.3 USE OF SITE CAPACITY The f o l l o w i n g statements r e l a t i v e to the use of s i t e c a p a c i t y are e x t r a c t e d from Osborn's g r a p h i c a l s y n t h e s i s of stand dynamics (Osborn, 1968b): 1. Optimum s i t e use occurs when the most e f f i c i e n t trec-s are present and stand d e n s i t y i s at a maximum with complete s t o c k i n g . 2. There i s a range over which stand d e n s i t y does not a f f e c t s i t e c a p a c i t y . D i s t r i b u t i o n of m o r t a l i t y 151 TABLE XXX SIZE OF OPENINGS IN QUADRAT SAMPLING CONSISTENT WITH RANDOMNESS1 AREA WITHOUT EXPECTED TREES OR SIZE TREES MEAN NO. QUADRAT PERMISSIBLE OF OPENING IN PER PER NO. OF SIZE NO. OF THE CANOPY ACRE QUADRAT QUADRATS (m i l a c r e ) OPENINGS (mi l a c r e ) 100 1 100 10 37 . 370 2 50 20 6.5 130 3 33 30 1.6 50 /+ 25 40 0.5 20 5 20 50 0.2 10 500 1 500 2 ' 185 370 2 250 32.5 130 3 166 6 8.3 50 k 125 8 2.5 20 5 100 10 1 10 1000 1 1000 1 370 370 2 500 2 65 130 3 333 16.7 50 k 250 h 5 20 5 200 5 0 t— 10 1Random means n e i t h e r r e g u l a r nor clumpy. I t a p p l i e s provided other c o n d i t i o n s of the Poisson s e r i e s are met, i . e . have p r o p o r t i o n of quadrats l e s s than or equal to the f o l l o w i n g , with the i n d i c a t e d number of t r e e s per quadrat: Number of Quadrat 5 k 3 2 1 0 Expected mean 1 .37 .37 2 .27 .27 .13 3 .23 .22 .15 .05 k .19 .19 .15 .07 .02 5 .17 .17 .14 .08 .03 .01 D i s t r i b u t i o n of m o r t a l i t y 152 5. Complete s t o c k i n g and minimum stand d e n s i t y should r e s u l t i n maximum growth r a t e . U. Stand y i e l d , on a s p e c i f i c s i t e at constant age, i n c r e a s e s l i n e a r l y with i n c r e a s e i n s t o c k i n g u n t i l crowns touch, then c u r v i l i n e a r l y with i n c r e a s i n g stand d e n s i t y . Stand increment i n c r e a s e s with i n c r e a s e d s t o c k i n g u n t i l crown c l o s u r e i s complete. 5. I n d i v i d u a l t r e e growth r a t e i s constant u n t i l s t o c k i n g i s complete. With these o b s e r v a t i o n s i n mind, an attempt was made at es t i m a t i n g to which extent t r e e s from the sample p l o t s s t u d i e d d i d use s i t e c a p a c i t y . The mean annual increment ( i n cubic volume) was compiled by diameter c l a s s , f o r each p l o t measurement. A method s i m i l a r to the one proposed by Stage (1969) was then a p p l i e d to c a l c u -late, what p r o p o r t i o n of the s i t e c a p a c i t y was, on the average, used by t r e e s i n each, diameter c l a s s . I t was assumed that the maximum s i t e c a p a c i t y was r e f l e c t e d by the mean annual gross increment of normal stands ( S t a e b l e r , 1955 a)« The p r o p o r t i o n of t h i s maximum annual increment per acre a c t u a l l y used by the average t r e e was c a l c u l a t e d f o r each one-inch diameter c l a s s . Table XXXI has been b u i l t with the r e s u l t s of t h i s a n a l y s i s i n pure n a t u r a l stands, and i n the Wind R i v e r spacing p l a n t a t i o n . On the average, n a t u r a l stands used 88 percent of the site-c a p a c i t y as d e f i n e d above, and p l a n t a t i o n s used 96 percent. The mean s i t e c a p a c i t y i n n a t u r a l stands was 189 (52 to 266) cubic f e e t per acre per year; i n p l a n t a t i o n , i t was 73 (46 to • 157). Table XXXI shows, on the average, how many t r e e s per acre D i s t r i b u t i o n of m o r t a l i t y 153 TABLE XXXI AVERAGE USE OF SITE CAPACITY IN NATURAL STANDS AND PLANTATIONS 1 ' A V E R A G E P R O P O R T I O N A V . N F O R A V . N A V E R A G E P R O P O R T I O N A V . N FOR D B H i n c h P E R A C R E (1) U S E D 5 (2) USED P E R T R E E^(3) F U L L U S E5 (4) P E R A C R E (1) U S E D (2) U S E D P E R T R E E (3) F U L L U S E (4) 1 163 .003 . 0000 — 10 .000 .0000 — 2 192 .030 .0001 10000 112 .013 . 0001 10000 3 36 .012 .0003 3333 188 .225 . 0011 910 4 30 .017 . 0005 2000 213 .310 . 0014 715 5 3k .031 .0009 1111 169 .204 . 0012 830 6 32 .035 . 0011 909 110 .190 .0017 590 7 29 .041 .0014 714 r r 00 .147 . 0022 455 8 27 .046 .0016 625 41 .112 .0027 370 9 23 .045 .0019 526 31 .101 .0032 314 10 21 .048 .0023 435 27 .105 .0033 265 11 19 .055 . 0028 357 18 .083 .0046 216 12 16 .047 .0030 333 19 .096 .0050 200 13 lk .048 .0033 303 12 .067 .0055 182 14 lk .052 .0037 270 4 .027 .0067 150 15 11 .048 . O O 4 4 227 7 .052 . 0074 135 16 11 .053 .0050 200 17 9 .052 .0056 178 18 9 .056 .0062 161 19 8 .057 .0068 147 20 6 .050 .0078 128 D i s t r i b u t i o n of m o r t a l i t y 154 TABLE XXXI (Continued) ? AV.N AV.H AVERAGE PROPORTION FOR AV.N AVERAGE PROPORTION FOR FULL PER " FULL DBH USED^ USED, PER USE5 ACRE USED USED PER USE i n c h (1) (2) TREE (3) (4) (4) (2) TREE (3) (4) 21 r o .048 . 0084 119 22 5 .045 .0090 i l l 23 5 .045 .9905 105 24 4 .040 . 0102 98 25 3 .038 . 0111 90 26 3 .039 . 0118 85 27 3 .036 .0124 81 28 2 .033 .0132 76 29 2 .032 .0152 66 30 2 .028 .0155 64 31 2 .030 .0166 60 32 2 .027 .0168 59 33 1 .026 .0185 54 34 1 .026 .0200 50 35 1 .023 .0209 48 36 1 .025 .0208 48 37 1 .023 .0230 43 38 1 .023 .0230 43 39 i .023 .0230 43 Based on 14 p l o t s i n Group I, and 14 p l o t s i n the Wind R i v e r Spacing T r i a l . Average number of t r e e s i n p l o t s s t u d i e d (rounded). Average r a t i o of a c t u a l mean annual increment i n cu. f t , to normal gross m.a.i. from S t a e b l e r ( I 9 5 5 a ) . Column 2 d i v i d e d by column 1. 100 d i v i d e d by column 3 f o r each DBH c l a s s . D i s t r i b u t i o n of m o r t a l i t y 155 d i d use a c e r t a i n p r o p o r t i o n of s i t e c a p a c i t y (columns 1 and 2); the a c t u a l average use of s i t e c a p a c i t y by i n d i v i d u a l t r e e s was obtained by d i v i d i n g t h i s p r o p o r t i o n by the number of t r e e s (column 3 ) . Then, the average number of t r e e s that would be r e q u i r e d to make f u l l use of the s i t e was computed by diameter c l a s s (column U). Trees i n p l a n t a t i o n d i d use a l a r g e r " part of s i t e c a p a c i t y than i n n a t u r a l stands. T h i s may be duo i i r p a rt to the lower average s i t e c a p a c i t y i n p l a n t a t i o n , or to a g r e a t e r average tre e e f f i c i e n c y coupled v/ith a b e t t o r s t o c k i n g and a lower stand d e n s i t y . Three f a c t s i n d i c a t e that the second hypothesis may be the r i g h t one. F i r s t , t r e e s wore on the average younger i n p l a n t a t i o n (32 compared to 53 y e a r s ) , i . e . more e f f i c i e n t ; second, they had on the average a l a r g e r crown width/diameter r a t i o p e r m i t t i n g f u l l e r s t o c k i n g with lower d e n s i t y ; and t h i r d , they were unif o r m l y spaced. That t r e e e f f i c i e n c y i s at i t s best between ages of 20 and 50 has been demonstrated by Ovington (1956). That t r e e s i n each diameter c l a s s had l a r g e r crowns i n pl.anta.tion than i n n a t u r a l stands can be v e r i f i e d by r e f e r e n c e to F i g u r e I I I - 2 (Appendix I I I ) with data from column L i n Table XXXI. From t h i s f i g u r e , we know, f o r i n s t a n c e , that 900 s i x - i n c h n a t u r a l or 600 planted t r e e s were needed to use the s i t e f u l l y , and that the s i x - i n c h n a t u r a l t r e e s grew i n stands of 0.9 r e l a t i v e stand d e n s i t y v/ith average crown width/diameter r a t i o of 1.1, whereas pla n t e d t r e e s grew at 0.5 r e l a t i v e d e n s i t y with average CW/D3H of about D i s t r i b u t i o n of m o r t a l i t y 156 1.4. Tables I I I and V i n d i c a t e that n a t u r a l stands grew at incomplete s t o c k i n g (CCCLO = 52 percent) while more complete crown c l o s u r e was observed i n p l a n t a t i o n s (CCCLO = 121), Since complete s t o c k i n g and minimum stand d e n s i t y r e s u l t i n maximum growth r a t e , p a r t of the g r e a t e r e f f i c i e n c y i n s i t e use by p l a n t a t i o n t r e e s i s thus e x p l a i n e d . That r e g u l a r l y d i s p e r s e d t r e e s support on the average l e s s competition ( e s p e c i a l l y i n e a r l y stages of stand development) and arc thus more e f f i c i e n t i s q u i t e obvious. In n a t u r a l stands, p a t t e r n s were more heterogeneous than i n p l a n t a t i o n . T h i s i s i l l u s t r a t e d by the average p a t t e r n ' s variance-mean r a t i o of 0.94 (0.43 to 2.48) i n n a t u r a l stands, compared to 0.53 (0.32 to 1.02) i n p l a n t a t i o n s (Tables I I I and V). These o b s e r v a t i o n s support the view that optimum use of s i t e c a p a c i t y would be a t t a i n e d by f o l l o w i n g open-to-normal regimes of stand d e n s i t y (Smith, 1963)> with t r e e s randomly or at best u n i f o r m l y d i s t r i b u t e d over the area. 5.5 CHAPTER SUMMARY D e s c r i p t i o n s of m o r t a l i t y d i s t r i b u t i o n s i n n a t u r a l stands are made to supplement s t u d i e s of amount and t i m i n g of m o r t a l -i t y . The hypothesis that dead t r e e s are d i s t r i b u t e d l i k e l i v i n g t r e e s was t e s t e d by a n a l y z i n g t h e i r diameter, height, and s p a t i a l arrangements i n the same stands. Percentage diameter d i s t r i b u t i o n s 'of dead t r e e s d i d not f i t to normal but to negative binomial d i s t r i b u t i o n s i n most D i s t r i b u t i o n of m o r t a l i t y 157 cases. S p a t i a l d i s t r i b u t i o n s showed aggregation i n very dense or clumpy stands, i n stands where some i r r e g u l a r m o r t a l i t y d i d occur, and i n mixed stands of Douglas f i r and other c o n i f e r s ; otherv/ise, dead t r e e s were randomly d i s p e r s e d i n most stands observed. Most diameter d i s t r i b u t i o n s of l i v e t r e e s were normal at low stand d e n s i t y ; otherwise they were skewed. A l l d i s t r i b u -t i o n s wore s t a b l e w i t h i n the same p l o t s over p e r i o d s of 10 to 30 y e a r s . Maximum t r e e s i z e was 1.95 times the average stand diameter i n n a t u r a l stands, and 1.75 times average i n p l a n t a -t i o n . •Height d i s t r i b u t i o n s were c l o s e l y r e l a t e d to diameter d i s t r i b u t i o n s , although more i r r e g u l a r . A h i g h degree of dumpiness i n t r e e d i s p e r s i o n was d i s c e r n e d only i n a few p l o t s . The m a j o r i t y of them showed random arrangements or a t l e a s t a trend towards i t . Average diameters of dead and l i v i n g t r e e s were r e l a t e d by a r e g r e s s i o n equation which shows that the r a t e of i n c r e a s e i n m o r t a l i t y diameter i s about one-half the r a t e of i n c r e a s e i n stand diameter. S i t e occupancy i n d i c a t o r s have been c o r r e l a t e d and theore-t i c a l l i m i t s to the p r o p o r t i o n of open spaces c o n s i s t e n t with randomness have been s e t . Trees i n p l a n t a t i o n were found to use s i t e c a p a c i t y more e f f i c i e n t l y than i n n a t u r a l stands be-cause they were younger, and because they grow i n a r e g u l a r arrangement, at lower d e n s i t y , i n b e t t e r stocked stands. 153 CHAPTER 6 PREDICTION OF MORTALITY Two d i f f e r e n t approaches can be taken to p r e d i c t m o r t a l i t y . The f i r s t one, the stand approach, i s c u r r e n t l y being considered when an o v e r a l l estimate of number of t r e e s , or of number per s i z e c l a s s per u n i t area i s needed. The second one, the t r e e  approach, i s considered when the s p a t i a l arrangement and the growth response of i n d i v i d u a l s to removal of competitors (by m o r t a l i t y or c u l t u r a l treatments) i s i n v e s t i g a t e d . U s u a l l y , the f i r s t method, much cheaper, l e s s accurate and l e s s informa-t i v e , serves f o r long-term p r o j e c t i o n s ; i t may r e q u i r e only a desk c a l c u l a t o r . The second method, which p r o v i d e s much more d e t a i l e d i n f o r m a t i o n f o r s h o r t e r p e r i o d s of time, r e q u i r e s a d i g i t a l computer. U l t i m a t e l y , both methods give an e v a l u a t i o n of a c t u a l f o r e s t growth and y i e l d . In t h i s regard, the t r e e approach i s more f l e x i b l e than the stand approach, and i t a l l o w s f o r s i m u l a t i o n to bo performed as w e l l . In t h i s d i s s e r t a t i o n , we a l r e a d y have computed much of the i n f o r m a t i o n needed to use o i t h e r approach i n p r e d i c t i n g r e g u l a r m o r t a l i t y of Douglas f i r . 6.1 THE STAND APPROACH 6.1.1 YIELD TABLES M o r t a l i t y i n n a t u r a l , oven-aged stands can be pre-d i c t e d d i r e c t l y i n cubic volume or b a s a l area from gross and net P r e d i c t i o n of m o r t a l i t y 159 y i e l d t a b l e s such as the ones prepared by St a e b l e r (1955&), C u r t i s (1967) and McArdle et a l . (1949). I t can a l s o be pre-d i c t e d i n number of t r e e s , by- t a k i n g the d i f f e r e n c e between normal numbers of l i v i n g t r e e s t a b u l a t e d f o r 10-year p e r i o d s . Normality, and trends i n normality must be measured ( B r i e g l e b and G i r a r d , 1943) or assumed. Methods of p r e d i c t i n g f u t u r e volumes have been explained by McArdle et a l . (1949), and John-son (1955). Figu r e 1 i l l u s t r a t e s the technique, 6.1.2 PREDICTION EQUATIONS When a l a r g e number.of permanent sample p l o t s i s a v a i l a b l e , m o r t a l i t y can be recorded ( i n s e v e r a l d i f f e r e n t u n i t s ) , and r e l a t e d to stand c h a r a c t e r i s t i c s by r e g r e s s i o n equations. These equations then serve- ( w i t h i n the range of data i n c l u d e d ) to p r e d i c t m o r t a l i t y i n s i m i l a r stands. The ones developed i n t h i s d i s s e r t a t i o n , v/ith q u i t e a v a r i e t y of Douglas f i r stands, are presented i n Tables XV to XVIII, and XXIV. 6.1.3 RELATIONSHIPS BETWEEN LIVE AND DEAD TREES When m o r t a l i t y i s p r e d i c t e d i n number of stems, or i n volume, the average s i z e , or volume of dead t r e e s can be determined by the r e l a t i o n s h i p stand d i a m e t e r - m o r t a l i t y diame-t e r , or average l i v e t r e e volume-average dead t r e e volume. Both are d e s c r i b e d f o r Douglas f i r i n F i g u r e s 17 and 18, and page 144. P r e d i c t i o n of m o r t a l i t y 160 6.1.4 PERCENTAGE DIAMETER DISTRIBUTIONS Regressions of annual m o r t a l i t y on average stand diameter and/or age can be coupled with percentage diameter d i s t r i b u t i o n s of dead t r e e s to make p r e d i c t i o n s of m o r t a l i t y i n number of t r e e s by diameter c l a s s i T h i s method was used by Lee (1969) i n lodgepole pine stands. F i g u r e 13 and Tables XV to XVIII provide t h i s type of estimates f o r Douglas f i r . 6.2 THE TREE APPROACH The tree approach has a l r e a d y boon used e x t e n s i v e l y i n t h i s d i s s e r t a t i o n (Appendix I) to evaluate s p a t i a l p a t t e r n s , growth, s t o c k i n g and m o r t a l i t y . T h i s i n f o r m a t i o n has a l s o served i n b u i l d i n g a stand model to simulate both m o r t a l i t y and growth. 6.2.1 SEMI-STOCHASTIC STAND MODEL A simple s e m i - s t o c h a s t i c stand model has been c r e -ated by which i t i s p o s s i b l e to p r e d i c t , f o r a r e s t r i c t e d num-ber of 10-year p e r i o d s , what w i l l be the f u t u r e g r o s s and net growth and y i e l d of a given f o r e s t based on a p l o t t a l l y of i n -d i v i d u a l t r e e diameters. The model e s s e n t i a l l y uses the t r e e approach, four r e g r e s s i o n equations, and three groups of mor-t a l i t y t a b l e s to p r e d i c t f u t u r e m o r t a l i t y and growth. 6.2.1.1 ANNUAL DIAMETER INCREMENT FUNCTIONS The increment was c a l c u l a t e d on a per P r e d i c t i o n of m o r t a l i t y 161 annum b a s i s because p l o t s s t u d i e d had been remeasured at i r r e g -u l a r i n t e r v a l s . Random increment valuos were chosen from each p l o t romeasurementj i n each group d e s c r i b e d i n Table I I . They are: (1) P l a n t a t i o n s ; (2) Pure Douglas f i r (Group I ) ; (3) Mixed D.F. (Group I I , 76-90% f i r ) ; (U) Mixed D.F. (Group I I I , 51-75% f i r ) ; (5) Mixed D.F. (Group IV, l e s s than 50% f i r ) . T h i s random s e l e c t i o n of measurements was performed because the t o t a l p o s s i b l e number of increment valuos a v a i l a b l e was much too l a r g e and unnecessary to b u i l d a cceptable r e g r e s s i o n equations. The o r i g i n a l i d e a was to develop a r e g r e s s i o n model that would i n c l u d e a l l , or p a r t of tho f o l l o w i n g s i t e , stand and tree c h a r a c t e r i s t i c s : 1) i n d i v i d u a l t r e e parameters (diameter and crown c l a s s ) ; 2) r e l a t i v e t r e e parameters ( t r e e diameter over stand diameter); 3) stand parameters ( d e n s i t y i n b a s a l area, average stand diameter, stand age, s t o c k i n g , v a r i a n c e -moan r a t i o of the s p a t i a l d i s t r i b u t i o n ) ; and U) s i t e parameters ( s i t e index, i n f e e t at 100 y e a r s ) . In the development of an acceptable r e g r e s s i o n model, s e v e r a l combinations of the aforementioned v a r i a b l e s were t e s -ted. With the Wind R i v e r p l a n t a t i o n data, i t was p o s s i b l e to account f o r 80 percent of the v a r i a t i o n i n annual diameter i n -crement with seven independent v a r i a b l e s . However, the best model, chosen f o r i t s s i m p l i c i t y , i n c l u d e d only 3 v a r i a b l e s , namely diameter, age and s i t e index. A s i m i l a r model was chosen f o r Group I, even i f a s l i g h t l y higher c o e f f i c i e n t of P r e d i c t i o n of m o r t a l i t y 162 determination was obtained with 8 independent v a r i a b l e s . In Groups I I , I I I , and IV, very l i t t l e advantage was gained with any other combination. Regression models i n c l u d i n g diameter, diameter squared and. crown c l a s s , or diameter, r e l a t i v e diame-t e r and s i t e index were e q u a l l y good. Simple r e g r e s s i o n s of annual diameter increment on tr e e diameter were very s a t i s f a c t o r y f o r Group l l ( r " = .53)> Group I I I ( r ^ = .68), and Group IV ( r 2 = .44). Regressions of diame-t e r increment on b a s a l area f o r Wind R i v e r (r = .38), or on crown c l a s s f o r Group I (rd = .38), f o r Group I I I ( r = .32), and f o r Group IV ( r 2 = .44) were a l s o s i g n i f i c a n t . The model i n c l u d i n g t r e e diameter (DBH), AGE, and s i t o i n -dex (SI) was chosen to run the stand model because i t was tho best equation common to a l l groups ( p l a n t a t i o n s , and pure or mixed n a t u r a l s t a n d s ) . Equations, and stand and t r e e charac-t e r i s t i c s are presented i n Table XXXII. The o v e r a l l range of val u e s s t u d i e d i s q u i t e l a r g e : mimima and maxima of annual diameter increment (DIN) were -.04 and .533. A c t u a l DBH v a r i e d between 1.1 and 38.3 inches; s i t e index ranged from 72 up to 207, and age from 18 to 87 years. The lowest c o e f f i c i e n t s of d e t e r m i n a t i o n were observed i n groups having c i t h e r a f a i r l y l a r g e c o e f f i c i e n t of v a r i a t i o n i n diameter increment (Group IV, 96%), or r e p r e s e n t i n g a wide range of c o n d i t i o n s (Group I, DBH from 1.2 to 38.3 i n c h e s ) . P r e d i c t i o n of m o r t a l i t y 163 TABLE XXXII ANNUAL DIAMETER INCREMENT FUNCTIONS FOR DOUGLAS FIR Independent V a r i a b l e s and « s ^ I n t e r c e p t Regression c o e f f i c i e n t s R (Inch.) DBH SI AGE (inch) ( f t . at (years) age 100) WIND RIVER (N=434) 1 0.172847 0.019391 0.001152 -0.008682 .73 .04 GROUP I (N=630) 0.246187 0.013019-0.000676 -0.003480 .39 .07 GROUP I I (N=179) 0.448678 0.025313-0.001429 -0.008525 .64 .07 GROUP I I I (N=125) 2 0.177289 0.020276-0.000368 -0.004060 .75 .0.7 GROUP IV (N=7D 0.2.04096 0.011269-0.000924 -0.002299 .45 .06 PLANTATION GROUP I TREE CHARACTERISTICS STAND CHARACTERISTICS DIN DBH SI AGE Inch per year Inches Feet at age 100 Years Mean 0.100 5.1 95.6 32.3 Min. -0.040 1.6 72.0 22.0 Max. 0.383 11.8 147.0 42.0 St. dev. .08 2.1 18,7 5-3 CV. {%) 80.0 41.0 19.0 16.0 Mean 0.115 12.7 157.2 56.7 Min. -0.033 1-2 113.0 18.0 Max. 0.533 38.3 207.0 87.0 St. dev. 0.09 6.1 24.5 13.6 CV. (%) 79.0 48.O 17.0 24.0 GROUP I I Moan Min. Max. St. dev. GROUP I I I Mean Min. Max. St. dev. CV. (%) GROUP IV Mean Min. Max. . St. dev. CV. P r e d i c t i o n of m o r t a l i t y TABLE XXXII (Continued) 164 TREE CHARACTERISTICS DIN DBH Inch per year 0.147 0 . 0 0 . 5 0 0.12 81 . 0 0.16 -0.12 0.54 0.14 92.0 0.087 0.0 0.44 0.08 96.0 10.2 2.9 26.8 4.2 41.0 11.0 1.0 32.5 6.2 56.0 10.7 3.4 24.7 5-0 47.0 STAND CIIARACTERISTIC S SI Inches Feet at agc 100 146.0 104.0 185.0 21.3 14.0 147.2 79.0 196.0 37.4 25.0 104.3 96.0 124.0 10.2 10.0 AGE Years 41.2 29.0 54.0 6.7 16.3 45.6 30.0 71.0 10.7 23.0 61.5 53.0 70.0 5.6 9.0 N = number of ob s e r v a t i o n s . Underlined v a r i a b l e s do not make a s i g n i f i c a n t c o n t r i b u t i o n to the r e g r e s s i o n at the 0.01 l e v e l . Symbols are i d e n t i f i e d i n Appendix IV. P r e d i c t i o n of m o r t a l i t y 165 6.2.1.2 HEIGHT FUNCTION S i m i l a r l y , a standard form of height equa-t i o n has been adopted. I t l i n e a r l y i n c l u d e s t r e e diameter, diameter squared and b a s a l area per a c r e . In most cases, the b a s a l area term i s not used, f o r i t does not improve g r e a t l y the p r e c i s i o n of estimates made with diameter and diameter squared. S e v e r a l equations are l i s t e d i n Appendix I I , which can be used by the stand.model i n the same way diameter i n c r e -ment i s . No height increment f u n c t i o n was c a l c u l a t e d ; i n s t e a d , i n -d i v i d u a l t r e e h e i g h t s were evaluated at the beginning of each p r e d i c t i o n p e r i o d . 6.2.1.3 VOLUME FUNCTION The volume equation used throughout t h i s work, and i n the stand model, i s the one developed by Browne (1962), which i s commonly r e f e r r e d to as "the B.C. F o r e s t Ser-v i c e l o g volume equation f o r immature Douglas f i r . " I t g i v e s t o t a l cubic volume i n s i d e bark of Coast imma.ture f i r , 2 to 60 i n c h e s i n diameter, and up to 1L0 years i n age. The e x p r e s s i o n i s the f o l l o w i n g : l o g VOL = -2.638025 + 1.739925 l o g DBH + 1.133187 l o g H DBH = diameter o.b. at b r e a s t height; H = t o t a l h e ight I t s accuracy i n e s t i m a t i n g i n d i v i d u a l t r e e volumes was t e s t e d a g a i n s t volume estimates given by Williamson (1963) f o r 10 one-acre sample p l o t s e s t a b l i s h e d i n w e l l - s t o c k e d stands. The P r e d i c t i o n of m o r t a l i t y 166 program YIELD (Appendix I) was used f o r that purpose. Answers were remarkably comparable, and t h e r e f o r e , t h i s form was pre-f e r r e d to other models d e s c r i b e d by Smith and Broaden (1964), and Honor (1965). 6.2.1.4 SPATIAL PATTERN FUNCTIONS Based on the a n a l y s i s of s p a t i a l p a t t e r n , a r e g r e s s i o n of variance-mean r a t i o on stand c h a r a c t e r i s t i c s was c a l c u l a t e d to p r e d i c t tho va.ria.nco-m.ean r a t i o of the spa-t i a l d i s t r i b u t i o n . The i d e a was to reproduce, by the same method used to analyze, the p a t t e r n of m o r t a l i t y most l i k e l y to be encountered i n a given f o r e s t at a given time. A f i r s t assumption was made, th a t dead t r e e s would be d i s -t r i b u t e d l i k e l i v i n g t r e e s , thus, a l l o w i n g f o r some parameters of the l i v i n g stand to be used i n p r e d i c t i n g tho d i s t r i b u t i o n of m o r t a l i t y . T h i s assumption i s j u s t i f i e d i n the l i g h t of d i s t r i b u t i o n a nalyses made i n Chapter 5« Seve r a l stand parameters i n c l u d i n g age, average diameter, r e l a t i v e d e n s i t y , square spacing, number of t r e e s , s i t e index, crown c l o s u r e , crown space occupation, and crown competition f a c t o r wore r e l a t e d by m u l t i p l e r e g r e s s i o n techniques to the variance-mean r a t i o (VARMEAN) of the d i s t r i b u t i o n of quadrats having tho same number of t r e e s . Quadrat s i z e was a l s o i n c o r -porated i n t o tho r e l a t i o n s h i p . Tho h i g h e s t simple c o r r e l a t i o n c o e f f i c i e n t s with VARMEAN were a t t r i b u t e d , i n p l a n t a t i o n , to crown competition f a c t o r P r e d i c t i o n of m o r t a l i t y 167 (.65 )3 square spacing and number of t r e e s per acre (.62), crown space occupation (.60), normality i n b a s a l area (.58), cumula-t i v e crown c l o s u r e (.57), r e l a t i v e d e n s i t y i n number of t r e e s (•45)) a n d quadrat s i z e (.39). In n a t u r a l stands, the order and the magnitude of the c o r r e l a t i o n were s l i g h t l y d i f f e r e n t , but the same v a r i a b l e s were important. In both cases, the best m u l t i p l e r e g r e s s i o n equation de-veloped to p r e d i c t variance-mean r a t i o of the s p a t i a l d i s t r i -b u t i o n i n c l u d e d square spacing and cumulative crown c l o s u r e , as f o l l o w s : P l a n t a t i o n VARMEAN = 0.6129 - 0.0283 SQSP + 0.0011 CCCLO N = 5 0 R 2 = .41 SEE = .11 (21%) Group I VARMEAN = 2.1383 - 0.0031 SQSP - 0.0244 CCCLO N = 6 5 R 2 = .29 SEE = .36 (40%) However, i n these r e l a t i o n s h i p s , a l o c a l crown width equation which i s seldom a v a i l a b l e would bo needed to evaluate cumula-t i v e crown c l o s u r e (CCCLO). Consequently, the two next best equations were chosen to i n c o r p o r a t e i n t o the stand model. In p l a n t a t i o n , the r e l a t i o n i n c l u d e s stand age and average stand diameter: VARMEAN = 0.3573 + 0.0157 AGE - 0.0623 AVDBH N = 5 0 R 2 = .33 SEE = .15 (27%) In pure Douglas f i r stands (Group I ) , r e l a t i v e d e n s i t y i n num-ber of t r e e s per acre, and age are i n c l u d e d . P r e d i c t i o n of m o r t a l i t y VARMEAN = l.i+691 - 1.1389 RDN + 0.0075 AGE N = 6 5 R 2 = .27 SEE = .36 (40%) In order to f a c i l i t a t e the r e p r o d u c t i o n of p a t t e r n s , a second assumption was made as to the number of p r o b a b i l i t y d i s t r i b u -t i o n s t h a t could be observed. The stand model was w r i t t e n (SUBROUTINE MAP) to generate only two s e t s of c o o r d i n a t e s a c c o r d i n g to the magnitude of the p r e d i c t e d variance-moan r a -t i o . When the p r e d i c t e d VARMEAN i s l e s s than 1.0, v a r i a t c s arc drawn from a Poisson d i s t r i b u t i o n ; when VARMEAN i s l a r g e r than 1.0, the c o o r d i n a t e s are p u l l e d out of a negative binomial d i s -t r i b u t i o n . The technique employed to generate- these v a r i a t c s i s w e l l explained i n Naylor et a l . (1966). In b r i e f , Poisson v a r i a t c s are obtained d i r e c t l y from a generator of u n i f o r m l y d i s t r i b u t e d random numbers (RAND 0.0). Negative b i n o m i a l v a r i a t c s on the other hand, are e s s e n t i a l l y the sum of k geometric v a r i a t c s . When k i s not an i n t e g e r (which i s the most usual case i n the stand model), the procedure i s an approximation which c o n s i s t s i n generating v a r i a t c s from a Poisson d i s t r i b u t i o n , whose s i n g l e parameter has a Gamma d i s -t r i b u t i o n . Much refinement i s needed before t h i s process of genera-t i n g the most l i k e l y l o c a t i o n s of dead t r e e s becomes r e l i a b l e . At present, i t i s l i t t l e more than a guess. Consequently, t h i s p a r t of the program should be considered as a f i r s t t r i a l aimed at g i v i n g an i d e a of the grouping of i n d i v i d u a l s that w i l l d i e . I t was decided to output dead t r e e l o c a t i o n s r a t h e r than l i v e P r e d i c t i o n o f m o r t a l i t y 159 t r e e s , m e r e l y t o s a v e on c o m p u t i n g t i m e . I n f a c t , t r e e c o o r d i n a t e s were n o t g e n e r a t e d , b u t t h e num-b e r o f t r e e s t h a t c o u l d d i e i n a number o f a d j a c e n t q u a d r a t s l a i d o v e r t h e a r e a ; q u a d r a t s i z e was d e t e r m i n e d by t h e t o t a l number o f d e a d t r e e s i n any g i v e n f o r e c a s t i n g p e r i o d . The s t a n d m o d e l , a s b u i l t , c a n a l s o l o c a t e d e a d t r e e s f o r w h i c h C a r t e s i a n c o o r d i n a t e s a r e a v a i l a b l e t o s t a r t t h o s i m u l a -t i o n . T h i s o f f e r s some a d v a n t a g e s a t t h e t e s t i n g s t a g e ; t h e p r e c i s e l o c a t i o n o f d e a d s u b j e c t s c a n bo compared w i t h l o c a -t i o n s a l l o c a t e d a c c o r d i n g t o t h e m a t h e m a t i c a l p r o c e s s e x p l a i n e d a b o v e , and t h e l a t t e r p r o c e s s i m p r o v e d . 6.2.1.5 MORTALITY ALLOCATION One o f t h e m a i n f e a t u r e s o f t h i s s t a n d m o d e l i s t h a t t r e e s a r c n o t k i l l e d by means o f some c a l c u l a t e d c o m p e t i t i o n i n d e x , b u t a c c o r d i n g t o p r o b a b i l i t y c u r v e s d rawn f r o m p a s t e x p e r i e n c e . Tho m a i n a s s u m p t i o n u n d e r l y i n g t h o a c c e p -t a n c e o f t h i s method i s t h a t , on t h e a v e r a g e , t r e e s h a v i n g t h e same r e l a t i v e c h a r a c t e r i s t i c s a s t h e i r p r e d e c e s s o r s by c o m p a r i -s o n t o t h e f o r e s t i n w h i c h t h e y grow, w i l l b ehave t h e same way. T h i s a p p r o a c h i s much s i m p l e r a n d , d e p e n d i n g on t h e q u a l -i t y o f t h e m o r t a l i t y t a b l e s , c a n be a s much o r more a c c u r a t e t h a n o t h e r m e t h o d s o f p r e d i c t i n g m o r t a l i t y . M o r e o v e r , m o r t a l -i t y t a b l e s o f f e r t h e p o s s i b i l i t y o f b e i n g i m p r o v a b l e a s e x p e r -i e n c e a c c u m u l a t e s . F o r t h e p u r p o s e o f t h i s d i s s e r t a t i o n , m o r t a l i t y t a b l e s P r e d i c t i o n of m o r t a l i t y 170 have been compiled i n such a way as to represent the l a r g e s t v a r i a t i o n of c o n d i t i o n s p o s s i b l e . However, i f such an approach to m o r t a l i t y p r e d i c t i o n s was judged worthwhile, a much f i n e r breakdown could be achieved when more data are pooled. For example, m o r t a l i t y t a b l e s could be b u i l t by c l i m a t i c r e g i o n s , by topographic c l a s s e s , by f o r e s t typos, a s s o c i a t i o n s , or species, by s i t e q u a l i t y and so on. Three of the s i x groups of m o r t a l i t y t a b l e s d e s c r i b e d i n Chapter L ( F i g u r e 3 to 1 2 ) , and i n Appendix V, were o r i g i n a l l y i n t r o d u c e d i n t o the stand model; the f i r s t one takes i n t o • account r e l a t i v e s i z e (diameter of one t r e e compared to average stand diameter), the second c o n s i d e r s i n d i v i d u a l t r e e height compared to stand top height, and the t h i r d one i n c l u d e s crown c l a s s e s . Tables based on increment were not considered because at l e a s t two p e r i o d s were needed to c l a s s i f y t r e e s i n any one c l a s s ; the t a b l e based on crown width was dropped due to a general l a c k of crown measurements i n c u r r e n t mensurational work (which could e v e n t u a l l y prevent tho model from being used). In the f i n a l stage of development, the crown c l a s s t a b l e was a l s o dropped because i t could not be used more than once i n the s i m u l a t i o n (crown c l a s s not being p r e d i c t a b l e ) . i . MORTALITY GENERATOR, MODEL I Tho f i r s t m o r t a l i t y generator t e s t e d i s based on the i d e a that no t r e e having more than a p r e - d e t e r -mined number of chances to d i e w i l l remain a i i v e at each P r e d i c t i o n of m o r t a l i t y 171 p r e d i c t i o n p e r i o d ( 1 0 - y e a r s ) . T h i s r i s k l e v e l i s meant t o be determined by tho f o r e s t manager. L e t us suppose t h a t a manager wants t o h a r v e s t t r e e s h a v i n g more than 50 p e r c e n t chances t o d i e of n a t u r a l m o r t a l i t y between the age of 40 and 50 y e a r s . A c c o r d i n g t o m o r t a l i t y t a b l e s , the r i s k o f l o s i n g t o m o r t a l i t y a l l the suppressed t r e e s (crown c l a s s 4)» or t r e e s s m a l l e r than 0.75 average stand d i a m e t e r , or s h o r t e r than 0.75 s t a n d top h e i g h t , would bo g r e a t e r than 50 p e r c e n t ( F i g u r e 1 9 ) . Thus, the f o r e c a s t e d v o l -ume of m o r t a l i t y can be computed, and a d e c i s i o n r e a c h e d as t o whether i t i s worth r e c u p e r a t i n g t h i s m a t e r i a l b e f o r e i t i s l o s t . I f , f o r some l o g i c a l r e a s o n , t h i s r i s k l e v e l i s judged t o be too low, the manager may d e c i d e t o keep o n l y t r e e s t h a t have l e s s than 20 p e r c e n t chances t o d i e . He w i l l t hen h a r v e s t a t age 40, the i n t e r m e d i a t e and suppressed t r e e s , and those which a r c s m a l l e r than average i n d i a m e t e r , and s h o r t e r than 0.75 stand top height.. T h i s model of m o r t a l i t y a l l o c a t i o n a l s o o f f e r s the p o s s i -b i l i t y o f t a k i n g tho p r o b a b i l i t y o f a c a t a s t r o p h e i n t o a ccount. An approach t o the c a l c u l a t i o n of t h i s p r o b a b i l i t y i s i n c l u d e d i n Appendix V I . For example, i f the 0.20 r i s k l e v e l was agreed upon as e c o n o m i c a l l y and s i l v i c u l t u r a l l y a c c e p t a b l e i n 40-year pure Douglas f i r s t a n d s i n which t h i n n i n g s or p r o - s a l v a g e o p e r a t i o n s would be contemplated, i t might become 0 .15, a f t e r the p r o b a b i l i t y of a c a t a s t r o p h e i s b e i n g e s t a b l i s h e d i n tho r e g i o n where these s t a n d s are l o c a t e d . I n t h a t case, one would Figure 19- Mortality Generator Model I- Periodic probability of mortality on relative size,relative height and crown class in natural stands* Prediction of mortality 173 be j u s t i f i e d to take out, over what was previously calculated, every tree smaller than average height (based on Figure 19). i i . MORTALITY GENERATOR, MODEL II The second mortality generator (which can be used simply by interchanging SUBROUTINE KILL i n the model) makes straight use of the f u l l p robability curve at any given period. If, during one prediction period, tho probabil-i t y that a tree of a given r e l a t i v e size or height w i l l dio i s 0.60, then on the average 60 trees, out of 100 having the same cha r a c t e r i s t i c s , s h a l l be k i l l e d i n the simulation. Negative exponential regression equations have been f i t t e d to the data (Appendix V) to avoid grouping trees into a r t i f i -c i a l classes. Figure 12 roughly i l l u s t r a t e s the shape of t h i s type of curve. In Model II, r e l a t i v e size (dbh/DBH) and r e l a -tive height (h/TOPH) only have been used to predict mortality. At any r e l a t i v e size or height, a tree i s given the highest probability of mortality determined by any one of those two curves. I t s actual death i s determined by a uniform random number chosen between 0 and 1. Whenever the value drawn at ran-dom f a l l s under tho mortality curve, the tree i s k i l l e d . Dif-ferent curves are used for d i f f e r e n t time periods. A l l their regression c o e f f i c i e n t s are included i n subroutine KILL. As an i l l u s t r a t i o n , l e t us assume that, at a given moment in time, a tree has a r e l a t i v e height X-^  equals to 0.79 and a r e l a t i v e size X, equals to O.96 (Figure 20). Since tho highest 174 Figure 20- Mortality Generator Model H- Periodic probability of mortality on relative size and relative height in natural stands* P 1 0 0 r T I I 1 1 0 * 6 5 ^ S . 1 | 1 0 * 6 3 0 * 6 0 0 5 0 | B 1 1 ^ 1 1 1 1 1 1 ^ ^ > \ d b h / D B H 1 1 1 1 h / T O P H ^ 0 * 0 0 1 1 1 1 I I 1 1 x n X d 0*79 0*96 Relative Size and Height P r e d i c t i o n of m o r t a l i t y 175 p r o b a b i l i t y of m o r t a l i t y i s determined by the h/TOPH curve at X_h = 0.69 ( p o i n t A), the random number generated w i l l be com-pared to P at that l e v e l (P = O.63). Before drawing the number, we know that 63 times out of 100, the random value w i l l f a l l below the curve. Thus each t r e e of t h i s p a r t i c u l a r s i z e and height w i l l have at that time e x a c t l y 63 chances to d i e and 37 chances to l i v e , out of 100. I f the number drawn i s .50 (B), the t r e e i s k i l l e d . I f the number i s .65 (C), the t r e e may or may not l i v e depending on the r i s k of a catastrophe (Appendix V I ) . In a p a r t i c u l a r r e g i o n where t h i s r i s k would be evaluated a t 0.05, the t r e e would be k i l l e d (0.65 - 0.05 = 0.60 or D l e v e l , v/hich i s under A). Somewhere e l s e , the r i s k of a catastrophe might be n e g l i g i b l e , and the t r e e would then l i v e and grow u n t i l the noxt p e r i o d . In Model I I , t r e e s of any s i z e can t h e o r e t i c a l l y d i e , whereas i n Model I, only t r e e s l o c a t e d beyond a determined t h r e s h o l d are a f f e c t e d . Thus Model I I i s s t o c h a s t i c , and reproduces the r e a l process with more f i d e l i t y . 6.2.1.6 DESCRIPTION OF THE STAND MODEL A t a l l y of t r e e diameters (one t r e e per card) i s fed i n t o the computer together with the sample p l o t dimensions ( r e c t a n g u l a r or square), stand age and d e s i r e d num-ber of p e r i o d s of p r e d i c t i o n for the s i m u l a t i o n run. Gross y i e l d of the a c t u a l p l o t i s then compiled on an acre b a s i s P r e d i c t i o n of mortality. 176 (SUBROUTINE YIELD) using height and volume f u n c t i o n s p r e v i o u s l y d e s c r i b e d . Then, each tr e e goes through the SUBROUTINE KILL, which generates m o r t a l i t y a c c o r d i n g to Model I or Model I I , de s c r i b e d i n s e c t i o n s i and i i above. An histogram of dead t r e e s i s p r i n t e d (SUBROUTINE HISTO) with a map (SUBROUTINE MAP) showing t h e i r l o c a t i o n , as explained i n s e c t i o n 6.2.1..'-:-. Coming back to SUBROUTINE YIELD, m o r t a l i t y f i g u r e s , not y i e l d , and gross and net growth are computed. L i v i n g t r e e s are then d i r e c t e d through tho diameter increment f u n c t i o n (Table XXXII), and t h e i r s i z e i s i n c r e a s e d 10 times by a c a l c u l a t e d annual i n -crement. And the procedure s t a r t s anew, u n t i l the pr e - d e f i n e d number of p r e d i c t i o n p e r i o d s i s reached. F i n a l l y , a complete growth and y i e l d t a b l e i s p r i n t e d . A flow c h a r t of the computer program, and an example of output from m o r t a l i t y generator Model I I ore i n c l u d e d i n Appen-d i x V I I . The' process by which the a c t u a l height and diameter i n -crement of each tr e e are p r e d i c t e d deserves some e x p l a n a t i o n . Most of tho time, when u t i l i z i n g r e g r e s s i o n equations f o r pre-d i c t i o n , i t i s accepted that the p r e d i c t e d v a r i a b l e (Y) i s the same each time the independent v a r i a b l e s (x'c) are given the same va l u e . T h i s way, with a given height-diameter r e l a t i o n -ship, every t r e e having the same diameter i s given the same height. However, while being acceptable to determine the average height of a l a r g e number of t r o o s i n a given diameter c l a s s , t h i s procedure i s not r e a l i s t i c enough whenever P r e d i c t i o n of m o r t a l i t y 177 i n d i v i d u a l h e i g h t s are p r e d i c t e d . Therefore, i n the stand model, use was made of the r e g r e s s i o n ' s standard e r r o r of es-timate (SEE) i n e v a l u a t i n g i n d i v i d u a l t r e e h e i g h t s and annual diameter increments. Making the assumption that the r e s i d u a l s had a normal d i s t r i b u t i o n around the r e g r e s s i o n l i n e , with standard e r r o r equal to the standard e r r o r of estimate, the value of the dependent v a r i a b l e (Y) was allowed to vary between (Y-SEE) and(Y+SEE), In each case, the exact value of Y was de-termined by u s i n g a generator of normally d i s t r i b u t e d random numbers (FUNCTION FORMAL). The two parameters of t h i s d i s t r i -b u t i o n were set at moan = Y and standard d e v i a t i o n = SEE/3. 6.2.1.7 TESTING OF THE STAND MODEL The stand model was t e s t e d with only one of the two m o r t a l i t y generators d e s c r i b e d because they are based e s s e n t i a l l y on the same p r i n c i p l e s . Model I I was chosen f o r i t does not r e q u i r e any e x t e r n a l i n t e r f e r e n c e to be used. The two sources of data were processed, i n c l u d i n g one sample p l o t from each of the four groups of n a t u r a l stands d e s c r i b e d i n Table I I , and three p l o t s from an independent 6 source. Number of t r e e s , average diameter, top height, and gross and net b a s a l area and cubic volume per acre were com-pared to a c t u a l stand c h a r a c t e r i s t i c s (as compiled with 'Made a v a i l a b l e by Crown Z e l l o r b a c h Corp., whose c o l l a b o r a t i o n i s g r a t e f u l l y acknowledged. P r e d i c t i o n of m o r t a l i t y 178 o r i g i n a l data, and programs YIELD and STOCK, both d e s c r i b e d i n Appendix I ) . Ton-year p r e d i c t i o n p e r i o d s were considered from time of f i r s t p l o t measurement. D e t a i l s of the comparison are given i n Appendix V I I I . R e s u l t s a f t e r a l i m i t e d amount of t e s t i n g i n d i c a t e that, i n most cases, 10-ycar p r e d i c t i o n s of m o r t a l i t y and growth can be made cheaply and with acceptable accuracy by u s i n g the stand model. Moreover, due to the s t o c h a s t i c process i n v o l v e d i n a l l o c a t i n g growth and m o r t a l i t y , the p r e c i s i o n i s somewhat improved by running the model more than once v/ith the same data. A l l simulated volume and b a s a l area estimates f o i l w i t h i n 11 percent of the measured c h a r a c t e r i s t i c s , the m a j o r i t y of thorn being w i t h i n p l u s or minus 9 percent of the a c t u a l value. Average stand diameter and top height estimates d i f f e r e d by l o s s than 5 percent (except i n one case), and number of t r e e s per acre by l o s s than 13 percent (except i n one c a s e ) . Diame-t e r d i s t r i b u t i o n s of dead t r e e s were skewed, and most s p a t i a l d i s t r i b u t i o n s of dead t r e e s generated d i d f i t to tho negative binomial probably d i s t r i b u t i o n , as observed i n nature (Chapter 5 ) . Tho p r e c i s i o n of these estimates compares f a v o r a b l y with those made from normal y i e l d t a b l e s by McArdle ot a l . ( 1 9 4 9 ) . They rep o r t e d standard e r r o r s i n 10-year f u t u r e cubic volume estimates of 7.3 percent f o r pure n a t u r a l stands of Douglas f i r . Moreover, they suggested that i f p r e d i c t i o n s wore c a r r i e d P r e d i c t i o n of m o r t a l i t y < 179 f u r t h e r i n t o the f u t u r e , the standard e r r o r should be m u l t i -p l i e d by l.A f o r 2 decades, by 1.7 f o r 3 decades, or by the square root of the number of decades. This seems to bo the case here; w h e n as many as f i v e 10-year p r e d i c t i o n s were made, based on the f i r s t p l o t measurement, the p r e c i s i o n of volume estimatec decreased. I t appears that the height-diameter r e l a t i o n s h i p u t i l i z e d would be the most important s i n g l e f a c t o r c o n t r o l l i n g the pre-c i s i o n of the output. I t has been noticed that b e t t e r r e s u l t s would be obtained with height-diameter f u n c t i o n s b u i l t from a wide range of stand c o n d i t i o n s , and i n c l u d i n g one v a r i a b l e that r e f l e c t s changes i n stand density with time (such as basal area per a c r e ) . P r a c t i c a l l y then, the p r e c i s i o n of the stand model de-creases as the number of p r e d i c t i o n periods i n c r e a s e s . This should t h e o r e t i c a l l y l i m i t the use of the stand model to annual p r e d i c t i o n s . I f i t were the case though, a e r i a l photography and remote sensing techniques ( H e l l e r , 1965) coupled with l i m i -ted ground sampling, would give f a r more accurate r e s u l t s , and e l i m i n a t e the need f o r stand models of t h i s type. However, r e s u l t s show that the model i s adequate f o r 10-year p r e d i c t i o n s of stand c h a r a c t e r i s t i c s and m o r t a l i t y . I t can p r e d i c t , with consistency, the development of a wide v a r i e t y of stands growing i n the Douglas f i r Region, based on a minimal amount of i n f o r m a t i o n . In f a c t , only a t a l l y of diameters and a height-diameter r e l a t i o n s h i p are needed. Moreover, the cost P r e d i c t i o n of m o r t a l i t y 1 8 0 of such p r e d i c t i o n s i s very low ( l o s s than 5 computer d o l l a r s f o r f i v e p r e d i c t i o n p e r i o d s , s t a r t i n g with 5 0 0 t r e e s ) . 6.2.2 THE IDEAL STOCHASTIC STAND MODEL In d e v e l o p i n g the model j u s t d e s c r i b e d , we have accumulated some experience i n the a r t of reproducing n a t u r a l processes. We have come to r e a l i z e t h a t almost a l l the e x i s t i n g models, ours i n c l u d e d , are qu i t e complicated, and yet c o n t a i n many s i m p l i s t i c assumptions, e s p e c i a l l y i n the s i m u l a t i o n phase of stand dynamics. Moreover, most of them a l l o w f o r only a l i m i t e d number of a l t e r n a t i v e s to be t e s t e d by s i m u l a t i o n . The c o n s i d e r a t i o n of these advantages and drawbacks i n e x i s t i n g models has l e d to some suggestions concerning the i d e a l stand model. T h i s i d e a l model should o f f e r the f o l l o w i n g f e a -t u r e s : 1) be f u l l y s t o c h a s t i c , 2) accept h y p o t h e t i c a l as w e l l as a c t u a l data, 3 ) provide p o s s i b i l i t i e s f o r h a n d l i n g s e v e r a l s p e c i e s simultaneously, or i n sequence, 4) i n t e g r a t e some non-mcnsurational o b s e r v a t i o n s , 5 ) evaluate competition and growth i n a mu l t i - d i m e n s i o n a l u n i v e r s e , 6) tend towards the i n t e g r a t i o n of complete biomass production, 7) allow f o r e x t e r n a l i n t e r f e r e n c e i n the s i m u l a t i o n process, P r e d i c t i o n of m o r t a l i t y l 8 l 8) produce many outputs i n g r a p h i c a l form, 9) c o n t a i n a s o l i - t e s t i n g mechanism, 1 0 ) o f f e r tho p o s s i b i l i t y of being coupled v/ith manage-ment games. Since i t i s p o s s i b l e to i n t r o d u c e p r o b a b i l i t y d i s t r i b u -t i o n s i n the study of p o p u l a t i o n s , i t should bo p o s s i b l e to use these p r o b a b i l i t i e s to reproduce the three b a s i c processes, i . e . s i m u l a t i o n of s p a t i a l p a t t e r n s , of competitive s t a t u s con-t r o l l i n g growth, and of m o r t a l i t y . The d i f f i c u l t y hero i s two-f o l d ; f i r s t i n f i n d i n g the a c t u a l p r o b a b i l i t y d i s t r i b u t i o n s , and second, i n u s i n g tho parameters of these d i s t r i b u t i o n s to reproduce f o r e s t s with s p e c i f i e d c h a r a c t e r i s t i c s . However, as most p r o b a b i l i t y d i s t r i b u t i o n s converge, the simpler forms could serve s i m u l a t i o n purposes. Thus, the i d e a l model should be b u i l t to v i r t u a l l y auto-generate, upon request, an u n l i m i t e d number of h y p o t h e t i c a l f o r e s t stands. T h i s phase i s p r e s e n t l y under i n v e s t i g a t i o n (Newnham and Maloley, 1 9 7 0 ) . At the same time, however, an e q u a l l y l a r g e number of p o s s i b i l i t i e s should be o f f e r e d f o r f e e d i n g a c t u a l data i n . The f i r s t f e a t u r e i s d e s i r a b l e f o r s i m u l a t i o n of a wide spectrum of a l t e r n a t i v e s , whereas tho second i s important i n answering r e a l q u e s t i o n s . In t h i s r e s p e c t , many stands present a mixture of s p e c i e s having, most of the time, q u i t e d i v e r g e n t c h a r a c t e r i s t i c s . The e a s i e s t v/ay would be f o r thorn to be considered i n sequence, r a t h e r than simultaneously. But the s t a t e of the a r t i s the P r e d i c t i o n of m o r t a l i t y 182 main l i m i t i n g f a c t o r here, i . e . i n t e r - s p e c i f i c i n t e r a c t i o n s are s t i l l f a r from being w e l l understood. U n t i l i t i s improved, the amount of a t t e n t i o n given to stand models by p o t e n t i a l users i s g r e a t l y diminished. Up-to-date, i t has been assumed that mensurational data are s u f f i c i e n t to e x p l a i n and p r e d i c t stand dynamics. However, i t i s w e l l known that s o i l c h a r a c t e r i s t i c s , humidity regimes, m i c r o c l i m a t e , m i c r o s i t e , physiography, and a host of other f a c -t o r s g r e a t l y i n f l u e n c e the development of t r e e p o p u l a t i o n s . Consequently, among them, the most important ones should at l e a s t be taken i n t o account, even i f only as l i m i t i n g f a c t o r s . E f f o r t s should a l s o be d i r e c t e d at s i m u l a t i o n of competi-t i o n and growth i n more than two dimensions. U l t i m a t e l y , up to s i x dimensions should form the p i c t u r e , three underground and three above. T h i s would help g r e a t l y i n reproducing a l a r g e r p a r t of the t o t a l biomass development. H o p e f u l l y , these and other t e c h n i c a l a m e l i o r a t i o n s would improve q u a l i t y , r e l i a b i l i t y and u s e f u l n e s s of f u t u r e stand models. Once they become pa r t of complete management games, they s h a l l be used i n s i m u l a t i n g not only growth of f o r e s t s per so, but t h e i r u t i l i z a t i o n as well.. 6.3 CHAPTER SUMMARY Tho second main o b j e c t i v e of t h i s d i s s e r t a t i o n i s reached here, v/ith the development of some methods f o r p r e d i c t i n g mor-t a l i t y . They arc subdivided i n t o two groups: some based on the P r e d i c t i o n of m o r t a l i t y 183 stand approach, some on a tree approach. Methods us i n g the, stand approach to m o r t a l i t y p r e d i c t i o n s are summarized only, s i n c e they have boon d e s c r i b e d i n d e t a i l i i i p r e c e ding chapters. A simple s e m i - s t o c h a s t i c stand model i s d e s c r i b e d , with which i t i s p o s s i b l e to make growth, y i e l d , and m o r t a l i t y es-timates f o r a l i m i t e d number of 10-year p e r i o d s . I t i s based on the t r e e approach, on four p r e d i c t i o n equations, and on a number of m o r t a l i t y t a b l e s . A t a l l y of troo diameters and a height-diameter equation only are needed to simulate stand de-velopment. I f the f i r s t v e r s i o n of the model i s used, one can got d i f f e r e n t answers by s e t t i n g d i f f e r e n t r i s k l e v e l s . Tho second v e r s i o n of tho model generates m o r t a l i t y i n a s t o c h a s t i c manner, a c c o r d i n g to t r e e o.ge, diameter, and h e i g h t . L i m i t e d t e s t s of tho second v e r s i o n i n d i c a t e tlicit the pre-c i s i o n of b a s a l area and cubic volume estimates s t a y s w i t h i n 9 percent of the a c t u a l valuos f o r a 10-year p r e d i c t i o n . Diameter d i s t r i b u t i o n s of dead t r e e s aro skewed, as observed i n nature. Based on t h i s experience i n modeling, some ob s e r v a t i o n s are made about what could bo the i d e a l stand model. CHAPTER 7 184 DISCUSSION 7.1 BASIC DATA The data s t u d i e d here are not f u l l y r e p r e s e n t a t i v e of the c o n d i t i o n s p r e v a i l i n g i n the C o a s t a l Douglas f i r f o r e s t s . In f a c t , the Douglas f i r Region covers approximately 10,000 square m i l e s (6.4 MM a c r e s ) ; i t s t r e t c h e s from c e n t r a l B r i t i s h Colum-b i a to c e n t r a l C a l i f o r n i a , west of the Cascade Range. Stands 170 years and o l d e r occupy 3 m i l l i o n a c r e s (McMahon, 1961), and much of the remaining area supports young-growth. Since our samples are not s c a t t e r e d u n i f o r m l y throughout t h i s r e g i o n , and s i n c e they cover only 27 a c r e s out of about 3 m i l l i o n acres of second-growth f o r e s t s , they cannot adequately p i c t u r e the e x i s t i n g c o n d i t i o n s . Nevertheless, the permanent sample p l o t s chosen represent a good p a r t of the stem-mapped i n f o r m a t i o n p e r t a i n i n g to Douglas f i r , and i n c l u d e some of the o l d e s t permanent p l o t r e c o r d s of n a t u r a l and pla n t e d stands kept i n the P a c i f i c Northwest ( s i n c e 1925). As such, they are most v a l u a b l e . But, heter o g e n e i t y i n p l o t s i z e s , d i v e r s i t y i n measurements taken, i n c o n s i s t e n c i e s i n the i d e n t i f i c a t i o n of m o r t a l i t y causes, and unequal l e n g t h s of time between remeasurements are major shortcomings. The best i n f o r m a t i o n came out of l a r g e p l o t s (one-half acre and l a r g e r ) , stem-mapped a t time of establishment, and remeasured r e g u l a r l y at f i v e - or ten-year i n t e r v a l s , d u r i n g p e r i o d s of 20 years or D i s c u s s i o n 185 more. H a l l (1959), studying the advantages of permanent p l o t s over temporary p l o t s i n e s t i m a t i n g growth, suggested that t h i s advantage was d i m i n i s h i n g with i n c r e a s e d i n t e r v a l s between ob-s e r v a t i o n s (up to 20 years) due to the d i m i n i s h i n g c o r r e l a t i o n between s u c c e s s i v e measurements, and to v a r i a b i l i t y induced by f l u c t u a t i o n s i n growth r a t e s , m o r t a l i t y and ingrowth. The use of p l o t remeasurements as independent o b s e r v a t i o n s i n most of the m u l t i p l e r e g r e s s i o n analyses c a r r i e d out de-serves some comments. C u r t i s (196?) was a l s o faced with t h i s problem of a u t o c o r r e l a t i o n between s u c c e s s i v e o b s e r v a t i o n s taken from the same p l o t s . The s e r i a l c o r r e l a t i o n of r e s i d u a l s would give i n e f f i c i e n t but unbiased estimates of parameters, and cause an underestimation of e r r o r s by co n v e n t i o n a l r e g r e s -s i o n methods. I t can be ap p r e c i a t e d by f i t t i n g a r e g r e s s i o n equation v/ith data from p l o t remeasurements, and by c a l c u l a t i n g a c o e f f i c i e n t of determination with s u c c e s s i v e r e s i d u a l s from each p l o t . When s i g n i f i c a n t a u t o c o r r e l a t i o n i s detected, only one o b s e r v a t i o n per p l o t must bo randomly chosen and used i n b u i l d i n g a new r e g r e s s i o n equation. In h i s study, C u r t i s (1967) showed that a u t o c o r r e l a t i o n was s i g n i f i c a n t i n y i e l d equations, but not i n growth equations. Therefore, tho problem was neglec-ted i n the present study by assuming that growth r a t e s and m o r t a l i t y r a t e s were e s s e n t i a l l y dependent on the same v a r i -a b l e s . D i s c u s s i o n Igg 7.2 NATURE OF MORTALITY That r e g u l a r m o r t a l i t y i s p a r t of the growth process has been demonstrated i n Chapter 4> i n which r e g r e s s i o n equations were c a l c u l a t e d to r e l a t e m o r t a l i t y r a t e s and growth r a t e s to the same variables,namely age, stand d e n s i t y and s i t e index. For example, annual percent m o r t a l i t y (APCM) = f(AGE, SI) annual volume of m o r t a l i t y (VOLMOR) = f(AGE, YIELDN, SI) gross volume growth r a t e (GVOINC) = f(AGE, BAN, SI) mean annual volume increment (MAIN) = f(AGE, BAN, SI) annual b a s a l area m o r t a l i t y (BAMOR) = f(AGE, SI) gross b a s a l area growth r a t e (GBAINC) = f(AGE, SI) mean annual b a s a l area increment (BAIN) = f(AGE, SI) However, i t has a l s o been shown that m o r t a l i t y i s indeed much more v a r i a b l e than growth; and yet, as l i t t l e i r r e g u l a r m o r t a l i t y as p o s s i b l e was entered i n t o the o r i g i n a l data. Lee (1969) suggested that p l o t s i n which the t o t a l number of dead t r e e s exceeds two standard d e v i a t i o n s from the mean number of dead t r e e s i n a l l p l o t s s t u d i e d should be d i s c a r d e d . T h i s c r i -t e r i o n would be a c c e p t a b l e only f o r p l o t s grouped by age Table XVII Table XVIII Table XX Table XXII Table XV Table XX Table XXII D i s c u s s i o n 187 c l a s s e s . Table XI shows that more e l a s t i c l i m i t s were t o l e r -ated i n our study; tho aim was to d i s c o v e r excessive l o s s e s . P l o t s were r e j e c t e d a f t e r p e r i o d i c amounts of m o r t a l i t y , i t s s p a t i a l arrangement and i n t e r v a l s between p l o t measurements were taken i n t o account, and a f t e r examination of m o r t a l i t y causes on t a l l y sheets. Rejected p l o t s had: 1) negative not p e r i o d i c annual increments both i n cubic volume and b a s a l area, 2) an u n u s u a l l y l a r g e p o r p o r t i o n of t r e e s k i l l e d d u r i n g short p e r i o d s of time (3 to 5 y e a r s ) , and 3) some d i s e a s e s or other i r r e g u l a r causes i d e n t i f i e d i n the t a l l y book, and c r e a t i n g contagious d i s p e r s i o n of dead t r e e s . A comparable method was used by S t a e b l e r (1953); he considered that i r r e g u l a r m o r t a l i t y had occurred i n p l o t s where the " p c r - a c r c por-decado" l o s s e s i n b a s a l area were so great that they would not bo expected to occur o f t e n e r than one decade i n 100 years. His r e j e c t e d p l o t s had s u f f e r e d seven times tho p r e d i c t e d r e g u l a r l o s s . The a n a l y s i s of i r r e g u l a r and c a t a s t r o p h i c m o r t a l i t y would have r e q u i r e d a s p e c i a l study by i t s e l f , not based on perma-nent p l o t r e c o r d s , but cn broad-scalo surveys, r e g i o n a l c l i -matic r e c o r d s , physiographic maps, and the l i k e . Since so l i t t l e has been done on that l i n e , some r u l e s of thumb were suggested i n Appendix VI by which r e g u l a r m o r t a l i t y e s t i m a t e s could be a d j u s t e d somewhat f o r these u n p r e d i c t a b l e events. I f they were not, much of t h e i r p r e c i s i o n would be l o s t because of tho known f a c t that 20 to 25 percent of annual l o s s e s i n the P a c i f i c Northwest are a t t r i b u t a b l e to f i r e , d i s e a s e s , i n s e c t s , D i s c u s s i o n 188 and other agents (Anon., 1967, M e t c a l f , 1968) . In view of the f a c t that the d i s t i n c t i o n between r e g u l a r and i r r e g u l a r m o r t a l i t y i s not based on completely o b j e c t i v e c r i t e r i a , and that c u r r e n t i n f o r m a t i o n from p l o t r e c o r d s does not help i n r e f i n i n g the c l a s s i f i c a t i o n , i t might be more ap p r o p r i a t e to d i f f e r e n t i a t e only between n a t u r a l and a c c i d e n -t a l m o r t a l i t y . T h i s might s i m p l i f y the i d e n t i f i c a t i o n both at the stand and at the t r e e l e v e l . 7.3 AMOUNT, TIMING AND DISTRIBUTION OF MORTALITY One of the most u s e f u l r e s u l t s of t h i s p r o j e c t i s presen-ted i n F i g u r e 13 where the percentage frequency d i s t r i b u t i o n of dead t r e e s i s r e l a t e d to average stand diameter and age. From i t , Table XXXIII has been b u i l t to d i f f e r e n t i a t e between mor-t a l i t y of ingrowth ( t r e e s 1 to A i n c h e s ) , growing stock ( t r e e s 5 to 10 i n c h e s ) and sawtimber ( t r e e s 11 i n c h e s +). T h i s break-down i s important because i t i n d i c a t e s the k i n d of stand i n which m o r t a l i t y i s , or would be, r e c o v e r a b l e , given c e r t a i n management and m e r c h a n t a b i l i t y l e v e l s . At the present time, i t i s known that i n the S t a t e s of Oregon and Washington, the annual m o r t a l i t y amounts to roughly l . i f b i l l i o n cubic f e e t c f growing stock, and to 6.9 b i l l i o n board f e e t (approx. 1.15 b i l l i o n cubic f e e t ) of sawtimber, of which 15 to 20 percent i s being harvested mainly among the saw-timber f r a c t i o n (Anon., 1967). Thus, from Table XXXIII, one can i n f e r that salvage i s mainly c a r r i e d out i n o l d e r immature D i s c u s s i o n 189 TABLE XXXIII BREAK DOWN OF REGULAR MORTALITY INTO INGROWTH, GROWING STOCK AND SAWTIMBER1 AVERAGE STAND DIAMETER STAND PERCENTAGE OF TOTAL MORTALITY IN (inches) AGE INGROWTH GROWING STOCK SAWTIMBER PURE NATURAL STANDS - GROUP I 10.6-14.5 14 75 11 14.6-18.5 7 63 30 16.5-22.5 0 23 77 50 16 70 14 60 21 68 11 70 9 58 33 80 6 30 64 DOUGLAS FIR IN MIXED STANDS - GROUPS I I , I I I , IV 10.6-14.5 12 88 none 16-35 94 6 none 36-55 29 71 none 56-75 21 79 none WIND RIVER SPACING PLANTATION 2.6-4-5 94 6 none 4.6-6.5 79 21 none 6.6-8.5 47 53 none 3.6-10.5 31 69 none 30 93 7 none 40 82 18 none Data from F i g u r e 13. Ingrowth = t r e e s 1 to 4 inches; growing stock = t r e e s 5 to 10 inches; sawtimber = t r e e s 11 inches + i n diameter measured ou t s i d e bark at bre a s t h e i g h t . D i s c u s s i o n 190 f o r e s t s , 18 or more in c h e s i n diameter, or 80 or more years o l d , where 64 to 77 percent of the t o t a l annual m o r t a l i t y has reached sawtimber s i z e . T h i s t a b l e can a l s o be r e f e r r e d to i n a p p r e c i a t i n g what p r o p o r t i o n of some p r e d i c t e d m o r t a l i t y s h a l l be i n small and l a r g e timber. For i n s t a n c e , p r e d i c t e d annual l o s s e s (from Tables XV, XVI or XVIII) i n pure stands c f Douglas f i r , 12 i n -ches i n diameter, should be composed of approximately 75 per-cent small t r e e s and 11 percent sawlogs; between the ages of 55 and65» l o s s e s should be composed of about 21 percent i n -growth, 68 percent growing stock and 11 percent sawlogs. In mixed stands, there should not be any sawtimber m o r t a l i t y among Douglas f i r t r e e s before the age of 75* In a d d i t i o n to c a l c u l a t e d amounts of m o r t a l i t y , a 4 per-cent r e d u c t i o n on the standing gross merchantable volumes should be made f o r decay ( s t a r t i n g p o i n t of the m o r t a l i t y p r o c e s s . . . ) , waste and breakage l i k e l y to be encountered i n h a r v e s t i n g saw-timber i n immature f o r e s t s . In mature f o r e s t s , up to 52 per-cent could be s u b t r a c t e d , depending on the e x t e r n a l appearance of s o - c a l l e d l i v i n g t r e e s (B.C.F.S., 1966). Such i n f o r m a t i o n completes what could be obtained from F i g u r e s 17 and 18 concerning average m o r t a l i t y diameter, and from S t a e b l e r (1953)> concerning the percentage of number of t r e e s expected to d i e i n some pr e - d e f i n e d crown c l a s s e s . His equations were devised to p r e d i c t the percentage of dominant and codominant, or of i n t e r m e d i a t e and suppressed t r e e s l i k e l y D i s c u s s i o n 191 to d i e i n the next ten-year p e r i o d . They are: APCM = 4.96 + 0.08 AGE - 0.41 DBH Do., Coco. R = 0.266 N = 72 f o r DBH = 6 to 30 inches; AGE = 30 to 90 years APCM = -13.01 + O.54 SI + 0.61 AGE - 7.83 DBH Int., Supn. R = 0.71 N = 72 f o r DBH = 2 to 16 inches; AGE = 30 to 90 y e a r s Information about m o r t a l i t y from time of germination on-ward was not i n c l u d e d i n the present study because there has been ( e s p e c i a l l y i n n a t u r a l stands) h a r d l y any systematic r e c o r d kept of amount, t i m i n g and d i s t r i b u t i o n of m o r t a l i t y comparable to the k i n d of data gathered i n o l d e r stands. G i l l (1950), Smith and Ker (1957), Persson (1964), and Smith et a l . (1965, 1966a) have d i s c u s s e d amounts and d i s t r i b u t i o n s of r e g e n e r a t i o n and m o r t a l i t y i n j u v e n i l e stands. A common assumption seems to be that much v a r i a t i o n can occur at e a r l y stages without appre-c i a b l y a f f e c t i n g subsequent stand development. Some open spaces are l i k e l y to be f i l l e d by n a t u r a l ingrowth or by " f i l l -i n " p l a n t i n g o p e r a t i o n s . Ingrowth was i n d i r e c t l y taken i n t o account by i n c l u d i n g i n our a n alyses a l l t r e e s 1.6 i n c h e s i n diameter and l a r g e r , which i s L i n c h e s below the lower l i m i t of what i s normally considered as growing stock. T h i s technique was recommended by Spurr (1952) and Husch (1963) f o r stand t a b l e p r o j e c t i o n s . D i s c u s s i o n 192 N i c h o l s (1966), i n v e s t i g a t i n g on growth of young Douglas f i r , found that the number of t r e e s added to the stand (ingrowth minus m o r t a l i t y ) i n c r e a s e d with d e c r e a s i n g stand d e n s i t y . P o s i t i v e c o r r e l a t i o n c o e f f i c i e n t s given i n Tables XII, XIII and XIV between annual m o r t a l i t y and number of t r e e s per acre i l l u s t r a t e tho same concept without d i r e c t r e f e r e n c e to ingrowth; i n that sense, i t was p r e v i o u s l y s t a t e d that ingrowth has been n e g l e c t e d . 7.4 SPATIAL PATTERNS Re c o g n i t i o n of s p a t i a l p a t t e r n s i s u s e f u l i n f o r e s t r y f o r d e v e l o p i n g e s t i m a t o r s of s t o c k i n g and stand d e n s i t y , f o r i n -d i c a t i n g the best number, s i z e and shape of samples i n inven-t o r y works, f o r d e s c r i b i n g and c l a s s i f y i n g p l a n t a s s o c i a t i o n s by presence, absence, abundance and cover, f o r c r e a t i n g a r t i f i -c i a l p o p u l a t i o n s , f o r s i m u l a t i n g f o r e s t growth, and even f o r d e s i g n i n g h a r v e s t i n g machinery ( P a i l l c ' and McGreevy, 1970, Kershaw, 1957, Newnham, 1965, 1968). B a s i c a l l y , two methods have been used i n order to evaluate p o p u l a t i o n d e n s i t y and to d e t e c t whether or not the i n d i v i d u a l s are randomly d i s p e r s e d , and, i f not, to which degree. The f i r s t method i s based on the count of p o i n t s f a l l i n g i n random or contiguous quadrats l a i d over an area that may r e p r e s e n t a complete p o p u l a t i o n or a sample of i t (Webb, 1969). The sec-ond method c o n s i s t s of a n a l y z i n g the d i s t r i b u t i o n of d i s t a n c e s measured from random p o i n t s or p l a n t s to t h e i r nearest D i s c u s s i o n 193 neighbour c r neighbours (Dice, 1952, M o r i s i t a , 1957, Mawson, 1968). In both, the mean number or tho average d i s t a n c e i s an i n d i c a t o r of d e n s i t y , and t h e i r v a r i a n c e i s an i n d i c a t o r of d i s p e r s i o n . Aggregation, heterogeneity, dumpiness or over-d i s p e r s i o n are i n d i c a t e d by a va r i a n c e l a r g e r than the mean; r e g u l a r i t y , u n i f o r m i t y , or u n d e r - d i s p e r s i o n i n arrangements aro i n d i c a t e d by a va r i a n c e l e s s than the mean. When the var i a n c e equals the mean, the assumption i s that the p a t t e r n of i n d i v i d u a l s was generated by a Poisson process, and there-f o r e , t h a t they are randomly d i s t r i b u t e d over tho area. Ran-domness hqs boon c l e a r l } ' d e f i n e d by C l a r k and Evans (1954) • In p r a c t i c e , t h i s hypothesis i s t e s t e d a f t e r f i t t i n g observed quadrat f r e q u e n c i e s or d i s t a n c e measures to a Po i s s o n d i s t r i b u -t i o n . When the hyp o t h e s i s i s r e j e c t e d , the f i t to some other d i s c r e t e p r o b a b i l i t y d i s t r i b u t i o n s i s analyzed, and the degree of non-randomness i s evaluated. More than 300 papers have d e a l t with some aspects of pat-t e r n e v a l u a t i o n s i n c e 1883; they were reviewed by Goodall (1952) and McGroevy (1969). Among those, few d e a l t with a c t u d p o p u l a t i o n s of t r e e s , and only two ( F o s t e r and Johnson, 1963a, and Payandeh, 1968) considered Douglas f i r stands. In the present study, the variance-mean r a t i o of quadrat frequency d i s t r i b u t i o n s (VARMEAN) was considered as an inde -pendent v a r i a b l e that could, together with s t o c k i n g and stand d e n s i t y (measured by complete enumeration), c h a r a c t e r i z e s i t e occupancy and account f o r part of the v a r i a t i o n i n growth and D i s c u s s i o n 194 m o r t a l i t y . On the one hand, i t was found to be h i g h l y c o r r e -l a t e d with s t o c k i n g and d e n s i t y : i n n a t u r a l stands, an i n c r e a s e i n VARMEAN ( v a r y i n g between 0.43 and 2.4-5) was r e l a t e d to a decrease i n s i t e occupancy (Table XXVIII), i . e . more hetero-geneous d i s p e r s i o n s were a s s o c i a t e d with a lower percentage crown c l o s u r e and a smaller b a s a l area per aero; i n p l a n t a -t i o n s , t h i s trend was overshadowed by age e f f e c t s (Table XXIX). M o r t a l i t y r a t e s were not a f f e c t e d by changes i n p a t t e r n (Tables XII and X I I I ) . On the other hand, because of t h i s c o r r e l a t i o n with stock-i n g and d e n s i t y , VARMEAN d i d not s i g n i f i c a n t l y improve growth or m o r t a l i t y r e l a t i o n s h i p s a l r e a d y i n c l u d i n g other stand para-meters much e a s i e r to measure. However, P a i l l e and McGreevy (1970) showed that, by not r e c o g n i z i n g s p a t i a l p a t t e r n s , extremely l a r g e v a r i a t i o n s of estimates could bo obtained. They sampled two one-acre p l o t s (Olympic 2 and R a i n i e r 9, i n Table II) with randomly l o c a t e d s u b p l o t s of f i v e d i f f e r e n t s i z e s . Both p l o t s were quite comparable, except f o r p a t t e r n . In Olympic 2 (VARMEAN = 1.0), estimates of number of t r e e s per acre v a r i e d between 223 and 64O ( a c t u a l value, 388); average stand diameter v a r i e d between 7-6 and 10.0 i n c h e s ( a c t u a l value, 9.1); i n R a i n i e r 9 (VARMEAN = 1.78), number of t r e e s f l u c t u a t e d between 50 and 2700 ( a c t u a l value, 413)? and diameter went from 7.0 to 17.2 i n c h e s ( a c t u a l value, 10.0). Thus, s p a t i a l p a t t e r n i s important to r e c o g n i z e i f v a r i -a t i o n s of 650 percent, as obtained above, are to be avoided i n D i s c u s s i o n I95 sampling experiments. Otherwise, i t i s not a c r i t i c a l v a r i a b l e h i g h l y r e l a t e d with growth and y i e l d of Douglas f i r . Webb (1969) suggested that the mean of the quadrat frequency d i s t r i -b u t i o n would be a dynamic stand parameter most u s e f u l f o r simu-l a t i n g f o r e s t stands because i t was found to f l u c t u a t e with age: here, the variance-mean r a t i o was adopted f o r that purpose because the means were not comparable due to v a r i a t i o n s i n quadrat s i z e s . VARMEAN was r e l a t e d to age and diameter or age and d e n s i t y (Chapter 6, s e c t i o n 6 . 2 . 1 . 4 ) . A f t e r c o n s i d e r i n g the d i f f i c u l t y of measurement and the l a c k of tendency f o r Douglas f i r to become e s t a b l i s h e d i n clumps, i t i s d o u b t f u l whether s p a t i a l p a t t e r n w i l l ever be measured except f o r r e s e a r c h purposes. The f a c t i s that ex-treme dumpiness, which a f f e c t s f o r e s t y i e l d s and t h e i r e s t i -mates, seems to be induced mainly be i r r e g u l a r m o r t a l i t y ; and, quit e o f t e n , i t can be r e a d i l y detected by o c u l a r o b s e r v a t i o n s , and taken i n t o account or c o r r e c t e d wherever f e a s i b l e . From a t h e o r e t i c a l p o i n t of view, i t might be i n t e r e s t i n g to compare the answers from t h i s study with those obtained a f t e r sampling with random r a t h e r than contiguous quadrats, or with s i m i l a r quadrat s i z e s f o r s u c c e s s i v e p l o t measurements. A major problem, e s p e c i a l l y when c o n s i d e r i n g dead, t r e e s , con-s i s t s i n determining the smaller number that can l o g i c a l l y be studied i n a given p a t t e r n a n a l y s i s . There appears to be no g u i d e l i n e s i n the l i t e r a t u r e on that t o p i c . D i s c u s s i o n 196 7.5 SITE OCCUPANCY The proposed method of measuring s i t e occupancy (Appendix I I I ) i n the computer r a t h e r than i n the f i e l d or from a e r i a l photographs i s h e l p f u l i n p r e d i c t i n g m o r t a l i t y . A c t u a l l y , i t was shown i n Table XV that, when coupled with age, cumulative crown c l o s u r e or crown space occupation s.re b e t t e r m o r t a l i t y e s t i m a t o r s than c o n v e n t i o n a l expressions of stand d e n s i t y . Although the use of these v a r i a b l e s i n b u i l d i n g m o r t a l i t y volume t a b l e s (by c l a s s e s of stand height and crown c l o s u r e of dead trees ) has not been t e s t e d here, they could e v e n t u a l l y serve that purpose and complete the i n f o r m a t i o n about m o r t a l i t y gathered from a e r i a l photographs (Wort, 1969). T h e i r e f f e c t i v e n e s s i n p r e d i c t i n g f o r e s t growth has not been demonstrated e i t h e r , but i t i s r e a l . For i n s t a n c e , crown space occupation made a s i g n i f i c a n t c o n t r i b u t i o n when added to a r e g r e s s i o n of y i e l d i n cubic volume on age, bas a l area and s i t e index. T h e o r e t i c a l l y , i t i s l o g i c a l to t h i n k that not only s t o c k i n g but a l s o some expressions of s p a t i a l d i s t r i b u t i o n and t r e e p o t e n t i a l can improve the r e l a t i o n s h i p . However, since these stand and s i t e measurements cannot f u l l y i n t e g r a t e a l l tho f a c t o r s r e s p o n s i b l e f o r tre e growth, and si n c e they are not a l l p e r f e c t l y independent, the e f f o r t i n v o l v e d i n combining them to e x p l a i n an i n c r e a s i n g l y l a r g e r p r o p o r t i o n of the growth process i s subject to the law of d i m i n i s h i n g r e t u r n s . In p r a c t i c e , a much l e s s s o p h i s t i c a t e d approach i s taken i n e v a l u a t i n g s i t e occupancy (McCormack, 1967, Sayn-Wittgenstein D i s c u s s i o n 197 and Aldred, 1 9 6 9 ) . In d i s c u s s i n g tho use of s i t e occupancy, one must r e a l i z e that l o s s e s due to r e g u l a r m o r t a l i t y of Douglas f i r are only h a l f of the t o t a l l o s s i n c u r r e d by l a c k of management. I t has been shown by s e v e r a l authors that gross y i e l d s could be 50 percent l a r g e r i n i n t e n s i v e l y managed stands than those com-monly c a l c u l a t e d f o r the C o a s t a l f o r e s t s ( B r i e g l e b , 1952, Karlborg, 1961, Staebler, 1963, Bruce, 1 9 6 9 ) . Therefore, i f t h i s new l i m i t was set as an optimum, the a c t u a l p r o p o r t i o n of s i t e c a p a c i t y being used f o r t r e e growth would be much smaller than c a l c u l a t e d i n Chapter 5, hut the r e d u c t i o n due to mortal-i t y would a l s o decrease i n p r o p o r t i o n . 7.6 PREDICTION OF MORTALITY I t was s t a t e d that there i s no sound procedure f o r making m o r t a l i t y p r e d i c t i o n s other than consecutive r e i n v e n t c r i e s of permanent sample p l o t s (Avery, 1 9 6 7 ) . In the l i g h t of the present study, i t i s evident that even when such p l o t s are a v a i l a b l e , m o r t a l i t y p r e d i c t i o n s are d i f f i c u l t to make with p r e c i s i o n f o r a number of t e c h n i c a l reasons mentioned e a r l i e r . One of the most s e r i o u s i s b e l i e v e d to be the r e d u c t i o n to a common annual b a s i s of o b s e r v a t i o n s taken over uneven p e r i o d s of time. Another c o n s i s t s i n drawing the l i n e between r e g u l a r and i r r e g u l a r m o r t a l i t y ; t h i s i s done more or l e s s a r b i t r a r i l y on the b a s i s of c r i t e r i a d i s c u s s e d i n Chapter 2. U n t i l r e -g i o n a l p r o b a b i l i t i e s of c a t a s t r o p h i c m o r t a l i t y are a v a i l a b l e , D i s c u s s i o n 198 though, one must be s a t i s f i e d with p r e d i c t e d r a t e s of r e g u l a r m o r t a l i t y based on stand and s i t e c h a r a c t e r i s t i c s , p l u s some r u l e s of thumb designed to account f o r ' i r r e g u l a r events. These c o n s i d e r a t i o n s prevented tho author from developing vory s o p h i s t i c a t e d r e g r e s s i o n models to r e l a t e m o r t a l i t y to f o r e s t parameters, but prompted the a n a l y s i s of the question on a p r o b a b i l i s t i c b a s i s . A s i m u l a t i o n model seemed the best means to t e s t the v a l i d i t y of that approach to m o r t a l i t y pre-d i c t i o n . Based on a l i m i t e d number of t r i a l s , tho f o l l o w i n g comments can be formulated about the model i t s e l f and the out-puts. 7.6.1 MORTALITY TABLES The r e l i a b i l i t y of each m o r t a l i t y curve i s not the same a c r o s s the range of age t e s t e d because they were drawn v/ith d i f f e r e n t numbers of o b s e r v a t i o n s . More i n f o r m a t i o n would bo needed e s p e c i a l l y i n age c l a s s e s 20-30 and 80-90 y e a r s . Even i f the c o e f f i c i e n t s of determination given i n Appendix V are very s i g n i f i c a n t , every curve l a c k s p r e c i s i o n i n the c r i -t i c a l r e g i o n , near the o r i g i n . T h i s l a c k of harmonization among and w i t h i n tho curves would bo tho major cause of v a r i -a t i o n i n the- r e s u l t s . 7.6.2 DIAMETER INCREMENT In d e v e l o p i n g the diameter increment f u n c t i o n s pre-sented i n Tabic XXXII, tho author d i d not f o l l o w the procedure D i s c u s s i o n 199 explained i n Husch (1963), which i m p l i e s that each t r e e must be bored at b r e a s t height to determine the increment d u r i n g a given p e r i o d of time i n s i d e bark. Instead, two s u c c e s s i v e diameter measurements ou t s i d e bark wore chosen on about 10 per-cent of the i n d i v i d u a l s i n each p l o t , and tho average annual increment computed by d i v i d i n g the d i f f e r e n c e by the l e n g t h of tho p e r i o d . T h i s average annual increment i n t r e e s i z e was then r e l a t e d to other v a r i a b l e s measured at the beginning of the p e r i o d . The equations obtained by m u l t i p l e r e g r e s s i o n techniques were not c o n d i t i o n e d through the o r i g i n of tho axes because a wide range of diameter was t e s t e d i n each group of sample p l o t s . The main source of e r r o r i n t h i s procedure seems to be the averaging of increment over uneven p e r i o d s of time ( f l u c t u a t i n g between 3 and 20 y e a r s ) . S e v e r a l i n v e s t i g a t o r s s t u d i e d the r e l a t i o n s h i p between diameter and stand parameters. Tho Douglas f i r second-growth Management Committee recognized four v a r i a b l e s r e l a t e d to diameter growth, namely s i t e index, crown c l a s s , diameter and age (Anon., 1947). Ken (1953) a s s o c i a t e d decadal r a d i a l growth to diameter, crown c l a s s , number of " s i d e s f r e e " of competition, and crown q u a l i t y . Warrack (1959) found that i n i -t i a l diameter was the best s i n g l e c r i t e r i o n f o r e s t i m a t i n g diameter growth; i n c l u s i o n of crown width d i d not improve tho r e l a t i o n s h i p . Smith ot a l . (1961) found that diameter, s i t e index, crown width and crown c l a s s wore tho most u s e f u l inde-pendent v a r i a b l e s . Newnham (1964) simulated 5-year r a d i a l D i s c u s s i o n 200 increments by u s i n g diameter at age 10, diameter at the begin-ning of the p e r i o d and age, L i n (1969) used seven v a r i a b l e s i n c l u d i n g growing space, diameter, s i t e index and age, Trimble (1969) st u d i e d diameter growth, of hardwood t r e e s i n r e l a t i o n to crown width, diameter, b a s a l area and s i t e q u a l i t y . Among the f a c t o r s chosen to run the stand model (diameter, s i t e index and age), diameter at tho beginning of the p e r i o d was, as i n a l l the s t u d i e s j u s t c i t e d , the most important var-i a b l e ; age had a negative i n f l u e n c e on increment. S i t e index was p o s i t i v e l y c o r r e l a t e d with t r e e increment but made a nega-t i v e c o n t r i b u t i o n i n every equation i n c l u d i n g age (except f o r Wind R i v e r p l a n t a t i o n ) ; i t was the l e a s t s i g n i f i c a n t f every f u n c t i o n l i s t e d i n Table XXXII. Improvements i n the present stand model could c e r t a i n l y be made by r e p l a c i n g the diameter f u n c t i o n s with others d e r i v e d from stem analyses, but, i t i s not c e r t a i n whether the g a i n i n p r e c i s i o n would bo a p p r e c i a b l e over the f u l l range of co n d i -t i o n s represented by tho f u n c t i o n s , u n l e s s a much gr e a t e r e f f o r t i s made. At the outset, any l o c a l increment f u n c t i o n could be used to run tho model. 7.6.3 MORTALITY GENERATORS In the present stand model, p r o b a b i l i t i e s of mor-t a l i t y r e p l a c e tho competition i n d i c e s b u i l t i n t o most other e x i s t i n g models. Since they have been c a l c u l a t e d f o r undis-turbed n a t u r a l stands, they may not apply i n s i m u l a t i n g D i s c u s s i o n 201 s i l v i l c u t u r a l treatments. However, there i s a l a r g e acreage of e x t e n s i v e l y managed second-growth f o r e s t s i n the Douglas f i r Region f o r which m o r t a l i t y estimates are needed, and vi'here m o r t a l i t y t a b l e s can be used. M o r t a l i t y generator Model I d e s c r i b e d i n s e c t i o n 6 . 2 . 1 . 5 could bo run p r o f i t a b l y to make d e c i s i o n s about the nature, timing, and extent of some contemplated t h i n n i n g o p e r a t i o n s . The m o r t a l i t y curves themselves or the output from the stand s i m u l a t i o n would be put to p r o f i t i n d e v i s i n g marking r u l e s . M o r t a l i t y generator Model I I has been t e s t e d hero merely to give an i d e a of the k i n d of answers that could be obtained by u s i n g tho tre e approach to m o r t a l i t y p r e d i c t i o n s . I t should be r e f i n e d before becoming f u l l y o p e r a t i o n a l . For i n s t a n c e , p r o b a b i l i t i e s of m o r t a l i t y based on past diameter increment should be b u i l t i n ; l i m i t a t i o n s as to the maximum s i z e of t r e e s that are l i k e l y to d i e should be imposed; p r e d i c t i o n p e r i o d s should be m o d i f i a b l e at w i l l . 7 . 6 . 4 PRECISION OF PREDICTIONS The p r e c i s i o n of the p r e d i c t i o n s can only be t e s t e d a g a i n s t some a c t u a l . s t a n d s which have many chances not to be f u l l y r e p r e s e n t a t i v e of the gen e r a l m o r t a l i t y p a t t e r n . More-over, the n e a r l y constant presence c f some i r r e g u l a r m o r t a l i t y , and the t h r e a t of c a t a s t r o p h i c l o s s e s makes the d i s c u s s i o n about the p r e c i s i o n of estimates q u i t e hazardous. Since a very l i m i t e d number c f outputs has been t e s t e d i n D i s c u s s i o n 202 each group of p l o t s , no s t a t i s t i c s v/ere computed to t e s t the p r e c i s i o n of the model other than a percent d e v i a t i o n between simulated and observed stand c h a r a c t e r i s t i c s (Appendix V I I I ) . 203 CHAPTER 8 CONCLUSION T h i s d i s s e r t a t i o n on r e g u l a r m o r t a l i t y of Douglas f i r , based almost e x c l u s i v e l y on the study of permanent sample p l o t s e s t a b l i s h e d f o r growth and y i e l d analyses, l e a d s to a number of c o n c l u s i o n s concerning the nature and p o s s i b i l i t i e s of p r e d i c -t i o n of the process. 1. Estimates of expected r e g u l a r m o r t a l i t y can be made with an acceptable degree of p r e c i s i o n , based s o l e l y on some stand and s i t e c h a r a c t e r i s t i c s . Hov/cvcr, due to the th r o a t of i r r e -g u l a r and c a t a s t r o p h i c m o r t a l i t y , much of the confidence i n such estimates i s o f t e n l o s t . Regular m o r t a l i t y i s best expressed i n absolute number of i n d i v i d u a l s per acre per year, as a f u n c t i o n of stand ago and stand d e n s i t y , age and s t o c k i n g , or ago and average t r e e s i z e . However, more than 36 percent of the v a r i a t i o n i n observed mor-t a l i t y i s not accounted f o r by these v a r i a b l e s . In general, the c o n s i d e r a t i o n of a l a r g o number of i n t e r c o r r e l a t o d stand and s i t e parameters i s not more meaningful, and does not account f o r an a p p r e c i a b l y l a r g e r p r o p o r t i o n of the t o t a l v a r i a t i o n i n m o r t a l i t y . 2. F a c t o r s a f f e c t i n g stand growth and y i e l d a l s o a f f e c t com-p e t i t i o n m o r t a l i t y . Gross growth r a t e s , net y i e l d s and Conclusion ZOU m o r t a l i t y i n b a s a l area and cubic volume were a l l expressed as l i n e a r f u n c t i o n s of stand age, stand d e n s i t y and s i t e index. 3. The r a t e of i n c r e a s e i n average diameter of dead t r e e s i s about one-half the r a t e of i n c r e a s e i n stand diameter. • Skewed diameter and height frequency d i s t r i b u t i o n s of l i v i n g t r e e s are a s s o c i a t e d v/ith h i g h l e v e l s of stand d e n s i t y . Although t h e i r c o e f f i c i e n t of v a r i a t i o n decreases with i n c r e a s i n g age and average t r e e s i z e , they remain skewed over l o n g p e r i o d s of time. On the other hand, a l l observed diameter d i s t r i b u t i o n s of t r e e s l o s t to competition between the ages of 20 and 90 years were skewed, whatever the d i s t r i b u t i o n of l i v e t r e e s . The sawtimbor-size p r o p o r t i o n of m o r t a l i t y d i d not become l a r g e r than the growing stock p r o p o r t i o n before the age of 80 years i n n a t u r a l stands, or before they reached an average diameter of 16 i n c h e s . In mixed stands, none of the m o r t a l i t y could bo converted i n t o sawlogs even at age 73; i n p l a n t a t i o n s , more than 30 percent of the annual m o r t a l i t y was s m a l l e r than 5 i n c h e s i n stands of 9 to 11 i n c h e s . U. In the absence of i r r e g u l a r causes of m o r t a l i t y , or of other s p e c i e s , l i v e and dead Douglas f i r t r e e s do not occur i n clumpy p a t t e r n s . A f t e r p l a n t i n g i n uniform arrangements or a f t e r major d i s t u r b a n c e s i n n a t u r a l stands, they tend to become randomly d i s p e r s e d with time, tho f a s t e s t r a t e of change occur-r i n g i n the youngest and densest stands. P o p u l a t i o n s of l i v e Conclusion 205 and dead t r e e s , sampled v/ith contiguous qua.drats v a r y i n g widely i n s i z e and number, give a high p r o p o r t i o n of negative binomial frequency d i s t r i b u t i o n s . Tho variance-moan r a t i o of those quadrat d i s t r i b u t i o n s was found to bo a u s e f u l parameter to generate m o r t a l i t y p a t t e r n s , 5. M o r t a l i t y t a b l e s based on a l a r g o number of t r e e s observed d u r i n g a l o n g p e r i o d of time have proved to bo a r e l i a b l e way of p r e d i c t i n g m o r t a l i t y . They serve i n q u a n t i f y i n g changes i n the r e l a t i v e c a p a b i l i t y of each t r e e as the stand ages. They are based on the assumption that t r e e s having the same r e l a t i v e c h a r a c t e r i s t i c s as t h e i r predecessors by comparison to the f o r -est i n which they grow w i l l behave the same way. 6. The stand model, created mainly f o r e v a l u a t i n g amount, ti m i n g and d i s t r i b u t i o n of m o r t a l i t y , can a l s o serve i n simula-t i n g f o r e s t growth. While not being more p r e c i s e than many other c u r r e n t methods of growth p r e d i c t i o n , i t i s more ver s a -t i l e i n that 1) i t can accommodate a c t u a l or h y p o t h e t i c a l data; 2) i t i s both d e t e r m i n i s t i c and s t o c h a s t i c ; 3) i t generates i n f o r m a t i o n concerning frequency d i s t r i b u t i o n s of dead t r e e s and t h e i r s p a t i a l arrangement; and L) i t pr o v i d e s f a c i l i t i e s f o r t a k i n g i r r e g u l a r m o r t a l i t y i n t o account. In t h i s study, i t was considered tha.t i r r e g u l a r m o r t a l i t y had occurred i n sample p l o t s , when 1) not p e r i o d i c annual Conclusion 206 increments both i n cubic volume and b a s a l area were negative, 2) t r e e s k i l l e d d u r i n g short p e r i o d s of time (3 to 5 yoars) represented an unusually l a r g e p r o p o r t i o n of the l i v i n g t r e e s , and 3) some d i s e a s e s or other causes were i d e n t i f i e d i n t a l l y books, and c r e a t e d contagious d i s p e r s i o n s of dead t r e e s . 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The past, present and fu t u r e of permanent sample p l o t s i n the f o r e s t s of B r i t i s h Columbia. Univ. of B.C., Fac. For., B.S.F. T h e s i s . 5 1 pp. TRIMBLE, G.R. J r . 1 9 6 9 . Diameter growth of i n d i v i d u a l t r e e s - the e f f e c t of c e r t a i n t r e e and environmental f a c t o r s on the growth of s e v e r a l s p e c i e s , U.S.D.A. F o r e s t S e r v i c e , NE. F o r e s t Expt. Sta., Upper Darby, Pa. 2 5 pp. TURNBULL, K.J., G.R. LITTLE, AND G.E. HOYER. 1 9 6 3 . Compre-hensive tree-volume t a r i f t a b l e s . Dept. of Nat. Res., State of Wash. 2 3 pp. WALLIS, G.W. AND G. REYNOLDS. 1 9 6 5 . The i n i t i a t i o n and spread of p o r i a w e i r i i r o o t r o t of Douglas f i r . Can Dept. For., For. Entomo. and Patho. Br., Contr. No. 1 0 8 1 . 9 PP. WALTERS, J . , A. KOZAK, AND P.G. HADDOCK. 1 9 6 6 . The e f f e c t of f e r t i l i z e r p e l l e t s on the growth of Douglas f i r . Univ. of B.C., Fac. For., Res. Notes No. 5 6 . 3 pp. WALTERS, W. 1 9 5 1 . F o r e s t f i r e insurance i n North America with s p e c i a l r e f e r e n c e to B.C. B.C. Lumberman. 6 pp. WARRACK. G.C. 1 9 5 2 . Comparative o b s e r v a t i o n s of the changes i n c l a s s e s i n a thinned and n a t u r a l stand of immature Douglas f i r . For. Chron. 2 8 : 4 6 - 5 6 . WARRACK, G.C. 1 9 5 9 . Crown dimension, i n i t i a l diameter and diameter growth i n a j u v e n i l e stand of Douglas f i r . For. Chron. 3 5 ( 2 ) : 1 5 0 - 5 3 . WARRACK, G.C. 1 9 6 7 . In B.C. Fo r e s t S e r v i c e F o r e s t Research Review. Thinning experiments i n Douglas f i r . 8 1 - 8 2 . WEAR, J.F., AND P.G. LAUTERBACH. 1 9 5 5 . Color photographs u s e f u l i n e v a l u a t i n g m o r t a l i t y of Douglas f i r . Pro-ceedings SAF meeting. 1 6 9 - 7 1 . WEAR, J.F., R.B. POPE, AND P.G. LAUTERBACH. . 1 9 6 4 . E s t i m a t i n g b e e t l e - k i l l e d Douglas f i r by a e r i a l photo and f i e l d p l o t s . Jour. For. 6 2 ( 5 ) . WEBB, D. 1 9 6 9 . An i n v e s t i g a t i o n of the s p a t i a l d i s t r i b u t i o n of t r e e s i n a f o r e s t stand. Univ. of B.C., Fac. For., B.S.F. T h e s i s . 37 pp. WEETMAN, G.F. 1 9 5 7 . Recommended methods f o r i n v e n t o r y and i n v e s t i g a t i v e r e p r o d u c t i o n surveys. PPRIC, Montreal. 6 PP. L i t e r a t u r e c i t e d 222 WERT, S.L. 1969. Revised a e r i a l volume t a b l e f o r e s t i m a t i n g spruce and f i r m o r t a l i t y i n Minnesota. Jour. For. 67(5): 334-36. WILLIAMSON, R.L. 1963. Growth and y i e l d r e c o r d s from w e l l -stocked stands of Douglas f i r . U.S.D.A. Fo r e s t Ser-v i c e . Pac. Northwest For. & Range Expt. Sta., Res. Paper PNW-4. 2A pp. WOMMACK, D.E. 1964. Temperature e f f e c t s on the growth of Douglas f i r s e e d l i n g s . Oregon State Univ., Ph.D. T h e s i s . WORTHINGTON, N.P. 1955. M o r t a l i t y can be salvaged. The Lumberman. A p r i l . 4 pp. WORTHINGTON, N.P., D.L. REUKEMA, AND G.R. STAEBLER. 1962. Some e f f e c t s of t h i n n i n g on increment i n Douglas f i r i n western Washington. Jour. For. 60: 115-19-WRIGHT, K.H., AND G.M. HARVEY. 1967. The d e t e r i o r a t i o n of b e e t l e - k i l l e d Douglas f i r i n western Oregon and Wash-in g t o n . U.S.D.A. Fo r e s t S e r v i c e , Pac. Northwest F o r e s t and Range Expt. Sta., Res. Paper PNW-50.'20 pp. WRIGHT, K.H., AND P.G. LAUTERBACH. 1958. A 10-year study of m o r t a l i t y i n a Douglas f i r sawtimber stand i n Coos and Douglas f i r Counties, Oregon. U.S.D.A. F o r e s t S e r v i c e . Pac. Northwest For. & Range Expt. Sta., Res. Paper No. 27. 29 pp. ZINKE, P.J. 1962. The p a t t e r n of i n d i v i d u a l f o r e s t t r e e s on s o i l p r o p e r t i e s . Ecology. 43: 130-33* APPENDIX I 223 COMPUTER PROGRAMS PROGRAM YIELD T h i s program (362 F o r t r a n statements) has "been w r i t t e n i n F o r t r a n IV f o r the IBM 360/6?, to analyze growth and y i e l d i n one permanent sample p l o t of any s i z e , remeasured up to 20 times, and c o n t a i n i n g up to 1000 t r e e s at the time of p l o t es-tablishment. There should not be more than a t o t a l o f 150 one-i n c h diameter c l a s s e s . A r e g r e s s i o n equation to p r e d i c t i n d i v i d u a l t r e e height must bo a v a i l a b l e . Volumes are computed i n the subroutine FUNCTION, u s i n g the B.C. F o r e s t S e r v i c e l o g volume equation f o r Douglas f i r . Any other form or t a b i c could be used by modify-i n g t h i s f u n c t i o n . The f o l l o w i n g v a l u e s are c a l c u l a t e d i n the program cn an acre b a s i s f o r each remeasurement: 1. A c t u a l stand: number of t r e e s ; gross and net b a s a l area; average diameter; t o t a l gross and net cubic volume. 2. Increment: mean annual gross and net increment i n bas a l area and cubic volume. 3. Number of t r e e s per acre, cubic volume, and mean annual increment of each one-inch diameter c l a s s . I f more than one measurement i s read i n , the f o l l o w i n g computations are a l s o performed: Appendix I ,224 4. Increment: p e r i o d i c annual gross and net increment i n b a s a l area and cubic volume. 5. M o r t a l i t y : number of dead t r e e s between two measurements, cumulative number and p r o p o r t i o n compared to the beginning number of l i v i n g t r e e s ; p e r i o d i c annual m o r t a l i t y and annual percentage f o r each p e r i o d ; p e r i o d i c annual and cumulative m o r t a l i t y i n b a s a l and cubic v o l m i e. NOTE: I t i s assumed that a l l the t r e e s are Douglas f i r . No ingrowth i s taken i n t o account; every t r e e must have a diame-t e r at f i r s t measurement. Program Yield Simplified Flow Chart- 2 2 5 \Height 7 Regression / Individual VOLUMES Cubic Volumes per acre Mean annual increment in BA and cu- ft-. Appendix I 226 PROGRAM STOCK T h i s program (373 F o r t r a n statements) has been w r i t t e n i n F o r t r a n IV f o r the IBM 360/67, to analyze some methods of measuring STOCKING i n one sample p l o t c o n t a i n i n g up to 1000 t r e e s , and romeasured up to 20 times. Regression equations of crown width on dbh, height on dbh and b a s a l area, l i v e crown l e n g t h or height to l i v e crown on height and ago, and crown volume on crown width and crown l e n g t h must be a v a i l a b l e . I t i s assumed that a l l t r e e s are Douglas f i r . The f o l l o w i n g values are c a l c u l a t e d f o r each p l o t measure-ment : 1. Number of t r e e s per acre, average height of dominant and codominant t r e e s , and top height (average height of tho 100 l a r g e s t t r e e s per a c r e ) . 2. E q u i v a l e n t square spacing of the l i v i n g t r e e s . 3. Times average diameter and times average diameter p l u s two standard d e v i a t i o n s , based on e q u i v a l e n t square spacing. h. Average crown width/diameter r a t i o of the p l o t ; tho crown width of every t r e e i s p r i n t e d to help i n p l o t t i n g the crown on a chart and c a l c u l a t e percentage crown c l o s u r e . 5. Cumulative crown c l o s u r e , expressed as the sum of a l l crown p r o j e c t i o n s i n percent of the p l o t area; same crown c l o s u r e a l s o Appendix I 227 c a l c u l a t e d without t a k i n g suppressed t r e e s i n t o c o n s i d e r a t i o n . When no crown c l a s s i s a v a i l a b l e , i t i o assumed that suppressed t r e e s have a diameter l o s s than AVDBH-2 sigma, and that domi-nant and c©dominant t r e e s have a diameter l a r g e r than AVDBH. 6. Crown space occupation. Tho p l o t crown space i s determined by p l o t boundaries, the average height of dominant and co-dominant t r e e s and average height to l i v e crown of the same t r e e s . P a r a b o l o i d , c o n i c a l or n e i l o i d crown shapes can be en-tered i n the crown volume equation. Tho percentage of t h i s space occupied by crowns i s c a l l e d the a c t u a l % crown space occupation. The optimum crown space occupation i s determined by the percent crown space occupied by the crowns of t r e e s that would be unif o r m l y d i s t r i b u t e d a c c o r d i n g to a square spacing, a l l of them having the average diameter of the stand under c o n s i d e r a t i o n and showing a CW-DBH r a t i o of 1, a H-CW r a t i o of 5, and a l i v e crown r a t i o of 3 (normal stand). The s i t e occu-p a t i o n f a c t o r c a l c u l a t e d i s the r a t i o of these two percentages. 7. The K r a j i c e k crown competition f a c t o r (CCF). 8. T o t a l dry weight (biomass) per acre as a f u n c t i o n of dbh squared times height (Kurucz, 1969). REFERENCE KURUCZ, J . 1969. Component weights of Douglas f i r , western hemlock and western red cedar biomass f o r s i m u l a t i o n of amount and d i s t r i b u t i o n of f o r e s t f u e l s . M.F. t h e s i s , Fac. For., Univ. of B.C. 116 pp. 228 • Program Stock Simplified Flow Chart-( Start } READ Plot dimensions No- measurements^  Individual OBH.CC Regression coefficients CW HEIGHT LCL or HLC Crown Volume Biomass ©-» 1 Per acre N,AVD8H,BA Square spacing SQRT (Plot/N) Times AVDBH SQSP/AVOBH d>-> AV- height do.co-do CW0LCLfCV Crown area Biomass / * I Sum crown area/plot area CCCLO Average live crown length AVLCL Crown space 1 CS Plot area x AVLCL Crown competition factor Sum crown volumes SUMCV Crown space occupation SUMCV/CS no Appendix I 2 2 9 PROGRAM SPACE T h i s program (453 F o r t r a n statements) has been w r i t t e n i n F o r t r a n IV f o r the IBM 360/67, to analyze s p a t i a l p a t t e r n and frequency d i s t r i b u t i o n of t r e e diameters and t r e e h e i g h t s measured i n sample p l o t s of any s i z e , A maximum of 5 p l o t s can bo analyzed i n the same run, each c o n t a i n i n g up to 1,000 t r e e s . Rectangular c o o r d i n a t e s and diameter of each t r e e at breast height can be road i n on any format, i n the f o l l o w i n g order: X, Y, DBH, or DBH, X, Y. The program c a l c u l a t e s : 1. Number of t r e e s per acre. Z. Basal a r e a per acre* 3. Diameter of the trc-e of average b a s a l area. 4. S p a t i a l p a t t e r n by means of tho contiguous quadrat method. Four quadrat s i z e s arc t e s t e d , t h e i r s i z e being determined by a s p e c i f i e d expected moan number of t r e e s per quadrat, Tho expected number has to be s p e c i f i e d f o r the f i r s t run; i t i s then i n c r e a s e d by one at each of the subsequent three runs. 5. The frequency d i s t r i b u t i o n of diameters and h e i g h t s . In 4 and 5 above, tho goodness of f i t to one of 5 standard p r o b a b i l i t y f u n c t i o n s i s t e s t e d by means of a Chi-square t o s t . These a r e : normal, Poisson, b i n o m i a l , negative b i n o m i a l and 1 Appendix I 2 3 0 and uniform d i s t r i b u t i o n s . When no diameters are entered ( f i r s t diameter oqual to zero ) , an a n a l y s i s of s p a t i a l p a t t e r n only i s performed. When no height r e g r e s s i o n i s a v a i l a b l e , the height frequency d i s t r i -b u t i o n a n a l y s i s i s not performed and the volume outputs are zeros. Trees without c o o r d i n a t e s can be entered. I f some of them have no c o o r d i n a t e s , thoy are ignored i n the p a t t e r n a n a l y s i s . I f a l l o f them have' no co o r d i n a t e s , zeros are p r i n t e d i n the p a t t e r n output. Program Space Simplified Flow Chart* 231 Height regression coefficients I \ \Expected number/ of trees per fc \ , quadrat (I) / START Title card Number of trees N Plot dimensions XMAX.YMAX Coordinates of each tree XX,YY Diameter of each tree DBH Number of quadrats Quadrat size Allocate trees to each quadrat-Subroutine FREQNC Put data into freq-classes*, calculate fit to known prob- distrib-<-d) , ® I FREQNC fdr DBH I—4 1 Basal area /acre Average DBH Individual heights FREQNC for height I S PRINT 2 frequency distribution of DBH and HEIGHT and 4 freqnc-distrib-of quadrats having sa me number of trees -6 tests of goodness of fit-N/acre, AVDBH, B A/acre, VOL/acre quadrat size-, expected and actual number of quadrats-( S T O P Q Appendix I 232 PROGRAM MORT Th i s program (435 F o r t r a n statements) has been w r i t t e n i n FORTRAN IV f o r the IBM 3 6 0 / 6 7 , to c a l c u l a t e i n d i v i d u a l t r e e p r o b a b i l i t i e s of death based on sample p l o t remeasurements. Up to 10 remeasurements can be entered as input, and up to 1000 t r e e s . Apart from the program c o n t r o l cards, a height r e g r e s s i o n equation and a crown width r e g r e s s i o n equation can be fed i n , together with i n d i v i d u a l t r e e crown c l a s s . These are not com-pu l s o r y ; the core i s fed i n t o tho program, the l a r g e r the output. At l e a s t two suc c e s s i v e measurements of diameters are necessary to o b t a i n any r e s u l t . I f three measurements are a v a i l a b l e , the t a b l e s based on increments w i l l be c a l c u l a t e d . The f o l l o w i n g i n f o r m a t i o n i s p r i n t e d : Input two measurements of dbh, or more: TABLE 1: Annual p r o b a b i l i t y .of i n d i v i d u a l t r e e death d u r i n g the p e r i o d based on the r a t i o of a c t u a l diameter to average diameter at the beginning of the p e r i o d . S i x t e e n c l a s s e s of diameter r a t i o s aro c a l c u l a t e d ( 0 . 0 , 0 . 2 5 0 , 0 . 5 0 0 , 0 . 7 5 0 , 1 0 0 0 , 1 . 2 5 0 , e t c . . . ) . Input two measurements of dbh or more, and a height r e g r e s s i o n : TABLE 2 : Annual p r o b a b i l i t y of i n d i v i d u a l t r e e death d u r i n g the p e r i o d based on the r a t i o of a c t u a l height to Appendix I 233 stand top height at the beginning of the p e r i o d . Input two measurements of dbh or more, and crown width r e g r e s s i o n : TABLE 3: Annual p r o b a b i l i t y of i n d i v i d u a l t r e e death d u r i n g the p e r i o d based on the r a t i o of tr e e CW-DBH to average stand CW-DBH r a t i o at the beginning of the p e r i o d . Input three measurements of dbh or more: TABLE i+: Annual p r o b a b i l i t y of i n d i v i d u a l t r e e death d u r i n g the p e r i o d by c l a s s e s of tr e e annual dbh increment r a t i o to average p l o t dbh increment; the increment i s c a l c u l a t e d over the f i r s t p e r i o d and the p r o b a b i l -i t y i s given f o r the second p e r i o d on. Input three measurements of dbh or more, and a height r e g r e s s i o n : TABLE 5: Annual p r o b a b i l i t y of i n d i v i d u a l t r e e death d u r i n g the p e r i o d by c l a s s e s of annual t r e e height increment r a t i o to p l o t top height increment. Input two measurements of dbh or more and crown c l a s s : TABLE 6: Annual p r o b a b i l i t y of i n d i v i d u a l t r e e death i n each • of four crown c l a s s e s f o r each measurement p e r i o d . Due to the use of pe r i o d s of v a r i a b l e l e n g t h , the proba-b i l i t y c a l c u l a t e d i n each of the 6 t a b l e s i s an annual Appendix I 234 p r o b a b i l i t y of death. I t i s i n f a c t the annual p r o p o r t i o n of the i n i t i a l number of t r e e s i n each c l a s s (of dbh, height, cw-dbh, or crown c l a s s ) that d i e s d u r i n g the p e r i o d as determined by two s u c c e s s i v e p l o t remeasurements. In Tables L and 5, two measurements are necessary to e s t a b l i s h the increment c l a s s e s and the p r o p o r t i o n of dead t r e e s i s e s t a b l i s h e d by l o o k i n g i n t o the t h i r d p l o t measurement to see whether t r e e s are dead or not. I f ton p l o t remeasurements are a v a i l a b l e , nine p r o b a b i l i -t i e s w i l l bo p r i n t e d i n each c l a s s of t a b l e s 1, 2, 3? snd 6; and e i g h t p r o b a b i l i t i e s w i l l be p r i n t e d i n each c l a s s of t a b l e s 4 and 5* Negative increments i n diameter (due to r e a l decrease i n g i r t h or i n a c c u r a c i e s i n measurements) w i l l be considered as zero increment and consequently w i l l be taken i n t o account i n the f i r s t c l a s s (0.000). The number of l i v e t r e e s at the beginning of tho p e r i o d i s a l s o p r i n t e d i n each c l a s s to allow f o r c a l c u l a t i o n of tho number of dead t r e e s , and the number of t r e e - y e a r s on which each p r o b a b i l i t y i s based. • Program Mort Simplified Flow Chart- 2 3 5 ( START ) No- plot measurements \ D B H .age, N / X B A . S I . T O P H / \ A V C W / D B H / T \ HandCW / degression / Ycoeff- / Number live +1 NL |Class+ 025 I class 1 * 1 class 2 « 2 class 3 « 3 class 4 « 4 Individual ratio dbh/DSH h/TOPH cw-dbh/CW-DBH Print tables Annual probability of mortality No- of live trees Individual ratio d in/Oin h in / TOPH in C STOP ) yes APPENDIX II 236 REGRESSION EQUATIONS FROM ORIGINAL DATA B.C. FOREST PRODUCTS COMPANY7  HEIGHT REGRESSION EQUATION H = -3.7867 + 3.512/+ DBH + 0.2157 EA - 0.0829 DBH 2 N = 9 0 R 2 = 0.95 SEE = 8.4 DATA MIN MAX MEAN ST.DEV. C.V. DBH 5.4 39.2 15.8 6.7 42 H 36.0 205. 100.6 38.0 37 BA 53.0 493. 193.8 108.9 56 CROWN WIDTH REGRESSION EQUATION CW = 14.2 +1.11 . DBH - 0.126 H R 2 = 0.65 SEE = 4.3 From Smith, Ker and Csizmazia (1961). LIVE CROWN LENGTH REGRESSION EQUATION LCL = 15.2 + 0.997 DBH + 0.193 H R 2 = 0.61 SEE = 14 From Smith, Ker and Csizmazia (I96I). I d e n t i f i c a t i o n of symbols i n Appendix I I I n d i v i d u a l t r e e : height i n f e e t (H); diameter at brea s t height i n i n c h e s (DBH); crown width i n fe e t (CW); l i v e crown l e n g t h i n f e e t ( L C L ) ; height to l i v e crown i n fe e t ( H L C ) . Net b a s a l area i n square f e e t per acre (BA) jpiiumberpOf obser-v a t i o n s (N); c o e f f i c i e n t of dete r m i n a t i o n ( r or R ); standard e r r o r of estimate (SEE); minimum (MIN); maximum (MAX); mean (MEAN); standard d e v i a t i o n (ST.DEV.); c o e f f i c i e n t of v a r i a -t i o n i n percentage ( C . V . ) . Appendix II 237 B.C. FOREST SERVICE HEIGHT REGRESSION EQUATIONS PLOTS EP 69, EP 84, EP 85, EP 209 H = 1.8156 + 8.8220 DBH - 0.1011 DBH 2 N = 281 R^ = 0.8786 SEE = 12.98 DATA MIN MAX MEAN ST.DEV. C.V. H 8.0 157.0 55.5 37.1 , 66 DBH 0.9 25.9 6.9 5.1 73 PLOT # 69 (90% F i r +) H = 1.1621 + 8.93193 DBH - 0.32995 DBH 2 N = 94 R 2 = 0.866 SEE = 5.68 DATA MIN MAX MEAN ST.DEV. C.V. DBH .9 12.0 4.3 2.67 61 H 8.0 66.0 31.6 15.36 48 PLOT #84 (83% F i r ) H = 7.0408 + 11.8079 DBH - 0.25374 DBH' N = 50 R 2 = 0.80 SEE = 14.47 2 DATA MIN MAX MEAN ST.DEV. C.V. DBH 2.7 25.9 12.8 5.7 44 K 32 157 108.5 32.0 29 (T h i s equation was not used, and r e p l a c e d by H f o r a l l p l o t s . ) PLOT #85 (58% F i r ) H = 32.4917 + 6.6419 DBH - 0.06735 DBH 2 N = 2 5 R 2 = 0.94 SEE = 5.8 DATA MIN MAX MEAN ST. DEV. C.V. DBH 4.0 19.0 11.2 4.5 39 H 56.0 137.0 97.2 23.6 24 Appendix I I B.C. FOREST SERVICE (cont'd) PLOT #209 (75% F i r ) DBH N = 112 R1" = 0.98292 H = 2.5872 + 9-9573 ,2 DATA DBH H MIN 1.0 11.0 MAX 13,8 77.0 0.348594 DBH' SEE = 2.63 MEAN 5.5 42.6 ST.DEV. 3.5 19.9 238 C V . 63 46 PLOT #65 (66% F i r ) H 2.555 + 13.0305 N = 58 R = O.936 DBH - 0.6753 DBH" SEE =4.5 DATA H DBH MIN 6.0 0.1 MAX 63.0 8.2 MEAN 31.4 2.8 ST.DEV. 17.5 2.0 C V . 55 71 PLOT #283 (87% F i r ) H 21.3254 + 9.5759 N = 64 DATA H DBH R^ = 0.947 MIN 23.0 1.7-DBH - 0.2186 DBH 2 SEE =6.8 MAX 118.0 15.2 MEAN 79.0 7.57 ST.DEV. 20.8 3.3 C V . 26 43 CROWN WIDTH AND LIVE CROWN LENGTH REGRESSION EQUATIONS  PLOT #69 CW = 2.72506 + 1.22247 DBH r ^ = 0.89 SEE =1.04 LCL = -0.6625 + 2.7489 DBH + 0.1135 H R^ = 0.83 SEE =3.7 These equations were developed f o r the Wind R i v e r Spacing t r i a l data, f o r the 4 x 4 spacing; c o n d i t i o n s more or l e s s as i n p l o t #69: as i n low s i t e , same age, e t c . . Appendix II 239 B.C. FOREST SERVICE (cont'd) PLOT #84, 85, 283 cw = 5.88 + 1.47 DBH - 0.133 H R 2 = 0.91 LCL = 12.0 + 3.63 DBH - 0.184 H R 2 = 0.84 These equations were developed by B r i e g l e b (1952). PLOT #209 CW = 3.33742 + 1.10692 DBH r 2 = 0.86 SEE = 1.15 LCL = 8.2043 + 2.7986 DBH + 0.1234 H N = 46 R 2 = 0.89 SEE = 2.96 (17%) These two equations were developed f o r the Wind R i v e r Spacing T r i a l (6 x 6 spaci n g ) ; same range of dbh; same SI; same age; same N/acre as i n p l o t #209. PLOT #65 CW = 2.72506 + 1.22247 DBH r 2 = 0.89 SEE = I.04 LCL = -0.6625 + 2.7489 DBH + 0.1135 H R 2 = 0.83 SEE = 3.7 These are from the Wind R i v e r Spacing T r i a l (4 x 4 spacing) i n which the c o n d i t i o n s are qu i t e s i m i l a r . CROWN ZELLERBACH C O M P A N Y HEIGHT REGRESSION EQUATION H = 33.159 + 7.294 DBH - 0.205 DBH 2 T h i s equation was given by the Company. Appendix I I LIVE CROWN LENGTH EQUATIONS  PLOT #5 LCL = -42.1156 - O.7269 DBH + 0.9509 H R 2 = 0.79 SEE = 3-46 DATA MIN MAX MEAN ST.DEV. DBH 4.5 13.4 7.5 2.2 H 62 94.O 75 8.5 LCL 9 40.0 24 7.3 PLOT #8 LCL = 80.1823 + 8.6987 DBH - 1.5971 H R 2 = 0.65 SEE = 4.38 ' DATA MIN MAX MEAN ST.DEV. DBH 3.2 12.3 8.3 2.3 H 54.0 92.0 78.5 9.0 LCL 13.0 41.0 26.9 7.3 CROWN WIDTH REGRESSION EQUATION CW = 5.88 + 1.47 DBH - 0.133 H ^ = 0.91 Th i s equation was developed by B r i e g l e b (1952). I t i s u fo r both P l o t s #5 and #8. U.B.C. FERTILIZATION TRIAL HEIGHT, CROWN WIDTH, AND HEIGHT TO LIVE CROWN REGRESSION  EQUATIONS H = 3.2545 + 9.3179 DBH - 0.7356 DBH 2 R 2 = 0.61 SEE = 2.61 N = 52 CW = 5.9355 + 1.4539 DBH r 2 = 0.2418 SEE = 1.98 N = 52 Appendix I I 241 U.B.C. FERTILIZATION TRIAL (cont «d) HLC = -7.2774 - 0.3934 DBH + O.4896 K R 2 = 0.369 SEE = 2.38 N = 52 DATA MIN MAX H 15.0 34.6 CW 5.3 18.0 HLC 0.0 10.9 DBH 1.4 5.2 MEAN 27.4 11.3 4.6 3.7 ST.DEV. 4.0 2.2 2.9 0.76 C.V. 15 20 63 20 U.B.C. RESEARCH FOREST PLOT #107 (72% F i r ) HEIGHT, CROWN WIDTH AND LIVE CROWN LENGTH REGRESSION, 0 H = 4.5 + 12.63 DBH - 0.513 DBH 2 T h i s equation was given i n the t a l l y book. CW = 5.88 + 1.47 DBH - 0.133 H From B r i e g l e b (1952). LCL = -6.6024 + 3-8844 DBH N = 51 r 2 = 0.81 SEE =3.9 DATA MIN MAX DBH 5.0 .14.5 LCL 13.0 49-0 MEAN 8.1 24.8 ST.DEV. 2.0 8.8 C.V. 25 35 PSP #22, 216, 214, 217, 218 HEIGHT, CROWN WIDTH AND LIVE CROWN LENGTH REGRESSION EQUATIONS H =-11.084 + 8.2710 DBH + 0.1605 BA . - 0.154 DBH 2 R^ = 0.94 N = 869 From Newnham (1964). Same c o n d i t i o n s p r e v a i l i n g . CW = 5.88 + 1.47 DBH - 0.133 K R 2 = 0.91 From B r i e g l e b (1952). Appendix I I 242 U.B.C. RESEARCH FOREST (cont'd) PSP #212, 216, 21A, 217, 218 (cont'd) LCL = 12.0 + 3.63 DBH - 0.134 H R 2 = O.84 From B r i e g l e b (1952). SPACING TRIAL (1968) HEIGHT, CROIVN WIDTH AND LIVE CROWN LENGTH REGRESSION EQUATIONS TRIAL #14 ( o r i g i n a l spacing: 9 x 9 f e e t ) H = 3.4798 + 7.5723 DBH - 0.6220 DBH 2 N = 35 R'? = 0.61 S E E = 1.59 CW = 5.76 + 1.3353 DBH N = 35 r 2 = 0.45 SEE = 1.13 LCL = -0.2144 - O.4626 DBH + 0.9749 H N = 35 R- =0.57 SEE = 1.17 DATA MIN MAX MEAN ST.DEV. C V . DBH 2.2 6.3 4.1 0 . 0 19 H 15.0 28.0 23.6 2.5 10 CW 6.6 14.2 11.2 1.5 13 LCL 14.0 26.0 20.9 2.4 12 TRIAL #15 ( o r i g i n a l spacing: 3 x 3 f e e t ) H = 2.7570 + 10.0391 DBH - O.6984 DBH 2 N = 4 5 R 2 = 0.72 SEE = 3.44 CW = 3.3819 + 1.4413 DBH N = 4 5 r 2 = 0.62 SEE = 0.94 LCL = -4.6449 + 1.3388 DBH + 0.7069 H N = 45 R 2 = 0.88 SEE = 2.03 Appendix I I 243 U.B.C. RESEARCH FOREST (cont'd) TRIAL #15 (cont'd) DATA MIN MAX MEAN ST .DEV. C.V. DBH 0.6 4.1 2.4 .8 34 H 9.0 34.0 22.5 6.4 28 CW 2.9 10.0 6.9 1-5 22 LCL 3.0 26.0 14.5 5,8 40 TRIAL #16 ( o r i g i n a l spacing: 12 x 12 f e e t ) H = 4.73H + 5.5910 DBH - 0.3167 H N = 160 R 2 = 0.71 SEE = 2.19 CW = 3.8900 + 0.7448 DBH + 0.2458 H N = 160 R 2 = 0.53 SEE = 1.69 LCL = -0.1147 + 0.0324 DBH + 0.9692 H N = 160 R 2 = 0.96 SEE = 0.76 DATA MIN MAX MEAN ST.DEV. C.V. DBH .8 12.7 4.1 -1.3 31 H 9.0 32.0 21.9 4.1 19 CW 3.0 19.0 12.3 2.5 20 LCL 9.0 32.0 2.1.2 4.0 19 TRIAL #17 ( o r i g i n a l spacing: 6 x 6 f e e t ) H = 11.3916 + 4.3627 DBH - 0.2738 DBH 2 N = 37 R 2 = O.58 SEE-= 1.83 CW = 6.4973 + 0.9183 DBH N = 37 r 2 = 0.45 SEE = 0.95 LCL = 8.4552 + 1.0580 DBH + 0.2594 H N = 3 7 R 2 = 0.33 SEE = 2.34 DATA MIN MAX MEAN ST.DEV. C.V. DBH 2.1 5.8 3.8 .9 24 H 19.0 29.0 23.9 2.7 11 CW 8.0 12.7 10.0 1.3 13 LCL 12.0 25.0 18.7 2.8 15 Appendix I I 244 U.B.C. RESEARCH FOREST (cont'd) TRIAL #18 ( o r i g i n a l spacing: 15 x 15 f e e t ) H = - 0 . 4 2 1 1 + 3 . 6 6 6 0 DBH - 0 . 7 4 5 0 DBH 2 N = 35 R 2 = 0 . 8 2 SEE = 1 .47 CW = 3 . 3 8 0 5 + 2 .1510 DBH N = 35 r 2 = 0 . 6 4 SEE = 1 .43 LCL = 0 . 7 3 5 8 + 0 . 2 0 2 2 DBH + 0 . 8 7 9 8 H N = 3 5 R 2 = 0 . 9 7 SEE = 0 . 5 8 DATA MIN MAX MEAN ST. DEV. C V . DBH 1 . 8 5 . 3 3 . 7 . 9 23 H 1 3 . 0 2 5 . 0 2 0 . 9 3 . 3 16 cw 5 . 0 1 5 . 0 1 1 . 3 2 . 3 21 LCL l l . o 2 4 . 0 1 9 . 9 3 . 1 16 U.S. FOREST SERVICE  VOIGHT CREEK HEIGHT, CROWN WIDTH AND LIVE CROWN LENGTH REGRESSION EQUATIONS H = 2 5 . 3 3 6 8 + 7 . 0 8 2 4 DBH - 0 . 1 2 2 1 DBH 2 N = 100 R 2 = 0 . 7 7 SEE = 9 . 2 DATA MIN MAX MEAN ST. DEV. C V . DBH 5 . 1 2 4 . 6 1 1 . 5 4 . 3 37 H 4 9 . 0 1 3 4 - 0 8 8 . 6 1 9 . 1 21 CW = 5 . 8 8 + 1 .47 DBH - 0 . 1 3 3 H R 2 = 0 . 9 1 LCL = 1 2 . 0 + 3 . 6 3 DBH - O.I84 H R 2 = 0 . 8 4 These two equations are from B r i e g l e b ( 1 9 5 2 ) . Appendix I I 245 WILLIAMSON'S PLOTS HEIGHT REGRESSION EQUATIONS SITE I ( P l o t : Siuslaw #9) H = 81.1109 + 6.0761 DBH - 0.06733 BA - 0.06178 DBH 2 N = kO R 2 = 0.878 SEE = 7.A7 DATA MIN MAX ST.DEV. DBH 12.1 36.3 5.82 BA 3.85. 236. 36.7 H 122. 20A. 20.6 SITE I I ( P l o t s : R a i n i e r " 1,2,A,5,7,8) H = -A6.3104 + 7.5236 DBH + 0.3079 BA - 0.1134 DBH 2 N = 127 R 2 = 0.89 SEE = 10.67 DATA MIN MAX ST.DEV. DBH A.9 35.8 7.0 H 52. 183. 121.5 BA 206. • 326. 31.8 SITE I I I ( P l o t s : R a i n i e r #9; Olympic #1, 2) H = -19.3101 + 9.1109 DBH + 0.13602 BA - 0.1769 DBH 2 N = 84 R 2 = 0.89 SEE = 8.74 DATA MIN MAX ST.DEV. DBH 3.8 27.5 5.5 H 27. 147. 26.1 BA 176. 290. 31.7 CROWN WIDTH AND LIVE CROWN LENGTH REGRESSION EQUATIONS  FOR ALL PLOTS CW = 5.88 + 1.47 DBH - 0.133 H R 2 = 0.91 LCL = 12.0 + 3.63 DBH - O.I84 H R 2 = 0.84 These two equations are from B r i e g l e b (1952). Appendix I I 246 U.S. FOREST SERVICE (cont'd) WIND RIVER SPACING TRIAL  HEIGHT REGRESSION EQUATIONS  4 x 4 SPACING ( P l o t s #1, 2, 3) H = 3.9680 + 9.3302 DBH - 0.2668 N = 135 R 2 = 0.85 SEE =4.79 DATA MIN MAX MEAN ST.DEV. C.V. DBH 1.5 11.5 4.1 1.7 41 H 72. 75. 37.4 12.4 33 5 x 5 SPACING ( P l o t s #4, 5, 6) H = 2.4182 + 8.7634 DBH - 0.2380 DBH N = 151 R 2 = 0.84 SEE = 4-57 Tr2 DATA MIN MAX MEAN ST.DEV. C.V. DBH 1.5 9.0 4.17 1.6 37 H 10. 64. 34.3 H.3 33 6 x 6 SPACING ( P l o t s #7, 8, 9) H = 4.6275 + 7.9371 DBH ~ 0.1771 DBH 2 N = 173 R 2 = 0.83 SEE = 4.38 DATA MIN MAX MEAN ST.DEV. C.V. DBH 1.7 H.O 4-38 1.7 35 H 14. 73. 38.6 11.7 30 8 x 8 SPACING ( P l o t s #10, 11, 17) 11 = 3.8912 + 6.7561 DBH - 0.0415 DBH 2 N = 128 R 2 = 0.88 SEE = 5-75 DATA MIN MAX MEAN ST. DEV. C.V. DBH 1.6 12.0 5.8 2.4 41 H 8.0 78.0 41.7 16.2 39 Appendix I I U.S. FOREST SERVICE (cont'd) WIND RIVER SPACING TRIAL  10 x 10 SPACING ( P l o t s #13, 14, 15) H = 2.2293 + 7.9471 DBH - 0.1324 N = 107 R 2 = 0.83 SEE = 7.3 DATA DBH H MIN 1.5 13.0 MAX 14.8 90.0 MEAN 6.9 49.9 DBH 2 ST.DEV. 2.8 13.1 247 C.V. 40 36 CROWN WIDTH REGRESSION EQUATIONS These equations have been computed by the For e s t S e r v i c e , and were used as such. Form: CW SPACING 4 x 4 5 x 5 6 x 6 8 x 8 10 x 10 a + b DBH a 2.72506 3.34767 3.33742 4.71675 3.97588 b 1.22247 1.12911 1.10692 1.04783 1.13506 R SEE .9455 1.04559 .9426 0.91559 .9274 1.14707 .9223 1.32171 .9340 1.54012 LIVE CROWN LENGTH REGRESSION EQUATIONS  4 x 4 SPACING LCL = -0.6625 + 2.7489 DBH + 0.1135 K N = 39 R 2 = 0.83 SEE = 3.7 F = 91 (2, 36) DATA DBH H LCL MIN 1.9 28.0 2.0 MAX 11.8 77.0 37.0 MEAN 5.9 51.7 15.4 ST.DEV. 2.4 14.2 9.0 C.V. 41 27 58 Appendix I I U.S. FOREST SERVICE (cont'd) WIND RIVER SPACING TRIAL  LIVE CROWN LENGTH REGRESSION EQUATIONS  5 x 5 SPACING LCL = -0.2496 + 2.8438 DBH + 0.0344 N = 3 3 R 2 = 0.82 SEE = 3.3 DATA MIN MAX MEAN DBH 1.9 9.5 5.6 H 18.0 64.O 46.I LCL 2.0 29.0 15.0 6 x 6 SPACING LCL = -8.2043 + 2.7986 DBH + 0.1234 N = 4 6 R 2 = 0.89 SEE = 2.96 DATA. MIN MAX MEAN DBH 2.3 11.8 6.6 H 20.0 82.0 53.6 LCL 3.0 37.0 16.9 8 x 8 SPACING LCL = -8.18106 + 1.1927 DBH + 0.2694 N = 44 R 2 = 0.87 SEE = 3.9 DATA MIN MAX MEAN DBH 1.9 12.4 7.1 H 17.0 84.0 55.4 LCL 4.0 41.0 20.5 Appendix I I 249 U.S. FOREST SERVICE (cont'd) WIND RIVER SPACING TRIAL  LIVE CROWN LENGTH REGRESSION EQUATIONS 10 x 10 SPACING LCL = -0.6433 + 2.5591 DBH + 0.0551 H N = 47 R 2 = 0.81 SEE =4.8 DATA MIN MAX MEAN ST. DEV. C V . DBH 2.2 15.4 9.3 3.4 37 H 17.0 92.0 71.0 17.8 25 LCL 3.0 52.0 27.0 10.8 40 WEYERHAEUSER COMPANY (Clemons Tree Farm) HEIGHT, CROWN WIDTH AND LIVE CROWN LENGTH REGRESSION EQUATIONS. H = 16.8591 + 6.9036 DBH - 0.10189 DBH 2 N = 70 R 2 = 0.81414 SEE = 10.9511 DATA MIN MAX MEAN ST. DEV. C V . DBH 4.3 37.3 14.4 6.87 47.7 H 23. 149.0 90.4 25.0 27.7 CW = 5.88 + 1.47 DBH - 0.133 H LCL = 12.0 + 3.63 DBH - O.I84 H These two equations are from B r i e g l e b (1952). REFERENCES BRIEGLEB, P.A. 1952. An approach to d e n s i t y measurements i n Douglas f i r . Jour. For. 50: 529~36. NEWNHAM, R.M. 1964. The development of a stand model f o r Douglas f i r . Ph.D. t h e s i s . Fac. For., Univ. of B.C., Vancouver. 2 0 1 pp. SMITH, J.H.G., J.W. KER, AND J . CSIZMAZIA. 1 9 6 I . Economics of r e f o r e s t r a t i o n of Douglas f i r , western hemlock, and western red cedar i n the Vancouver F o r e s t D i s t r i c t . Fac. For., Univ. of B.C. B u l l . No. 3. 144 pp. APPENDIX I I I 250 MEASUREMENT OF SITE OCCUPANCY 1. STOCKING S i t e has been d e f i n e d by the Soci e t y of American F o r e s t e r s (1950) as "an area, considered as to i t s e c o l o g i c a l f a c t o r s v/ith r e f e r e n c e to c a p a c i t y to produce f o r e s t s or other vegeta-t i o n . " In simpler terms, s i t e was a l s o d e f i n e d as: "an area or l o c a l i t y t h a t supports t r e e growth (Spurr, 1952)," or as the "place of f o r e s t p r o d u c t i o n ( H i l l s , i 9 6 0 ) . " Crov/n c l o s u r e or the p r o p o r t i o n of s i t e (area) occupied by the v e r t i c a l p r o j e c t i o n of t r e e crowns i s a two-dimensional measure of s i t e occupancy. I t i s expressed i n percentage and of t e n r e f e r r e d to as s t o c k i n g . The d e n s i t y of the f o l i a g e above the ground o f t e n i s termed canopy d e n s i t y . Although these are important parameters, they are seldom d i r e c t l y token i n t o account i n e s t i m a t i n g grov/th and y i e l d , due to the c o n s i d e r a b l e amount of s u b j e c t i v i t y and v a r i a t i o n i n v o l v e d i n t h e i r evalu-a t i o n , e i t h e r from a e r i a l photos, or from the ground. Methods of measurement and t h e i r p r e c i s i o n have been s t u d i e d by Meyer (1930), Lemmon (1956), Pope (i960), Chiaa (1967), Bonner (1968), and many others . Cumulative crown c l o s u r e (CCCLO) i s suggested here as an a l t e r n a t i v e , and more o b j e c t i v e method of measuring percent crown c l o s u r e , or s t o c k i n g . T h i s technique c o n s i s t s of c a l -c u l a t i n g a crov/n width-diameter r e l a t i o n s h i p from i n d i v i d u a l Appendix I I I 251 t r e e o b s e r v a t i o n s . Then, by u s i n g the c a l c u l a t e d equation, i n d i v i d u a l crown widths are p r e d i c t e d , and the sum of t h e i r crown p r o j e c t i o n a l area i s obtained. T h i s a l l o w s s t o c k i n g to be measured i n a computer, from a t a l l y of t r e e diameter, and to evaluate s e p a r a t e l y the degree of crown c l o s u r e a t t r i b u t a b l e to any p o r t i o n of the canopy that i s of i n t e r e s t (e.g. by u s i n g t r e e crown c l a s s ) . When the assumption of c i r c u l a r crown p r o j e c t i o n a l area i s met, the minimum f u l l s i t e occupancy l e v e l can be d e f i n e d t h e o r e t i c a l l y as the 78.5 percent l e v e l of cumulative crown c l o s u r e , due to the r e l a t i o n s h i p of a square to a c i r c l e ( F i g -ure I I I - l ) . Such a lower l i m i t to f u l l s t o c k i n g has a l s o been suggested by G i n g r i c h (.1964) f o r hardwoods. An a l t e r n a t i v e method of s t o c k i n g e v a l u a t i o n i s the per-centage crown space a c t u a l l y occupied by t r e e crowns (CSO). T h i s three-dimensional measure of s i t e occupancy a l s o permits the e v a l u a t i o n of canopy d e n s i t y . I t i m p l i e s that both crown width and crown l e n g t h can be measured. Crown space i s de-f i n e d as the product of a p l o t area by tho d i f f e r e n c e i n height between average height to green crowns and stand top height ( F i g u r e I I I - l ) . The p r o p o r t i o n of t h i s space that can be occupied i s e s s e n t i a l l y a f u n c t i o n of crown shape, and canopy d e n s i t y i s a f u n c t i o n of the degree of crown o v e r l a p p i n g . In the open, Douglas f i r t r e e s e x h i b i t a pyramidal (cone) crown shape, whereas i n f o r e s t stands, a t y p i c a l ovate ( p a r a b o l o i d a l ) shape can be observed most of the time (Ronay, 1961). Both Figure IE-I- Minimum f uii site occupancy ievel by cumulative crown closure or percent crown space occupation* Crown Space =40x30x20= 240Mcu ft Crown Space Occupation when Crown Shape is< Paraboloid 93 Mcuft (397o) Conical 62Mcuft(26%) Neiloid 48Mcuft(20%) Appendix I I I 253 forms have been used i n previous r e s e a r c h (Smith et a l . , 1961, C u r t i s and Reukema, 1969). A minimum of 20 to 30 percent of the crown space would a c t u a l l y have to be occupied by t r e e crowns to o b t a i n f u l l s i t e occupancy, depending on the crown shape adopted (Figure I I I - l ) . At t h i s minimum l e v e l of f u l l occupancy, canopy d e n s i t y would be at a minimum, and a l l space would be occupied by t r e e s ; but ample room would be l e f t a v a i l a b l e f o r the development of crop t r e e s . T h i s puts tho concept of f u l l s t o c k i n g i n t o i t s true p e r s p e c t i v e d e f i n e d i n the f o r e s t terminology (SAF 1950) as a l e v e l of stock d e s i r a b l e f o r best growth and management. I t i s a l s o t h e o r e t i c a l l y p o s s i b l e to d e f i n e a maximum l e v e l of f u l l s t o c k i n g . E m p i r i c a l data are needed f o r t h i s purpose. When number of t r e e s and b a s a l area per acre are c a l -c u l a t e d from sample p l o t s or y i e l d t a b l e s , and p l o t t e d on a graph a g a i n s t average stand diameter, there i s a range w i t h i n which a l l o b s e r v a t i o n s f a l l around each average diameter l i n e ( Figure III-2). I f we assume that every t r e e i s of average diameter, then percentages of crown c l o s u r e can be c a l c u l a t e d f o r pre-detormined crown width-diameter r a t i o s (or d e n s i t y l e v e l s ) . G i n g r i c h (1964) d e f i n e d s t o c k i n g l i m i t s d i f f e r e n t l y by de-ter m i n i n g the maximum area one t r e e could u t i l i z e ( e q u i v a l e n t to CW/DBH = 2.0 i n Fi g u r e III-2) and tho minimum area that he would r e q u i r e to su r v i v e ( u s i n g the t r e e - a r e a r a t i o method de-veloped by Chisman and Schumacher (1940)). Figure m-2 Theoretical full stocking range for Douglas-fir related to stand density' Basal Area per Acre gq.ff. Average Tree dbh (inches) 1816 14 12 10 9 8 7 6 5 Relative Density Scale C W / O B H » | . Q ^ CW/DBH«20 Absolute density is based on empirical data-Percent stocking calculated by assuming that all the trees are average size and have the same CW/DBH ratio-8 9 10 II 12 13 14 15 16 17 18 19 20 Hundred Trees per Acre ro 4k Appendix I I I 255 Fig u r e I I I - 2 i l l u s t r a t e s the range i n s t o c k i n g f o r stands of 3 to 18 inc h e s i n diameter. I t appears that at any age, on any s i t e , Douglas f i r stands ( n a t u r a l or planted) would theore-t i c a l l y occupy the s i t e f u l l y w i t h i n the d e f i n e d l i m i t s of crown c l o s u r e , tho low being 78 and the high 132 percent when diameter i s 3 in c h e s . As diameter i n c r e a s e s , t h i s range would become much narrower, e s p e c i a l l y over 10 i n c h e s . Measurements of crown c l o s u r e i n f u l l y stocked stands, by B a s k e r v i l l o (1965) and Williamson ( 1 9 7 0 ) , i n d i c a t e that these t h e o r e t i c a l l i m i t s are not u n r e a l i s t i c . F i g u r e I I I - 2 i l l u s t r a t e s a l s o that f o r a given average diameter, f u l l s t o c k i n g could be obtained with roughly the same percentage crown c l o s u r e , whatever the sta n d . d e n s i t y . In prac -t i c e however, data from Douglas f i r stands showed that percen-tage s t o c k i n g i n c r e a s e s with stand d e n s i t y w i t h i n tho same diameter c l a s s , i n some cases w e l l o u t s i d e the range t h e o r e t i -c a l l y e s t a b l i s h e d . T h i s can be a t t r i b u t e d to tho f a c t s 1) that a l l t r e e s do not have the same diameter, or show a skewed diameter d i s t r i b u -t i o n , 2) that t r e e s are not, or do not remain uniformly d i s t r i -buted on a given s i t e , and 3) that t r e e s of the same diameter do not have e i t h e r a c i r c u l a r crown p r o j e c t i o n , or the same crown width (due to stand composition, genetic f a c t o r s , compe-t i t i o n p a t t e r n s , e t c . ) . These are tho f a c t o r s that could be manipulated, together with stand d e n s i t y , i n order to o b t a i n maximum tre e growth and optimum f o r e s t y i e l d with a minimum of s t o c k i n g . Appendix I I I 256 2. STAND DENSITY Stand d e n s i t y r e f l e c t s the i n t e n s i t y of s i t e occupancy by c h a r a c t e r i z i n g the average degree of crowding w i t h i n the,por-t i o n of s i t e occupied by t r e e s (Smith and B a i l e y , 1 9 6 4 ) . I t can be expressed i n absolute, e f f e c t i v e , or r e l a t i v e terms. Number of t r e e s and b a s a l area per acre are the most common expressions of absolute stand d e n s i t y ; n o r m a l i t y i n b a s a l area or cubic volume i s a r e l a t i v e d e n s i t y measurement. In stands with complete crown c l o s u r e and random or sys-tematic stem s p a t i a l d i s t r i b u t i o n , absolute d e n s i t y i s equiva-l e n t to e f f e c t i v e d e n s i t y because the whole area i s occupied with t r e e s . In f a c t , at e a r l y stages of stand development, i n e x c e s s i v e l y crowded stands, and i n multiple-canopy f o r e s t s , the sum of crown p r o j e c t i o n a l areas can be twice as l a r g e as the growing space (200 percent crown c l o s u r e ) . In these cases, y i e l d per u n i t area i s a f f e c t e d by d e n s i t y alone, and not by st o c k i n g ; thus, no conver s i o n f a c t o r has to be a p p l i e d to ab-so l u t e d e n s i t y measurements ( F i g u r e III-3). In understocked stands, however,-* (that can reach and A number c f d e n s i t y i n d i c e s have a l s o been developed. They were e x p l a i n e d and d i s c u s s e d by B i c k f o r d et a l . (1957)? Ging-r i c h ( 1 9 6 4 ) , Vezina ( 1 9 6 4 ) and Osborn ( 1 9 E 5 7 . T h i s does not apply to r e g e n e r a t i o n s t o c k i n g which i s estab-l i s h e d on the b a s i s of stocked quadrats by number of i n d i v i -d u a l s . FigureHI-3 Effective and relative density evaluation-Area 3 0*1 acre Area * 0-1 acre Stocking 8 50 % CCCLO Stocking ° 101 % CCCLO Absolute Density Absolute Density usual way* 11 x 10 8 110 trees /acre usual way * 25 x 10 8 250 trees/acre effective den*> II /(50/IOO)x 108 220 trees/acre effective den « absolute density Average Diameter 8 15" Average Diameter815" Standard Density 8194 trees/acre Relative Density 8 220/194 »1-13 Relative Density 8 250/194 81-29 Appendix I I I 25o maintain f u l l s t o c k i n g l e v e l s ) , absolute d e n s i t y has to be ex-pressed i n e f f e c t i v e terms, i f s t o c k i n g i s to be taken i n t o account. T h i s means that holes i n the canopy have to be taken out before e s t i m a t i n g the degree of crowding i n s i d e the area occupied. T h i s i s done by d i v i d i n g absolute d e n s i t y f i g u r e s by the s t o c k i n g f r a c t i o n ( F igure I I I - 3 ) . Pope (i960) termed t h i s process the "cramming method" f o r e s t i m a t i n g crown c l o s u r e on a e r i a l photographs. When crowns do not completely cover the s i t e , they have to be o c u l a r l y crammed before an estimate of the percentage cover i s made. T h i s i s e s s e n t i a l l y what i s being done by c a l c u l a t i n g CCCLO, as i l l u s t r a t e d i n F i g u r e I I I -3. When CCCLO equals 100%, there are no more ho l e s i n the can-opy, and absolute stand d e n s i t y i s the same as e f f e c t i v e den-s i t y . E f f e c t i v e stand d e n s i t y i s sometimes being measured by s u b t r a c t i n g a given p r o p o r t i o n of the area from the gross f o r e s t acreage, to account f o r gaps i n the f o r e s t canopy, rock outcrops, roads, e t c . (McArdle et a l . , 1949, Bradley et a l . , 1966). When compared to the e l a b o r a t e procedures a p p l i e d to c a l c u l a t e growth and y i e l d , t h i s method i s q u i t e rudimentary. Most of the time, i t i s ignored anyhow, and absolute d e n s i t y i s c a l c u l a t e d as i f the area were f u l l y occupied by t r e e s . R e l a t i v e d e n s i t y , as of now, has been c a l c u l a t e d mainly by comparing absolute d e n s i t y to normal d e n s i t y f o r a given age, on a given s i t e . Consequently, i t i s b e t t e r known as normality, which i s a comparative e v a l u a t i o n of a c t u a l d e n s i t y Appendix I I I 259 with optimum or normal d e n s i t y , measured i n normal f u l l y stocked s t a n d s . U s e f u l values are t a b u l a t e d i n numerous so-c a l l e d "normal y i e l d t a b l e s . " The n o r m a l i t y i s s u e with r e f e r -ence to normal y i e l d t a b l e s i s well documented; i t has been analyzed and d i s c u s s e d by Schumacher (1928), Meyer (1930, 1933), B r i e g l e b (1942), S t a e b l e r (1949), Watt (1950), B e l l (1964), John (1964), Nelson and Bennett (1965), Smith.(1965), and C u r t i s (1969). I t appears from these p r e s e n t a t i o n s that r e l a t i v e d e n s i t y estimates made by comparing a c t u a l stand den-s i t y to the d e n s i t y of f u l l y stocked, unmanagod, pure, n a t u r a l , oven-aged stands of a given s p e c i e s arc biased, .and no longer as d e s i r a b l e as they were. An a l t e r n a t i v e stand d c n s i t 5 r standard i s needed. I t i s suggested here that STANDARD DENSITY could be do-f i n e d as the number of t r e e s per acre when square spacing i s equal to average stand d i a m e t e r . T h e standard d e n s i t y func-t i o n i s i l l u s t r a t e d i n F i g u r e I I I - 4 . I t shows the minimum Both concepts of f u l l s t o c k i n g and normal f o r e s t s arc c l o s e l y a s s o c i a t e d ; they have r e f l e c t e d f o r a l o n g time the i d e a l goal to achieve under the presumption that maximum volume i n -crement would be obtained with normal f u l l s t o c k i n g ( B i c k f o r d et a l . , 1957). The term s t o c k i n g i n the sense of " i d e a l s t o c k i n g f o r best growth" a l s o o r i g i n a t e d from the same source, and i s s t i l l widely recognized, although tho c r i t e r i a to estimate the " i d e a l " are changing. Number of t r e e s , diameter and spacing as measurements of den-s i t y have a t t r a c t e d c o n s i d e r a b l e a t t e n t i o n i n the 1930's (Eeineke, 1933, Matthews, 1935, Lexen, .1939, Chisman and Schumacher, 1940, M i t c h e l l , 1943, S t a h c l i n , 1949). Use of crown width-diameter r a t i o s i n r e l a t i o n with these parameter has been promoted by Smith et a l . , 1961; Davis (1966) made a summary of those and other d e n s i t y measurements. 2 6 0 Figure HE-4- Standard density as a function of diameter and number of trees per acre Standard Density Features Spacing 3 square C W - D B H » I 0 BA«237sq f t -VARMEAN"10 CSO* 39-3% CCL0»78*5% CC«100% Based on relationship' N« 43,560/ D B H 2 15 DBH (inches) Example* DBH» 1*0 DBH-100 DBH-200 N» 43,560 N« 436 N» 109 BA«237sq- ft* BA«237sq*ft-BA« 237 sqft* Appendix III 2 6 l number of t r e e s of a given s i z e that would be r e q u i r e d to com-p l e t e l y occupy any s i t e under t h i s assumption. F i g u r e I I I - l i l l u s t r a t e s how cumulative crov/n c l o s u r e or crown space occupa-t i o n are evaluated at standard d e n s i t y . When cumulative crown c l o s u r e equals 78.5%, and when the average crown width-diameter r a t i o equals 1.0, tho height-crown width r a t i o should be 5.0, and the h e i g h t - l i v e crown l e n g t h r a t i o should be 3.0 (Smith et a l . , 1961, Smith, 1963). Assuming t h a t crowns have a para-b o l o i d a l shape, crown space occupation (CSO) would then be 39.3%. 1 2 T h i s standard of d e n s i t y i s not proposed as a goal towards which management should s t r i v e , but r a t h e r as a general stan-dard and t h e o r e t i c a l p o i n t of r e f e r e n c e ( l i k e water i n ph y s i c s ) to which any stand, of any composition could be compared at any age, on any s i t e . F i g u r e III-3 shows how r e l a t i v e d e n s i t y could be computed by t a k i n g t h i s standard i n t o account; and Fig u r e III-2 i l l u s t r a t e s how the decimal s c a l e of r e l a t i v e d e n s i t y i s r e l a t e d to absolute d e n s i t y . CW = DBH H = 5 x CW = 5 x DBH LCL = H/3 = 5/3 DBH CV = XxCWxCWxLCL = XxDBHxDBHx5/3 DBH N = A/CWxCW = A/DBHxDBH SUM CV = (5/3 XxDBHxDBHxDBH) (A/DBHxDBH) = 5/3 AxXxDBH CS = 5/3 AxDBH PARABOLOID = X = 1/2 (DBHxDBIlA) = 0.393 CSO = X (5/3 AxDBH)/(5/3 AxDBH)100 = 39.3% REFERENCES Appendix I I I 262 BASKERVILLE, G.L. 1965. Dry matter p r o d u c t i o n i n immature b a l -sam f i r stands. For. S c i . Mono. 9. 42 pp. BELL, J.F. 1964. Trends toward n o r m a l i t y of subnormal stands of Douglas f i r - A case study. Proceedings SAF meeting, Denver, Colorado. 227-28. BICKFORD, C.A., F.G. WILSON, ANDF.S. BAKER. 1957. Stocking, normality, and measurements of stand d e n s i t y . J . For. 55 ( 2 ) : 99-104. BONNER, G.M. 1968. A comparison of photo and ground measure-ments of canopy d e n s i t y . For. Chron. AA ( 3 ) : 12-16. BRADLEY, R.T., J.M. CHRISTIE, AND D.R. JOHNSTON, i960. F o r e s t management t a b l e s . F o r e s t r y Commission, London. Booklet No. 16, 218 pp. BRIEGLEB, P.A. 1942. Progress i n e s t i m a t i n g trend of nor m a l i t y percentage i n secondary growth Douglas f i r . J . For. 40: 785~93. CHIAM, Y.C. 1967, The use of a e r i a l photographs to d i s t i n g u i s h between s t o c k i n g and d e n s i t y of western hemlock stand on the U n i v e r s i t y of B r i t i s h Columbia Research F o r e s t Haney, B.C. MF t h e s i s , Fac. For., Univ. of B.C. 234 pp. CHISMAN, H.'H. AND F.X. SCHUMACHER. 1940. On the t r e e r a t i o and c e r t a i n of i t s a p p l i c a t i o n s . J . For. 38: 311-17. CURTIS, R.O. 1969. Growth and y i e l d p r e d i c t i o n : y i e l d t a b l e s past, present, and f u t u r e . Paper presented at SAF annual meeting, Miami, F l a . 21 pp. DAVIS, K.P. 19&6. F o r e s t management: r e g u l a t i o n and v a l u a t i o n . 2nd Ed., McGraw-Hill Book Co., N.Y. 519 pp. GINGRICH, S.F. 1964. C r i t e r i a f o r measuring s t o c k i n g i n f o r e s t stands. Proceedings SAF meeting, Denver, Colorado. 198-201. HILLS, G.A. I960. Regional s i t e r e s e a r c h . For. Chron. 36 (4): 401-23. JOHN, F.B. 1964. Trends towards normality of sub-normal stands of Douglas f i r . A case study. Proceedings SAF meeting, Denver, Colorado. Appendix I I I 263 LEMMON, P.E. 1956. A s p h e r i c a l densiometer f o r e s t i m a t i n g f o r e s t o v e r s t o r y d e n s i t y . For. S c i . 2:314-20. LEXEN, B. 1939. Space requirement of ponderosa pine by t r e e diameter. USDA F o r e s t S e r v i c e , Southwest, For. Range Expt. Sta., Res. Mote 63. 4 PP. MATTHEWS, D.M. 1935. Management of American F o r e s t s . McGraw-H i l l Book Co., N.Y. 495 PP. McARDLE, R.E., W.H. MEYER, AND D. BRUCE. 1949. The y i e l d of Douglas f i r i n the P a c i f i c Northwest. USDA, Fo r e s t S e r v i c e Tech. B u l l . No. 201 (Revised). 74 pp. MEYER, W.H. 1930. A study of the r e l a t i o n between a c t u a l and normal y i e l d s of immature Douglas f i r f o r e s t s . Jour. A g r i c . Res. 41: 635-65. MEYER, W.H. 1933. Approach of abnormally stocked stands of Douglas f i r to normal c o n d i t i o n s . J . For. 31 ( 4 ) : 400-406. MITCHELL, H.C. 1943. R e g u l a t i o n of farm woodlands by r u l e of thumb. J . For.: 243-48. NELSON, T.C. AND F.A. BENNETT. 1965. A c r i t i c a l l o o k at the n o r m a l i t y concept. J . For. Gj>: 107-9. OSBORN, J.E. 1968. C l a s s i f i c a t i o n , concepts, and uses of v a r i a b l e s d e s c r i b i n g s t o c k i n g and stand d e n s i t y . D i r e c t e d study. Fac. For., Univ. of B.C. 96 pp. POPE, R.B. i960. Ocular e s t i m a t i o n of crown d e n s i t y on a e r i a l photos. For. Chron., Tech. Note. 36 ( 1 ) : 89-90. REINEKE, L.H. 1933. P e r f e c t i n g a stand d e n s i t y index. Jour. A g r i c . Res. 46: 627-58. REUKEMA, D.L. 1969. F o r t y - y e a r development of Douglas f i r stands p l a n t e d eit v a r i o u s spacings. USDA Fo r e s t S e r v i c e , Pac. Northwest For. Range Expt. S t a t i o n (Under p r e p a r a t i o n ) . RONAY, A. 1961. Study of crown shapes of Douglas f i r , western hemlock, and western red cedar as an a i d i n the i d e n -t i f i c a t i o n of these s p e c i e s on a e r i a l photographs. MF T h e s i s . Fac. For., Univ. of B.C. 116 pp. SOCIETY OF AMERICAN FORESTERS. 1950. F o r e s t terminology. Washington. 93 PP« Appendix I I I 264 SCHUMACHER, F.X. 1928. Concerning normal s t o c k i n g of even-aged stands. J . For. 26: 608-17. SMITH, J.H.G., J.W. KER, J . CSIZMAZIA. 1 9 6 l . Economics of r e -f o r e s t a t i o n of Douglas f i r , western hemlock and wes-t e r n red cedar i n the Vancouver F o r e s t D i s t r i c t . Fac. For., Univ. of B.C. B u l l . No. 3. 144 pp. SMITH, J.H.G. 1963. A n a l y s i s of crown development can e s t a b l i s h b i o l o g i c a l and economic l i m i t s to growth of t r e e s and stands. Comm. For. Review A2 ( 1 ) : 27-33. SMITH, J.H.G., AND G.R. BAILEY. 1964. Inf l u e n c e of s t o c k i n g and stand d e n s i t y on crown widths of Douglas f i r and lodgepole p i n e . Comm. For. Rev. ( 3 ) : 243-45. SMITH, J.H.G. 1965. Comments on "A c r i t i c a l l ook at the nor-m a l i t y concept." J . For. 63 ( 9 ) : SPURR, S.H. 1952. F o r e s t Inventory. The Ronald Press Co., N.Y. 476 pp. STAEBLER, G.R. 1949. P r e d i c t i n g the volume and no r m a l i t y of re p r o d u c t i o n of Douglas f i r . J . For, 47 ( 1 0 ) : 823-33. STAHELIN, R. 1949. Thinning ged l o b l o l l y and s l a s h pine stands to s p e c i f i e d d e n s i t i e s . J . For. 47: 538-40. VEZINA, P.E. 1964. An a n a l y s i s of measures of d e n s i t y i n even-aged balsam f i r and jack pine stands. For. Chron. 46 ( 4 ) : 474-81. WATT, R.F. 1950. Growth i n understocked and overstocked stands. USDA F o r e s t S e r v i c e , Northern Roc. Mt. For. Range Expt, Sta., Res. Note 78. 3 pp. WILLIAMSON, R.L. 1970. Stand d e n s i t y - b a s a l area growth r e -l a t i o n s h i p f o r 70 to 150- y e a r - o l d Douglas f i r . -Paper presented at the Ann. Meeting, Northwest S c i e n t i f i c Assoc., Salem, Oregon (Not f o r p u b l i c a -t i o n ) . 265 . APPENDIX IV IDENTIFICATION OF SYMBOLS AGE Stand age from seed, i n p l a n t a t i o n s and i n n a t u r a l stands. AMOR P e r i o d i c annual m o r t a l i t y i n number of stems per acre. APCM P e r i o d i c annual percent m o r t a l i t y i n number of stems per a c r e . AVDBH Average stand diameter, i n i n c h e s . See DBH. BAIN Mean annual net b a s a l area increment i n square f e e t per a c r e . BAMOR P e r i o d i c annual m o r t a l i t y i n square f e e t of b a s a l area per a c r e . BAN Net b a s a l area i n square f e e t uer acre; BANSQ = square of BAN. BASTOC E f f e c t i v e b a s a l area i n square f e e t per acre, i . e . BAN/STOCGR. BNT T o t a l number of t r e e s per acre a t the b e g i n n i n g of a measurement p e r i o d as opposed to FNT, number measured at the end of a p e r i o d . CC Tree crov/n c l a s s (dominant, codominant, i n t e r m e d i a t e , suppressed). CCF Crown competition f a c t o r , a b s o l u t e v a l u e . CCCLO Cumulative crown c l o s u r e , or sum t o t a l of c i r c u l a r p r o j e c t i o n a l area of dominant, codominant and i n t e r -mediate t r e e s , expressed i n percent of p l o t a r ea. CSO Percent of the p l o t crown space volume a c t u a l l y occupied by tr e e crowns. E x p l a i n e d i n Appendix I I I . C.V. C o e f f i c i e n t of v a r i a t i o n i n percentage. CW Tree crown width i n f e e t (average measure at widest p o r t i o n of crown). CW/DBH R a t i o of crov/n width i n f e e t to diameter i n i n c h e s (absolute v a l u e ) . Appendix IV 266 DBH dbh dbh. n, GBAINC GVOINC h, H HLC LCL MAIN MDBH N RDN SDBH SEE SI SQSP ST.DEV. Average stand diameter, or diameter of the t r e e of average b a s a l area i n inch e s ; or, i n d i v i d u a l t r e e diameter a t bre a s t height i n i n c h e s . DBHSQ = square of t h i s term. I n d i v i d u a l t r e e diameter at breast height outside bark i n i n c h e s . DIN, DBH_. „ Annual increment i n t r e e s i z e at bre a s t height i n i n c h e s . in P e r i o d i c annual gross b a s a l area increment i n square f e e t per a c r e . P e r i o d i c annual gross increment i n cubic f e e t per acr e . I n d i v i d u a l t r e e t o t a l height i n f e e t . Height to l i v e crown i n f e e t . L i v e crown l e n g t h i n f e e t . Mean annual increment i n cubic f e e t per a c r e . Average diameter of dead t r e e s i n i n c h e s (same as DBH f o r l i v e t r e e s ) . Number of o b s e r v a t i o n s i n r e g r e s s i o n a n a l y s i s ; other-wise, number of t r e e s per ac r e . Simple and m u l t i p l e c o e f f i c i e n t of de t e r m i n a t i o n . R e l a t i v e d e n s i t y c a l c u l a t e d as : (N/(/+3560/DBHSQ)). Average diameter of l i v e t r e e s i n i n c h e s (same as DBH). Standard e r r o r of estimate; same u n i t as Y (.%) -(SEE/?) 100, S i t e index, or height of dominant and codominant t r e e s at age 100, Square spacing, or square r o o t of p l o t area d i v i d e d by number of t r e e s i n p l o t , expressed i n f e e t . Standard d e v i a t i o n ; same u n i t as the v a r i a b l e . Appendix IV 2 6 7 STOCGR Ra t i o of mean annual net volume increment i n cubic f e e t per acre to normal mean annual gross increment from gross y i e l d t a b l e s , i n percentage. TOPH Average height of the 1 0 0 l a r g e s t t r e e s per acre i n f e e t . VARMEAN Variance-mean r a t i o of the d i s t r i b u t i o n of t r e e s per quadrat i n a contiguous quadrat sampling o p e r a t i o n . VOLMOR P e r i o d i c annual m o r t a l i t y i n cubic f e e t per acre, Y Estimate of the dependent v a r i a b l e i n r e g r e s s i o n equations. YESTOC E f f e c t i v e y i e l d expressed as the r a t i o of YIELDN to STOCGR, i n cubic f e e t per acre. YIELDN Net y i e l d i n cubic f e e t per acre ( i n c l u d e s a l l t r e e s 1.6 i n c h e s i n dbh and over, u n l e s s s p e c i f i e d other-wise ). APPENDIX V MORTALITY TABLES Appendix V 269 ANNUAL PROBABILITY OF MORTALITY FOR GROUPS I, I I , I I I , I V 1 Ratio of a c t u a l t r e e diameter to average stand diameter AGE 20 30 AO 50 60 70 80 90 .250 TY 1 TO P 711 154 .042 891 107 .055 70 13 .069 275 53 .043 774 164 .030 838 113 .027 110 23 .030 .500 789 172 .015 2967 353 .057 1750 213 .061 3285 547 .039 3543 761 .032 4097 554 .034 503 100 .023 .750 798 175 .003 4278 506 .022 4051 487 .027 6895 977 .017 5768 1127 .013 6852 926 .009 1050 210 ,010 75 15 .013 1.000 642 139 .001 3533 412 .003 3942 473 .007 5811 919 .005 5406 1034 .003 6749 908 .003 1245 249 .004 125 25 .000 1.250 555 122 .001 2736 317 .002 2853 332 .003 3920 623 .003 3850 740 .002 4595 624 .002 715 143 .006 45 9 .000 1.500 288 65 .001 1791 209 .001 1734 206 .002 1380 306 .001 1791 349 .001 1857 260 .001 265 51 .004 5 1 .000 1.750 186 LO .000 778 91 , 000 704 86 ,001 655 100 .001 878 173 .001 716 96 .001 80 16 .000 2.000 87 20 .000 302 41 .000 295 36 .000 316 55 .000 330 65 .000 222 30 ,000 20 4 .000 2.250 12 3 .000 85 11 .000 147 18 .000 110 22 .000 157 31 .000 127 17 .000 2.500 48 10 .000 61 7 .000 17 3 .000 50 10 .000 73 14 .000 33 5 .000 Appendix V 270 Rat i o of a c t u a l t r e e diameter to average stand diameter AGE . 20 30 AO 50 60 70 80 90 2.750 10 4 15 15 32 2 1 3 3 4 .000 .000 .000 .000 .000 3.000 10 26 2 3 .000 .000 TREE-4116 17432 .15567 23212 22595 26144 3988 250 R a t i o of a c t u a l t r e e height to stand top height AGE 20 30 40 50 60 70 80 0.250 TY TO P 1221 260 .032 1002 114 .051 90 9 .100 270 45 .044 419 90 .029 419 .008 210 30 .000 0.500 1323 285 .008 3278 474 .030 2245 251 .061 3598 539 .036 3998 822 .030 2530 407 .028 265 53 .021 0.750 1098 247 .001 4577 757 .012 6021 819 .018 9091 1405 .014 8549 1639 .010 8004 1175 .012 1020 204 .010 1.000 417 96 .000 3251 544 .003 3955 649 .002 7043 1328 .002 9226 1847 .001 10820 1398 .003 1940 388 . 001 1.250 57 12 .000 704 109 .000 605 129 .000 687 129 .000 716 141 .000 791 102 .001 50 10 .000 1.500 65 11 .000 37 10 .000 6 2 .000 Appendix V 271 R a t i o of a c t u a l t r e e height to stand top height AGE  20 30 AO 50 60 70 30 1.750 5 1 .000 TREE-YEARS 4116 12882 12953 20695 22908 22564 3485 Annual diameter increment/average annual DBH. increment dbhin/DEH AGE 20 30 40 50 60 70 80 90 YEARS 0.000 TY 1 TO P 153 51 .060 424 85 .055 1359 202 .068 2033 302 .050 3352 o44 .038 5281 673 .042 945 191 .025 20 14067 4 .050 0.25 312 104 .019 566 114 .023 1426 210 .022 1467 248 .017 4745 907 .014 5306 7?2 .009 805 161 .009 103 15235 12 .000 0.50 177 59 .003 500 101 .006 1364 191 .010 1482 203 .005 3071 571 .001 5020 o7o .001 730 146 .003 70 14 .000 12414 0.75 180 60 .005 395 85 .001 1143 185 .007 1203 93 .006 3046 581 .000 3695 491 .002 750 150 .001 50 10 .000 9867 1.00 108 36 .000 256 53 .000 998 162 .001 1096 199 .000 1551 293 .001 2366 327 .000 420 86 .002 30 6 .000 6825 1.25 141 47 .000 78 17 .000 536 88 .000 664 115 .001 1342 260 .000 1648 211 .000 175 35 .000 15 3 .000 4599 1.50 75 25 .000 57 12 .000 246 42 .002 576 97 .000 531 103 .000 806 104 .000 95 19 .000 2386 Appendix V 272 Annual diameter increment/average annual DBH increment dbhin/DBH , AGE 20 30 40 50 60 70 80 90 YEARS 1.75 66 22 .000 10 2 .000 186 33 .000 258 48 .000 276 54 .000 346 42 .000 35 n .000 5 l .000 1182 2.00 24 8 .000 4 1 .000 61 10 .000 226 46 .000 88 17 .005 97 l l .000 5 1 .000 505 2.45 39 13 .000 44 9 .000 86 18 .000 45 9 .000 10 1 .000 224 2.50 3 l .000 15 4 .000 65 13 .000 5 1 .000 15 2 .000 103 Crown C l a s s TREE-AGE 4 3 1 YEARS 20 TY 1 TO 1242 271 .033 1059 222 .002 1278 283 .001 513 120 .001 4092 30 1551 312 .056 1173 236 .009 1294 262 .002 1137 229 .000 5155 40 722 117 .060 1482 278 .022 1410 262 .006 1662 287 .001 5276 50 3386 564 .041 6114 1066 .009 4176 778 .002 1600 332 .001 15276 60 3275 651 .032 5173 979 .010 4687 873 .002 2280 419 .000 15415 Appendix V Crown c l a s s 273 TREE-AGE 4 3 2 1 YEARS 70 3603 495 .036 5766 812 .010 6460 910 .002 3330 463 .001 19159 80 290 58 .033 420 35 .021 825 165 .007 410 82 .009 1945 90 45 9 .022 125 25 .000 8o 16 .000 250 TY = t r e e - y e a r s ; P = p r o b a b i l i t y ; TO = t o t a l number of t r e e s observed. Pure and mixed n a t u r a l stands combined. D e t a i l s of computation i n Chapter 4. Appendix V 274 ANNUAL PROBABILITY OF MORTALITY FOR WIND RIVER SPACING PLANTATION R a t i o o f a c t u a l t r e e d i a m e t e r t o average s t a n d d i a m e t e r AGE 20 30 40 .250 TY TO P 36 6 .017 96 27 .018 168 56 .036 .500 432 72 .004 4367 960 .013 2689 598 .029 .750 724 129 .000 8720 1930 .005 5524 1149 .018 1.000 1164 194 .000 9647 2134 .002 5726 1195 .010 1.250 677 117 .000 6430 1422 .001 4020 839 .003 1.500 234 39 .000 2220 494 .001 1503 313 .001 1.750 24 4 .000 495 109 .000 300 62 .000 2.000 6 1 .000 37 8 .000 27 0 .000 2.250 10 2 .000 9 2 .000 TREE-YEARS 3297 32022 19966 Appendix V 275 Ra t i o of a c t u a l t r e e height to stand top height AGE 20 30 AO 0.250 TY TO P 54 9 .011 635 142 .020 507 105 .028 0.500 768 128 .002 9963 2215 .008 6214 1295 .025 0.750 1536 256 .000 15406 3420 .002 8954 1924 .010 1.000 90 151 .000 5575 1239 .001 3983 823 .001 1.250 108 18 .000 315 70 .000 231 48 .004 TREE-YEARS 2556 31894 19889 Annual diameter increment/average annual DEH increment dbhin/DBHin AGE .008 .020 3375 2933 828 707 .004 .018 TREE-30 40 YEARS ,000 TY 94 3237 3331 TO 95 681 P .018 .025 .225 1367 2399 3766 340 495 .500 6308 Appendix V 276 Annual diameter increment/average annual DBH increment dbhin/DBHin AGE TREE-30 40 YEARS .750 1.000 1.250 1.500 1.750 2.000 2.250 2.500 2.750 3.000 3639 2839 6478 870 588 .002 .010 3411 3773 7184 806 781 .000 .008 2408 1535 3943 578 316 .000 .006 1475 1610 3085 359 338 .000 .003 348 466 814 86 95 .000 .001 88 651 739 22 135 .000 .001 26 49 75 7 10 .000 .000 16 175 191 4 36 .000 .003 10 10 2 .050 58 58 12 .000 Appendix V Crown c l a s s 2 7 7 AGE 20 30 40 1 TY 1 TO P 162 27 . 0 0 0 4176 966 .001 4620 958 .002 2 330 35 . 0 0 0 5056 1218 .002 6713 1386 .008 3 318 53 .002 3378 815 .002 5229 1092 .014 4 42 n .000 1772 438 .010 3488 743 .034 TREE-YEARS 852 14382 20050 Crown width-diaaeter/average crown width-diameter r a t i o AGE 20 30 40 .750 TY 1 216 759 367 36 161 74 P . 0 0 0 .001 . 0 0 0 1 .000 2088 20677 12055 348 4585 2619 . 0 0 0 .002 .007 1.250 750 8045 4704 125 1786 987 . 0 0 0 .007 .024 1.500 221 2114 1166 37 465 305 .005 .016 .029 Appendix V 273 Crown width-diameter/average crown width-diameter r a t i o AGE  20 30 40 1.750 72 91 160 12 89 34 .017 .011 .023 2.000 24 13 17 4 3 4 .000 .033 .025 TREE-YEARS 3371 31699 I8469 TY = t r e e - y e a r s ; P = p r o b a b i l i t y ; TO = t o t a l number of t r e e s observed. Appendix V 279 EQUATIONS FOR PREDICTING PERIODIC PROBABILITY OF INDIVIDUAL TREE MORTALITY GROUPS I , I I , I I I , IV (PURE AND MIXED NATURAL STANDS COMBINED) Equation: Y = A * X * * B * C * * X or l o g Y = l o g A + B l o g X + X l o g C X = ( r a t i o of a c t u a l t r e e diameter to average stand DBH) Regression C o e f f i c i e n t s A G E A O C R 2 2 0 7 . 2 1 1 0 0 0 E + 0 0 1 1 . 0 3 5 0 3 0 E + 0 0 1.953000E-05 . 8 4 30 1.329000E+03 3 . 9 2 7 2 2 0 E+00 4 . 2 1 0 0 0 0 E - 0 5 . 8 6 40 3 . 0 4 9 0 0 0 E + 0 2 3.502970S+00 3.414000E-04 . 8 5 50 8.775000E+01 2.839390E+00 7.915000E-04 .87 60 2 . 4 2 0 0 0 0 E + 0 1 2.028870E+00 2 . 0 2 5 0 0 0 E - 0 3 . 8 5 70 7 . 1 0 7 0 0 0 E + 0 0 1.236520E+00 I . 4 5 4 O O O E - O 3 . 5 1 80 4.057000E+01 2 . 8 6 2 0 0 0 E+00 1.454000E-03 .51 X = ( r a t i o of a c t u a l t r e e height to s t and top height) 2 0 3.469000E+07 8.500390E+00 1.558000E-.12 .93 30 1 . 2 7 2 0 0 0 E + 0 7 9 . 0 6 5 4 1 0 E + 0 0 6 . 7 0 4 0 0 0 E - 1 0 .78 40 2 . 0 7 2 0 0 0 E+11 1.574710E+01 4.515000E-14 . 8 9 50 1.467000E+09 1.283700E+01 7.238000E -12 . 9 4 60 3.378000E+08 1 . 1 6 9 2 4 0 E + 0 1 1 . 9 5 7 0 0 O E - 1 O .90 70 2.481000E+03 5 . 2 0 2 5 0 0 E + 0 0 1.287000E-05 . 9 8 8o 3 . 4 1 5 0 0 0 E + 1 7 2 . 9 1 5 1 5 0 E + 0 1 2.456000E-20' .97 X E + 00 r 1 0 ° . Appendix V 280 GROUPS I,II,III,IV (cont'd) X = (rat i o of actual diameter increment to average annual diameter increment) Regression Coefficients R 2 AGE A B C 20 5.856000E-01 - l . 5 1 1 8 0 O E - 0 2 1.365000E-03 .79 30 8.660000E+00 2 . 4 1 2 0 0 0 E - 0 1 3 .219000E-05 .86 AO 1 . 9 0 3 0 0 0 E + 0 0 9.086800E-02 2.471000E-03 .75 50 6 . 0 1 6 0 0 0 E - 0 1 1 . 2 3 1 1 4 0 E - 0 2 2.534000E-03 .71 60 1.631000E-01 -8 . 4 8 8 7 2 0 E - 0 2 1.231000E-03 .62 70 6.652000E-01 3 . 8 7 5 9 7 0 2 - 0 2 3.929000E-04 .84 80 8 . I 1 7 0 0 0 E - 0 1 1 . 0 4 9 1 2 0 E - 0 1 1.989000E-03 .79 WIND RIVER SPACING PLANTATION X = (ra t i o of actual tree dbh to average stand DBH) Regression Coe f f i c i e n t s AGE A B C R 2 30 1.307000E+01 2.343100E+00 1.798000E-03 . 7 4 AO 6.727000E+03 6.397760E+00 1.366000E-05 .83 X = (ratio of actual tree height to stand top height) 30 1.552000E+01 1.877660E+00 3 . 4 9 5 0 0 0 E - 0 4 .77 A-o 1.649000E+03 4 . 2 2 5 2 0 0 E + 0 0 9.496OO0E-O6 .88 X = (rat i o of tree diameter increment to diameter increment) average stand 30 2 . 0 3 4 0 0 0 E + 0 0 1 . 4 3 3 5 9 0 E - 0 1 I . I 8 8 O O O E - O 4 . 7 4 40 4 . 6 7 9 0 0 0 E - 0 1 5.612000E-02 I . 4 6 6 O O O E-OI .78 APPENDIX VI 2 8 1 IRREGULAR AND CATASTROPHIC MORTALITY INTRODUCTION Great care i s taken i n sampling f o r e s t s i n order to e v a l -uate e i t h e r gross or net growth of t r e e s and stands, with an acceptable degree of p r e c i s i o n . In t h i s d i s s e r t a t i o n , the em-ph a s i s has been put on the e s t i m a t i o n of the d i f f e r e n t i a l be-tween gross and net values, i . e . r e g u l a r m o r t a l i t y . Regression equations of annual l o s s on stand parameters, and p r o b a b i l i t y t a b l e s based on tre e parameters have been b u i l t to p r e d i c t r e g u l a r m o r t a l i t y with accuracy. I r r e g u l a r and c a t a s t r o p h i c causes of m o r t a l i t y have been purposely l e f t out because they cannot be analyzed with the same methods. However, they must be taken i n t o account, f o r the v a r i a t i o n they create i n the system i s l a r g e enough to render a l l p r e d i c t i o n s completely i n a c c u r a t e . He r e a f t e r , i r r e g u l a r and c a t a s t r o p h i c causes of m o r t a l i t y w i l l not be d i f f e r e n t i a t e d . I t w i l l be assumed that a l l causes, except competition, are i n v o l v e d , and they w i l l be broken down i n t o four c l a s s e s as f o l l o w s : 1 . F i r e 2 . I n s e c t s 3. Diseases A. Weather, animals, and other s . Appendix VI 2o2 RULE OF THUMB Due to the sporadic and u n p r e d i c t a b l e nature of i r r e g u l a r m o r t a l i t y , simple r u l e s of thumb might p o s s i b l y be acc e p t a b l e . One such r u l e would be to c o n s i d e r annual estimates of r e g u l a r m o r t a l i t y as 80 percent of t o t a l , s i n c e i r r e g u l a r l o s s e s rep-r e s e n t on the average 20 percent of the t o t a l l o s s i n the State s of Oregon and Washington (Metcalf, 1968). T h i s estimate can be d i v i d e d i n t o causes a f t e r c o n s i d e r a -t i o n of trends i n l o s s e s over a l o n g p e r i o d of years. T h i s has been done i n the Timber Resources Report f o r the Douglas f i r subregion (Anon., 1958) i n which the p r o p o r t i o n of l o s s e s i n grov/ing stock a t t r i b u t e d to the four c l a s s e s was the f o l l o w i n g : TABLE VI-1 LOSS BY CAUSES P r o p o r t i o n of Percentage of Cause I r r e g u l a r Loss T o t a l Loss  F i r e .06 1 . 2 0 I n s e c t s .41 8 . 2 0 Diseases .11 2 . 2 0 Weather & Others . A 2 8.40 1 . 0 0 2 0 . 0 0 Appendix VI 283 Therefore, r e g u l a r m o r t a l i t y p r e d i c t i o n s could bo adjusted by a v a r i a b l e p r o p o r t i o n , a c c o r d i n g to the r e l a t i v e importance of each f a c t o r . We recognize that the r e l a t i v e importance of each cause i s not the same everywhere and may vary a great d e a l depending on whether i t i s considered over short or l o n g p e r i o d s of time, e.g. l o s s e s r e p o r t e d by the B.C. F o r e s t S e r v i c e (B.C. F.S., 1957) d i f f e r c o n s i d e r a b l y from the ones presented i n Table VI-1. T h i s r u l e could e a s i l y be a p p l i e d to m o r t a l i t y r a t e s or to m o r t a l i t y t a b l e s , f o r p r e d i c t i o n on a t r e e b a s i s . Annual r a t e s or annual p r o b a b i l i t i e s of r e g u l a r m o r t a l i t y would have to be m u l t i p l i e d by a f a c t o r of 1.23 (or 100/80) to account f o r a l l causes of i r r e g u l a r l o s s , or by 1.01, 1.08, 1.02, or 1.09 r e s p e c t i v e l y f o r i n d i v i d u a l causes as presented i n Table VI-1. When a p p l i e d to m o r t a l i t y t a b l e s , t h i s would have the e f f e c t of i n c r e a s i n g the p r o b a b i l i t y of m o r t a l i t y of smaller t r e e s more than b i g t r e e s , due to the shape of m o r t a l i t y curves. T h i s assumption seems r e a l i s t i c e s p e c i a l l y i n c o n s i d e r i n g weather-damages which k i l l more small t r e e s than l a r g e r ones among the growing stock. PROBABILISTIC APPROACH I f more r e f i n e d p r e d i c t i o n s must be made by causes, then a more s o p h i s t i c a t e d method can be adopted. I t could i n v o l v e 1) the p r e d i c t i o n of the average damage from one cause, 2) the det e r m i n a t i o n of the p r o p o r t i o n of the t o t a l l o s s due to that Appendix VI 284 cause and to others, and 3) the e s t i m a t i o n of the p r o b a b i l i t y of t o t a l l o s s from a l l causes. In a s p e c i f i c f o r e s t r e g i o n , f i r e damages are the e a s i e s t to evaluate. In the Vancouver F o r e s t D i s t r i c t , f o r i n s t a n c e , each year between 1937 and 1968, 21,000 ac r e s of p r o d u c t i v e f o r e s t out of 10 m i l l i o n were burnt on the average (PaiUe', 1968). Thus, the annual p r o b a b i l i t y of each acre c f f o r e s t to be burnt i n the f u t u r e , i f the trend were s t a b l e , would be 0.0021. But, a c c o r d i n g to Turner (1970), i n the past f i v e decades, the trend was downward, i . e . the acreage burnt was halved i n any 15-year p e r i o d . I t i s eJ_so a f u n c t i o n of t o t a l hours of sunshine between May 1 and August 31? i . e . the acreage burnt a n n u a l l y would double f o r each 77 hours of sunshine accumulated d u r i n g the summer p e r i o d . The o v e r a l l r e l a t i o n s h i p f o r the 50-year p e r i o d , between 1921-1970, can be w r i t t e n as: Log A = 0.597 - 0.0201 (Y - 1921) + 0.003835 S where, A = t o t a l acreage burnt a n n u a l l y i n a c r e s Y = calendar year S = t o t a l hours of sunshine T h i s trend could be taken i n t o account to a d j u s t the o v e r a l l p r o b a b i l i t y of 0.0021. Within the same r e g i o n , there i s a l s o a tremendous v a r i -a t i o n i n the p r o b a b i l i t y of an acre to be burnt a c c o r d i n g to the nature of i t s f o r e s t cover. For the Vancouver F o r e s t D i s -t r i c t , t h i s i s r e f l e c t e d i n the f o l l o w i n g breakdown from Smith (1969): Appendix VI 285 TABLE VI-2 AVERAGE ANNUAL AREA BURNED (1959-1968) AS A PERCENTAGE OF AREA IN TYPE NOT ' NON NON SATISFACTORILY COMMERCIAL PRODUCTIVE ALL MATURE IMMATURE RESTOCKED COVER SITES FORESTS 0.022 O.OA/4, 0.328 0.756 0.015 0.080 Therefore, a c c o r d i n g to the a c t u a l f o r e s t cover (or absence of co v e r ) , the average annual p r o b a b i l i t y of f i r e f o r each acre would vary from O . O O O 4 4 to 0.00756 around the mean of 0.0021 (si n c e we are not i n t e r e s t e d here by the mature f o r e s t or the non p r o d u c t i v e a r e a s ) . Within the same r e g i o n and type, the chance of having a f i r e at a p a r t i c u l a r time of the year would vary roughly as i n d i c a t e d below: TABLE VI-3 PERCENTAGE OF NUMBER OF FIRES BY MONTH (Vancouver F o r e s t D i s t r i c t , 1932-1968) March A p r i l May June J u l y August September October TOTAL 0.9 3.2 14.0 13.4 31.4 26.6 9.2 1.3 100 Appendix VI 286 I f we assume that l o s s e s i n acreage are d i r e c t l y propor-t i o n a l to l o s s e s i n volume, and that a l l l o s s e s are a d d i t i v e , then the p r o b a b i l i t y of s u f f e r i n g damages from other causes can be computed from the i n f o r m a t i o n on f i r e damage , as f o l l o w s : TABLE VI -k ANNUAL PROBABILITY OF DAMAGE FROM ALL CAUSES Causes Percentage From Table IV-1 M u l t i p l i e r Annual P r o b a b i l i t y of Damage Per Acre Range f o r V a r i -a t i o n i n Cover F i r e 6 1.0 0 . 0 0 2 1 0.00044-0.00756 In s e c t s 41 6.8 0.0143 O.OO299-O.O514I Diseases 1 1 1.8 0.0038 0.00079-0.01361 Others 42 7.0 0.0147 0.00308-0.05292 TOTAL 0.0349 0.00730-0.12550 These p r o b a b i l i t i e s could h a r d l y be a p p l i e d to annual mor-t a l i t y estimates, but could d i r e c t l y be added to m o r t a l i t y t a b l e s (Chapter 4)' to adjust the r i s k l e v e l f o r i r r e g u l a r and c a t a s t r o p h i c events. CONCLUSION Th i s example a p p l i e s to the Vancouver F o r e s t D i s t r i c t . However, there i s a tremendous i n t e r - r e g i o n v a r i a t i o n . For i n s t a n c e , some r e g i o n s can be d e c l a r e d d i s a s t e r areas, e.g. Appendix VI 287 the Columbia Gorge Counties (Anon., 1970), whereas other w i l l be u n l i k e l y to s u f f e r any damage at a l l . The s e t t i n g of a r i s k l e v e l i n each r e g i o n would be the r e s p o n s i b i l i t y of l o c a l managers. In any case, s i m i l a r or more comprehensive approaches should be taken, i f v a l u a b l e e f f o r t s i n p r e d i c t i n g r e g u l a r l o s s e s are to be of any value. A t t e n t i o n should a l s o bo given to r e g u l a r and c y c l i c v a r i a t i o n s i n cl i m a t e , to the p r o b a b i l i t y of l o s i n g s p e c i e s , e.g. white pine (Pinus i n o n t i c o l a Dougl), and grand f i r (Abies grandis Dougl.) L i n d l . , and to the p o s s i -b i l i t y of r e d u c i n g the p r o b a b i l i t y of l o s s by salvage opera-t i o n s or by more i n t e n s i v e management. REFERENCES ANON. 1958. Timber Resources f o r America's f u t u r e . U.S.D.A., Fo r e s t S e r v i c e , For. Res. Rep. No. 1 L . 713 PP. ANON. 1970. Columbia Gorge Counties dec l e a r e d d i s a s t e r areas. Western F o r e s t e r . 15 (8): 7. B.C. FOREST SERVICE. 1957. Continuous f o r e s t i n v e n t o r y of B r i t i s h Columbia. Dept. Lands and F o r e s t s , V i c t o r i a , B.C. 223 pp. METCALF, M.E. 1968. Timber m o r t a l i t y i n Western United S t a t e s . Western F o r e s t Pest C o n d i t i o n s . 3 pp. FAILLE, G. 1968. Economics of i n t e n s i f i c a t i o n of f o r e s t man-agement i n the Vancouver F o r e s t D i s t r i c t . Fac. For., Univ. of B.C. 130 pp. SMITH, J.H.G. 1969. The fu t u r e of p r e s c r i b e d burning i n western North America. Fac. For., Univ. of B.C. Mimeo. ZU pp. TURNER, J.A. 1970. Hours of sunshine and f i r e season s e v e r i t y over the Vancouver F o r e s t D i s t r i c t . For. Chron. U6 (2): 106-11. APPENDIX VII 288 STAND MODEL - DESCRIPTION, FLOW CHART, OUTPUT Th i s stand model (700 F o r t r a n statements) i s based on pro-b a b i l i t y t a b l e s (dbh/DBH, h/TOPH), a l o g volume equation, a diameter increment f u n c t i o n and a p r e d i c t i o n equation of the s p a t i a l p a t t e r n ' s variance-mean r a t i o . M o r t a l i t y i s c o n t r o l l e d by a pre-determined r i s k l e v e l i n Model I, and a m o r t a l i t y curve i n Model I I . A y i e l d t a b l e , and a map showing the l o c a -t i o n of dead t r e e s are p r i n t e d f o r up to seven 10-year pe r i o d s , together with an histogram of dead t r e e s . The number of f o r e -c a s t i n g p e r i o d s i s l i m i t e d only by the a v a i l a b i l i t y of m o r t a l -i t y t a b l e s . The histogram shows 1-inch dbh c l a s s e s i f there are l e s s than 20 c l a s s e s , and 2-inch c l a s s e s when there are more than 20 c l a s s e s . When u s i n g Model I, p r o b a b i l i t i e s have to be read i n t o the model f o r ten-year p e r i o d s , together with a r i s k l e v e l and a p r o b a b i l i t y of catast r o p h e . There i s no map output when the number of dead t r e e s d u r i n g one p e r i o d i s l e s s than 10. More than 10 quadrats are always formed with an expected mean number of t r e e s per quadrat of at l e a s t one. The maximum number of quadrat p o s s i b l e i s 225. Enclosed i s an example of output from Model I I . The source l i s t i s not reproduced, but i t i s a v a i l a b l e on request. Appendix 3ZH- Simplified Flow Chart, Douglas-fir Stand Model-( START ) 1 MAP dead trees VARMEAN Poisson, Neg- no Binomial I Generates X s \ Regre Vof H, \ VAI s- coeff jDBH inc-RMEAN PRINT Dead tree locations 7 yes \ D A T A 7 \ DBH,CC,X,Y / DATA 7 DBH,X,Y j DBH increment L n o X LAST PERIOD. YIELD Basal area Height Volume I MORTALITY CLASS of dbh/DBH, I no h/HTOP.CC below RISK \PRINT I YIELD / TABLE / MORTALITY CURVE Random generator PRINT HISTOGRAM LDEAD TREES f STOP ) 290 DIAMETER DISTRIBUTION OF DEAD TREES FOR PERIOD 1 B.C. FOREST SERVICE PLOT 69-GROUP I-SI:120 AGE=l8.0 RISK=0.200 PCAT=0.0 PLOT SIZE=0.4l6 ACRE FREQUENCY 44 30 3 NEG. BINOMIAL DISTRIBUTION DEAD TREE LOCATIONS AND NUMBERS FOR PERIOD 1 B.C. FOREST SERVICE PLOT 69-GROUP I-SI: 120 NUMBER OF QUADRATS 16 QUADRAT SIZE (SQ.FT.) 953.73 (MILACRE) 21.89 NUMBER OF DEAD TREES 77 QUADRAT LIMIT o.o 30.9 61.8 92.6 123.5 0.0 1 2 1 5 30.9 6 3 3 3 61.8 3 3 3 5 92.6 3 3 3 0 44 43 * 42 41 39 38 * 37 36 * 35 34 33 32 * 31 30 •tt * 29 •K-28 * 27 * 26 25 * 24 * * 23 * •X-22 * * 21 tt 20 * 19 * 18 45-17 •it * 16 * 15 14 -tt 13 12 * X-11 * 10 •X- -9 -X-8 •K- * 7 * 6 * * 5 4'-4 * * 3 # 2 * 1 * DIAMETER 1 2 5 CLASS LIMIT 1.5 2.5 3.5 DIAMETER DISTRIBUTION OF DEAD TREES FOR PERIOD B.C. FOREST SERVICE PLOT 69-GROUP I-SI:120 AC-E=28.0 RISK=0.200 PCAT=0.0 PLOT SIZE=0.Al6 ACRE 291 FREQUENCY 0 47 kh 26 12 2 47 * 46 * It c * 44 * 43 * tt 42 tt 41 - X -40 * 39 * tt 38 tt 37 tt * 36 * 35 tt tt 34 tt 33 tt tt 32 * 31 -* 30 29 28 * * 27 * tt 26 tt * 25 * • x -24 - x - * 23 tt * * 22 * tt * 21 * 20 tt tt 19 * tt 18 * tt * 17 tt tt 16 tt 15 tt tt 14 •* tt 13 * *K- * 12 •ire -K-11 tt 10 "A* * * 9 8 * * 7 • s e - tt tt * 6 tt * 5 tt tt 4 tt 3 tt 2 * * * * 1 * tt * * * DIAMETER 1 2 3 4 5 6 CLASS LIMIT 1.52.5 3.54.55.56.5 NEG. BINOMIAL DISTRIBUTION DEAD TREE LOCATIONS AND NUMBERS FOR PERIOD 2 B.C. FOREST SERVICE PLOT 69-GROUP I-SI: 120 NUMBER OF QUADRATS 25 QUADRAT SIZE (SQ.FT.) 566.28 (MILACRES) 13.00 NUMBER OF DEAD TREES 131 QUADRAT LIMIT 0.0 23.8 47.6 0.0 23.8 47.6 71.4 95.2 7 4 3 7 2 4 3 3 5 6 71.4 6 5 95.2 5 4 4 7 2 119.0 6 8 4 7 4 DIAMETER DISTRIBUTION OF DEAD TREES FOR PERIOD 3 B.C. FOREST SERVICE PLOT 69-GROUP I-SI:120 AGE=38.0 RISK-0.200 PCAT=0.0 PLOT SIZE=0./+l6 ACRE FREQUENCY 0 1 H 11 22 12 2 2 0 1 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 r— n 4 -7 2 I *_ DIAMETER 1 2 3 4 5 6 7 8 9 10 CLASS LIMIT 1.5 2. NEG. BINOMIAL DISTRIBUTION DEAD TREE LOCATIONS AND NUMBERS FOR PERIOD 3 B.C. FOREST SERVICE PLOT 69-GROUP I-SI: 120 NUMBER OF QUADRATS 12. QUADRAT SIZE (SQ.FT.) 1132.56 (MILACRE) 26.00 NUMBER OF DEAD TREES 6-5 •X-*L tt * * tt * * tt tt tt * * tt * tt * * •K- tt .V. * # * * * tt * * * * tt tt * * tt tt * tt tt tt * -Se- tt tt tt tt * tt * tt tt * * -* * tt * * 3 4 5 6 7 Co ! 3.5 4.5 5-5 6.5 7.5 8 QUADRAT LIMIT 0.0 33-7 67.3 101.0 134.6 0.0 6 9 4 5 53-7 6 3 8 5 67.3 4 3 6 5 293 GROWTH AND YIELD TABLE B.C. FOREST SERVICE PLOT 69-GROUP I-SI:120 PLOT SIZE=0.Z+16 RISK=0.200 PCAT=0.0 PER ACRE VALUES N 0.11346E 04 0.94952E 03 0.63462E 03 0 , i ( -7837E 03 AGE 0 . 1 8 0 0 0 E 02 0 . 2 8 0 0 0 E 02 O . 3 8 O O O E 02 0.Z+&000E 02 AVDBH 0.32288E 01 O.Z+77H-2E 01 0.66523E 01 0 . 8 2 6 3 9 E 01 TOPH 0 . / + 3 0 5 1 E 02 0.50310E 02 0.56110E 02 O.58O8/+E 02 A—MORTALITY DBH 0.1630AE 01 0.32237E 01 O.50OA5E 01 PERIODIC ANNUAL MORTALITY N 0.18510E 02 0 . 3 1 A 9 0 E 02 0.15625e 02 BA 0.26836E 00 0 . 1 7 9 0 5 E 01 0 . 2 1 3 4 3 E 01 VOL 0.23196E 01 0.23887S 02 0.36433E 02 PERIODIC ANNUAL PERCENT MORTALITY' 0.16314E 01 0.33163E 01 0.2A621E 01 CUMULATIVE MORTALITY N 0.18510E 03 0.50000E 03 0 .65625E 03 BA 0.26836E 01 0 .20588E 02 0 .A1932E 02 VOL 0.23196S 02 0.26207E 03 G.626AOE 03 B—YIELD GROSS YIELD CU.FT. 0.94358E 03 0.20A91E OA 0 . 3 2 4 6 3 E OA O.A3365E 0/+ SQ.FT. 0 . 64516E 02 0 . 1 2 0 7 2 E 03 0.17376E 03 0.22011E 03 NET YIELD CU.FT. 0 .9Z1558E 03 0.20239E Ok 0.29843E OA 0.37101E Ok SQ.FT. O.6A516E 02 0 . 1 1 8 0 A E 03 0.15317E 03 0.17818E 03 C—GROWTH PERIODIC ANNUAL GROSS INCREMENT CU.FT. O.IIO36E 03 0.11972E 03 0 . 1 0 9 0 2 E 03 SQ.FT. 0.56208E 01 0.53037E 01 O.A6349E 01 PERIODIC ANNUAL NET INCREMENT CU.FT. 0.1080/+E 03 0 . 9 5 8 3 2 E 02 0.72S8AE 02 SQ.FT. 0.53524E 01 0 . 3 5 1 3 2 E 01 0 . 2 5 0 0 6 E 01 MEAN ANNUAL GROSS INCREMENT CU.FT. 0.52552E 02 0.73183E 02 0.83430E 02 0.903W 02 SQ.FT. 0.35842E 01 0 . / + 3 1 1 5 E 01 0 .A5726E 01 O .A5856E 01 MEAN ANNUAL NET INCREMENT CU.FT. 0.52532E 02 0.72355E 02 0 . 7 8 5 3 3 E 02 0.77294E 02 SQ.FT.. 0 . 3 5 8 4 2 E 01 0./+2157E 01 O.Z+0308E 01 0.37120E 01 Appendix VIII ACTUAL AND SIMULATED STAND CHARACTERISTICS 1 294 GROUP I STAND: Na t u r a l (100% F i r ) , PLOT: R a i n i e r 2 , PLOT SIZE: 1 . 0 acre SITE INDEX (100 years) : 165 f e e t AGE 2 Years 50 PEF 60 l ACRE^ 70 So N Trees 294 (294) 0 255 (247) ±5_ (135) 162 (128) - 2 1 DBH Inches 1 1 . 8 ( 1 1 . 8 ) 0 1 4 . 2 ( 1 3 . 5 ) -5 . ( 1 7 . 4 ) 1 8 . 5 ( 1 8 . 7 ) +1 TOPH Feet 110 (111) +1 129 (124) (119) 151 (130) GROSS BA S q . f t . 221 (221) 0 275 (262) - 2 (294) 351 (324) - 8 NET BA Sq.Ft. 222 (222) 0 257 (246) zit_ (222) 301 (244) -19 GROSS YIELD C u . f t . 8521 (8615) -1 .12317 (11292) - 8 (12258 13116-)(14325) -2.1 NET YIELD Cu . f t . 8521 (8615) +1 11655 (10760) - 8 (9562) 16107 (11317) - 3 0 Output from m o r t a l i t y generator Model I I . Si m u l a t i o n c a r r i e d out by us i n g the a c t u a l t a l l y of t r e e diameters at age 5 0 , and two equations: Ann. DBH i i i c r . = O .246 + 0 . 0 1 3 DBH - 0 . 0 0 0 ? SI - 0 . 0 0 3 AGE; H = 4 6 . 3 1 0 + 7 . 5 2 4 DBH + O .308 BA - 0 .113 DBEr Stand age (AGE); number of t r e e s (N); t r e e of average diameter at b r e a s t height (DBH); top height (100 l a r g e s t t r e e s per acre) (TOPH); gross and net b a s a l area; gross and net y i e l d , i n cubic volume ( 1 . 5 inches p l u s ) . ^ F i r s t l i n e : a c t u a l value; second l i n e : simulated value; t h i r d l i n e : d e v i a t i o n i n percent of a c t u a l v a l u e . Appendix VIII ACTUAL AND SIMULATED STAND CHARACTERISTICS' 295 GROUP I I STAND: Nat u r a l ( 9 0 % ) , PLOT: B.C i.F.S. 8 4 , PLOT SIZE: 0.ZF0 acre SITE INDEX (IOC ) years) : 130 f e e t AGE 2 Years PER ACRE-' 2k 2k 8/£ N Trees 760 (760) 0 548 (475) =12 378 (320) -15 (272) 222 (150) z22 (140) DBH Inches 5 . 9 ( 5 . 9 ) o 7 . 8 ( 7 . 9 ) +1 1 0 . 1 ( 9 . 8 ) _ ^ ( 1 0 . 7 ) 1 4 . 6 ( 1 2 . 6 ) - I V . ( 1 2 . 6 ) TOPH Feet 75 (75) o_ 90 (87) - 3 103 (97) - 6 (104) 124 (105) (102) GROSS BA S q . f t . 146 (146) 0 201 (192) =it_ 249 (223) - 1 0 (239) 341 (246) - 2 8 ( 2 4 D NET BA S q . f t . 146 (146) p_ 183 (163) -11 210 (167) - 2 0 (172) 258 (131) zkl (120) GROSS YIELD C u . f t . 3491 (3587) +5 5698 (5409) - 5 8046 (6874) -14 (7818) 13077 (8302) - 3 6 (8189) NET YIELD C u . f t . 3491 (3587) +3 5388 (4874) - 9 7337 (5754) - 2 1 (6439) 11210 (5465) - 5 1 (5155) Output from m o r t a l i t y generator Model I I . Sim u l a t i o n c a r r i e d out by u s i n g the a c t u a l t a l l y of tre e diameters at age 3 4 , and two equations: Ann. DBH i n c r . = 0 . 4 4 9 + 0 . 0 2 5 DBH - 0 . 0 0 1 SI - 0 . 0 0 8 AGE; H = 1 . 8 l 6 + 8 . 8 2 2 DBH - 0 . 1 0 1 DBH . p See f i r s t t a b l e f o r i d e n t i f i c a t i o n . - ^ F i r s t l i n e : a c t u a l value; second l i n e : simulated value; t h i r d l i n e : d e v i a t i o n i n percent of a c t u a l value. Douglas f i r t r e e s only are taken i n t o account. Appendix VIII ACTUAL AND SIMULATED STAND CHARACTERISTICS' 296 1 GROUP I I I STAND: Natu r a l (75% F i r ) , PLOT: U.B.C.R.F. 107 , PLOT SIZE: 0 . 1 5 acre SITE INDEX (100 y e a r s ) : 89 f e e t -Z ,r r, 2^ ^ PER ACRE^ AGE Years 64 N Trees 606 606 0 DBH Inches 7 . 2 ( 7 . 2 ) g TOPH Feet 75 (75) p_ GROSS S q . f t . 172 BA (172) 0 NET BA S q . f t . 172 (172) 0 559 7 . 9 76 196 190 GROSS Cu . f t . 4931 5682 YIELD (4993) +1 NET C u . f t . 4931 5533 YIELD (4993) +1 (431) ( 8 . 1 ) (75) (186) (156) (5394) (4600) 80 404 8it (377) 9 . 0 75 213 178 6180 5277 ( 8 . 3 ) (75) (189) (141) (5363) (4067) T Output from m o r t a l i t y generator Model I I . Si m u l a t i o n c a r r i e d out by using the a c t u a l t a l l y of t r e e diameters at age 6 4 , and two equations: Ann. DBH i n c r . = 0 . 177 + 0 . 0 2 0 D3H P- 0 . 0 0 0 4 SI - 0 . 0 0 4 AGE; H = 4 . 5 1 2 . 6 3 0 DBH - O .513 DBH "See f i r s t t a b l e f o r i d e n t i f i c a t i o n . F i r s t l i n e : a c t u a l value; second l i n e : simulated value; t h i r d l i n e : d e v i a t i o n i n percent of a c t u a l v a l u e . Douglas f i r t r e e s o n l y are taken i n t o account. Appendix VIII ACTUAL AMD SIMULATED STAND CHARACTERISTICS' 297 GROUP IV STAND: Na t u r a l ( 5 0 % F i r ) , PLOT: BCFPRO 2 0 3 , PLOT SIZE: 0 . 5 0 acre SITE INDEX ( 1 0 0 y e a r s ) : 1 6 5 AGE 2 Years 3 1 !£ 52 -< & 2 1 M PER ACRE-3 N Trees 2 0 9 1 5 3 ( 2 1 4 ) ( 1 5 4 ) ( 1 2 4 ) ( 1 0 8 ) ( 8 6 ) ( 8 2 ) +2 +1 DBH Inches 1 1 . 0 1 4 . 0 ( 1 1 . 0 ) ( 1 3 . 4 ) ( 1 5 - 3 ) ( 1 7 . 0 ) ( 1 9 . 4 ) ( 2 0 . 5 ) 0 - 4 TOPH Feet 8 5 9 7 ( 8 6 ) ( 9 2 ) ( 9 6 ) ( 1 0 1 ) ( 9 6 ) ( 9 7 ) +1 ^5 GROSS S q . f t . 1 3 7 1 7 6 BA ( 1 4 0 ) ( 1 6 5 ) ( 1 8 6 ) ( 2 0 8 ) ( 2 2 8 ) ( 2 4 7 ) +2 - 6 NET BA S q . f t . 137 163 ( 1 4 0 ) ( 1 5 2 ) ( 1 5 9 ) ( 1 7 2 ) ( 1 7 6 ) ( 1 8 9 ) +2 - 7 GROSS C u . f t . A209 6 0 5 0 YIELD ( 4 9 9 5 ) ( 5 4 5 4 ) ( 6 4 1 D ( 7 4 5 4 ) ( 8 3 1 2 ) ( 9 2 7 9 ) +4 - 1 0 NET C u . f t . 4 2 0 9 5 6 7 8 YIELD ( 4 3 9 5 ) ( 5 1 3 0 ) ( 5 6 2 3 ) ( 6 3 9 D ( 6 7 3 7 ) ( 7 4 8 1 ) +4 - 9 "^Output from m o r t a l i t y generator Model I I . S i m u l a t i o n c a r r i e d out by u s i n g the a c t u a l t a l l y of t r e e diameters at age 33? and two equations: Ann. DBH i n c r . = 0 . 2 0 4 + 0 . 0 1 1 DBH - 0 . 0 0 0 9 SI - 0 . 0 0 2 AGE; H = - 3 . 7 8 7 + 5 . 5 1 2 DBH + 0 . 2 1 6 BA - 0 . 0 8 3 DBH 2 p See f i r s t t a b l e f o r i d e n t i f i c a t i o n . ^ F i r s t l i n e : a c t u a l value; second l i n e : simulated value; t h i r d l i n e : d e v i a t i o n i n percent of a c t u a l v a l u e . Douglas f i r t r e e s only taken i n t o account. Appendix VIII ACTUAL AND SIMULATED STAND CHARACTERISTICS 1 298 INDEPENDENT DATA STAND: N a t u r a l (100% F i r ) , PLOT: Crown ML1 A, PLOT SIZE: 0 . 2 5 acre SITE INDEX (100 y e a r s ) : 160 f e e t AGE N DBH GROSS BA GROSS YIELD NET YIELD Years Trees Inches S q . f t . NET BA S o . f t . Cu. f • C u . f t . 42 62 72 540 (540) o 9 . 5 ( 9 . 3 ) o 256 (256) 0 256 (256) 0 8109 ( 8 l 8 l ) +1 8109 ( 8 l 8 l ) +.1 PER ACRE-432 (424) - 2 1 1 . 2 ( 1 0 . 9 ) Z2 313 (309) o 295 (277) -b 9589 (9649) +1 9038 (8640) - 5 (392) ( 1 2 . 0 ) (356) (307) ( 3 6 4 ) ( 1 2.6) (390) (313) (10656) (11244) (9128) (9092) Output from m o r t a l i t y generator Model I I . Simulation c a r r i e d out by u s i n g the a c t u a l t a l l y of t r e e diameters at age 4 2 , and two equations: Ann. DBH i n c r . = O .246 + 0 . 0 1 3 DBH - p 0 . 0 0 0 6 SI - 0 . 0 0 3 AGE; H = 11 .767 + 1 2 . 3 1 2 DBH - 0 . 5 1 2 DBH . 2 See f i r s t t a b l e for i d e n t i f i c a t i o n . • ^ F i r s t l i n e : a c t u a l value; second l i n e : simulated value; t h i r d l i n e : d e v i a t i o n i n percent of a c t u a l value. Appendix VIII 299 ACTUAL AND SIMULATED STAND CHARACTERISTICS 1 INDEPENDENT DATA STAND: Na t u r a l (100% F i r ) , PLOT: Crown MLI B, PLOT SIZE: 0.25 acre SITE INDEX: (100 y e a r s ) : 160 f e e t AGE Years i±2 PEE ^ 3 ! ACRE^ 62 Z 2 N Trees 504 (504) o 356 (404) ±12 (336) (288) DBH Inches 9.7 (9.7) o 11.4 (11.1) -3 (12.6) (13.3) GROSS BA S q . f t . 257 (257) o 301 (310) ±1_ (355) (387) NET BA S q . f t . 257 (257) q 252 (272) +8 (292) (279) GROSS YIELD C u . f t . 7951 (7997) +1 8996 (9359) (10161) (10622) NET YIELD C u . f t . 7951 (7997) +1 7625 (8275) +8 (8281) (7794) Output from m o r t a l i t y generator Model I I . Sim u l a t i o n c a r r i e d out by u s i n g the a c t u a l t a l l y of t r e e diameters at age 42, and two equations: Ann. DBH i n c r . = O.246 + 0.013 DBH - P 0 . 0 0 0 6 SI - 0.003 AGE; H = 11.767 + 12.312 DBH - 0.512 DBHr. See f i r s t t a b l e f o r i d e n t i f i c a t i o n . ' F i r s t l i n e : a c t u a l value; second l i n e : simulated value; t h i r d l i n e ; d e v i a t i o n i n percent of a c t u a l value. Appendix VIII ACTUAL AND SIMULATED STAND CHARACTERISTICS' 300 1 INDEPENDENT DATA STAND: Natur a l (100% F i r ) , PLOT: Crown ML 2, PLOT SIZE: 0.10 acre SITE INDEX (100 years) : 150 f e e t AGE 2 Years 2it PER tcRE^ 6ji N Trees 2020 (2020) 0 800 (1850) +131 (1200) (790) (710) (420) DBH Inches 3-4 (3.4) 0 6.3 (4.4) -JV (5.8) (7.1) (7.7) (9.0) GROSS BA S q . f t . 125 (125) 0 206 (198) (254) (293) (319) (332) NET BA S q . f t . 125 (125) 0 174 (194) +11 (218) (217) (231) (185) GROSS YIELD C u . f t . 3209 (3235) +1 5832 (5473) -6 (7342) (8618) (9376) (9657) NET YIELD C u . f t . 3209 (3235) +1 5182 (5413) (6518) (6723) (7146) (5629) "Output from m o r t a l i t y generator Model I I . S i m u l a t i o n c a r r i e d out by u s i n g the a c t u a l t a l l y of tr e e diameters at age 24, and two equations: Ann. DBH i n c r . = O.246 + 0.013 DBH - 0.0006 SI - 0.003 AGE; K = 11.767 + 12.312 DBH - 0.512 DBH2 'See f i r s t t a b l e f o r i d e n t i f i c a t i o n . F i r s t l i n e : a c t u a l value; second l i n e : simulated value; t h i r d l i n e : d e v i a t i o n i n percent of a c t u a l v a l u e . 

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