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The development of a stand model for Douglas fir Newnham, R. M. 1964

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THE DEVELOPMENT OP A STAND MODEL FOR DOUGLAS F I R by R v M f NEWNHAM B. S c . , U n i v e r s i t y o f W a l e s , 1956 M. P., U n i v e r s i t y o f B r i t i s h C o l u m b i a , 1958 A THE S I S SUBMITTED I N PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OP DOCTOR OP PHILOSOPHY i n t h e D e p a r t m e n t o f FORESTRY We a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF B R I T I S H COLUMBIA O c t o b e r , 1964 In presenting this thesis i n p a r t i a l fulfilment of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library shall make i t freely available for reference and study. I further agree that per-mission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. I t i s understood that copying or publi-cation of this thesis for f i n a n c i a l gain sh a l l not be allowed without my written permission® Department of F o r e s t r y The University of B r i t i s h Columbia, Vancouver 8, Canada Date 1 s t September, 1 9 6 4 The University of British Columbia FACULTY OF GRADUATE STUDIES PROGRAMME OF THE FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY of ROBERT MONTAGUE NEWNHAM B.Sc.3 The University of Waless 1956 M.F., The University of British Columbia, 1958 IN ROOM 2353 FORESTRY AND GEOLOGY BUILDING WEDNESDAY, SEPTEMBER 23rd 1964, AT 3s00 P.M. . COMMITTEE IN CHARGE Chairman; I. McT. Cowan Charlotte Froese D.P. Ormrod B.G. G r i f f i t h J.H.G. Smith P.G. Haddock R.W. Wellwood External Examiner: J.W. Ker University of New Brunswick THE DEVELOPMENT OF A STAND MODEL FOR DOUGLAS FIR ABSTRACT A mathematical model has been developed to describe the growth of trees i n stands of Douglas f i r (Pseudotsuga.  menziesii (Mirb.) Franco) from age ten to age 100 years, An i n i t i a l square pattern of spacing was assumed. At age ten years the trees were assumed to be open-grown, that is,.growing i n diameter at breast height at a maximum rate. A regression of d.b.h. on age was obtained from eighteen open-grown, Douglas f i r trees measured on the Saanich Peninsula, Vancouver Island. The r e l a t i o n s h i p derived from these data agreed with further data c o l l e c t e d elsewhere i n the coastal regions of B r i t i s h Columbia and Washington and i n the i n t e r i o r of B r i t i s h Columbia. The d.b.h, growth of i n d i v i d u a l trees was predicted by f i v e -year periods. Relationships between crown width and d.b.h, were calculated from data on 426 open-grown, Douglas f i r trees. There was a close c o r r e l a t i o n between crown width and root spread for open-grown trees. A multiple regression equation was obtained for height of 869 trees on d.b.h. and basal area per acre. A l l regression equations calculated for use i n the model, were highly s i g n i f i c a n t s t a t i s t i c a l l y . , The model i s i n i t i a t e d with a matrix of 15 x 15 trees (or tree " l o c a t i o n s ' ) . The i n i t i a l , d.b.h, of each tree i s s p e c i f i e d and, from the crown' width/d.b.h. regressions, the crown width of each tree i s calculated. As long as the tree remains free of competition, t h i s c a l c ulated crown width i s reduced by 40 per cent by the reduction factor "REDFAC", to give the "competitive" crown width. This was because i t was found that, i n young Douglas f i r plantations, there could be considerable overlapping of the crowns before d.b.h. growth was reduced. As soon as competition sets i n the o r i g i n a l 40 per cent reduction i s systematically reduced. The proportion of the circumference of each tree that i s occupied by the crowns of surrounding competitors i s then calculated. This proportion indicates the amount of competition to which the tree i s being subjected and varies between zero, i f the tree i s open-grown, and one or more, i f the tree i s completely enclosed by the surrounding competitors. I f the reduction i s s u f f i c i e n t l y great, continued s u r v i v a l of the tree i s considered u n l i k e l y s and the tree i s assumed to have died. The p e r i o d i c d„b.h. growth of the surviving trees i s calculated at five-year i n t e r v a l s to age 100 years„ A l l c a l c u l a t i o n s are performed using am I.B.M. 7090 e l e c t r o n i c computer. A summary of the structure of the stand can be printed out at the end of each five-year period i f required. Height growth can be described by modifying the stand model by including an appropriate regression equation. Similarly., volume growth can be estimated by modifying the basic stand model. The mathematical model developed here s a t i s f a c t o r i l y describes the growth of Douglas f i r stands on an i n d i v i d u a l tree b a s i s s over a wide range of s i t e c o n d i t i o n s s stand d e n s i t i e s , amounts and d i s t r i b u t i o n s of mortality and thinning regimes. F i e l d data cannot be secured to evaluate the accuracy of a l l the tests made. However, there are no gross errors i n absolute values and r e s u l t s are accurate proportionately. The model described here can a i d the forester i n managing Douglas f i r stands i n the P a c i f i c Northwest. By simulating the growth of his stands from age ten to age 100 years i n a. few minutes he can study questions that would otherwise require several, human generations to evaluate. GRADUATE STUDIES F i e l d of Study: Forestry S t a t i s t i c a l Methods i n Forest Research J.H.G. Smith Problems i n S t a t i s t i c a l Methods J.H.G. Smith S i l v i c u l t u r e P.G. Haddock General Forestry Seminar The S t a f f Related F i e l d s : Programming and Numerical Algorithms Computer Programming Mathematical S t a t i s t i c s J.R.H. Dempster Charlotte Froese Lorraine Schwartz PUBLICATIONS Newnham, R.M. 1956. A simple height measuring instrument (A modification of Smythies dendrometer). Quart. J. For., 50a 208-212. Newnham, R.M. 1965. Stem form and the variation of taper with age and thinning regime. Forestry, 3_7. (Accepted for publication.) i ABSTRACT Su p e r v i s o r : P r o f e s s o r J . H. G. Smith A mathematical model has been developed to d e s c r i b e the growth of t r e e s i n stands of Douglas f i r (Pseudotsuga  m e n z i e s i i (Mirb.) Franco) from age ten t o age 100 y e a r s . An i n i t i a l square p a t t e r n of spacing was assumed. At age ten years the t r e e s were assumed to be open-grown, t h a t i s , growing i n diameter at b r e a s t h e i g h t at a maximum r a t e . A r e g r e s s i o n of d. b. h. on age was obtained from eighteen open-grown, Douglas f i r t r e e s measured on the Saanich Penin-s u l a , Vancouver I s l a n d . The r e l a t i o n s h i p d e r i v e d from these data agreed w i t h f u r t h e r data c o l l e c t e d elsewhere i n the c o a s t a l r e g i o n s of B r i t i s h Columbia and Washington and i n the i n t e r i o r of B r i t i s h Columbia. The d. b. h. growth of i n d i v i d u a l t r e e s was p r e d i c t e d by f i v e - y e a r p e r i o d s . R e l a t i o n s h i p s between crown width and d. b. h. were c a l c u l a t e d from data on 426 open-grown, Douglas f i r t r e e s . There was a c l o s e c o r r e l a t i o n between crown width and r o o t spread f o r open-grown t r e e s . A m u l t i p l e r e g r e s s i o n equation was obtained f o r h e i g h t of 869 t r e e s on d. b. h. and b a s a l area per acre. A l l r e g r e s s i o n equations c a l c u l a t e d f o r use i n the model were h i g h l y s i g n i -f i c a n t s t a t i s t i c a l l y . The model i s i n i t i a t e d w i t h a matrix of 15 x 15 t r e e s (or t r e e " l o c a t i o n s " ) . The i n i t i a l d. b. h. of each t r e e i s s p e c i f i e d and, from the crown width/d. b. h..regres-s i o n s , the crown width of each t r e e i s c a l c u l a t e d . As long i i a s t h e t r e e r e m a i n s f r e e o f c o m p e t i t i o n , t h i s c a l c u l a t e d crown w i d t h i s r e d u c e d by 4-0 p e r c e n t by t h e r e d u c t i o n f a c t o r "REDFAC", t o g i v e t h e " c o m p e t i t i v e " crown w i d t h . T h i s was b e c a u s e i t was f o u n d t h a t , i n y o u ng D o u g l a s f i r p l a n t a t i o n s , t h e r e c o u l d be c o n s i d e r a b l e o v e r l a p p i n g o f t h e crowns b e f o r e d. b. h. g r o w t h was r e d u c e d . As s o o n as c o m p e t i t i o n s e t s i n t h e o r i g i n a l 40 p e r c e n t r e d u c t i o n i s s y s t e m a t i c a l l y r e d u c e d . The p r o p o r t i o n o f t h e c i r c u m f e r e n c e o f e a c h t r e e t h a t i s o c c u p i e d by t h e crowns o f s u r r o u n d i n g c o m p e t i t o r s i s t h e n c a l c u l a t e d . T h i s p r o p o r t i o n i n d i c a t e s t h e amount o f c o m p e t i -t i o n t o w h i c h t h e t r e e i s b e i n g s u b j e c t e d and v a r i e s between z e r o , i f t h e t r e e i s open-grown, and one o r more, i f t h e t r e e i s c o m p l e t e l y e n c l o s e d by t h e s u r r o u n d i n g c o m p e t i t o r s . The f i v e - y e a r d . b . h. g r o w t h o f e a c h t r e e i s t h e n d e t e r -mined f r o m t h e d . b . h./age r e g r e s s i o n d e s c r i b e d a b o v e . D. b. h.• i n c r e m e n t i s r e d u c e d i n v a l u e by t h e p r o p o r t i o n o f t h e crown o c c u p i e d by c o m p e t i t o r s . I f t h e r e d u c t i o n i s s u f f i c i e n t l y g r e a t , c o n t i n u e d s u r v i v a l o f t h e t r e e i s con-s i d e r e d u n l i k e l y , and t h e t r e e i s assumed t o have d i e d . The p e r i o d i c d. b. h. g r o w t h o f t h e s u r v i v i n g t r e e s i s c a l c u l a t e d a t f i v e - y e a r i n t e r v a l s t o age 100 y e a r s . A l l c a l c u l a t i o n s a r e p e r f o r m e d u s i n g an I . B. M. 7090 e l e c t r o n i c c o m p u t e r . A summary o f t h e s t r u c t u r e o f t h e s t a n d can be p r i n t e d o u t a t t h e end o f e a c h f i v e - y e a r p e r i o d i f r e q u i r e d . H e i g h t ; g r o w t h can be d e s c r i b e d by m o d i f y i n g t h e s t a n d , model by i n c l u d i n g an a p p r o p r i a t e r e g r e s s i o n e q u a t i o n . S i m i l a r l y , volume g r o w t h can be e s t i m a t e d by i i i m o d ifying the b a s i c stand model. The mathematical model developed here s a t i s f a c t o r i l y d e s c r i b e s the growth of Douglas f i r stands on an i n d i v i d u a l t r e e b a s i s , over a wide range of s i t e c o n d i t i o n s , stand d e n s i t i e s , amounts and d i s t r i b u t i o n s of m o r t a l i t y and t h i n n i n g regimes. F i e l d data cannot be secured t o ev a l u a t e the accuracy of a l l the t e s t s made. However there are no gross e r r o r s i n a b s o l u t e values and r e s u l t s are ac c u r a t e propor-t i o n a t e l y . The model d e s c r i b e d here can a i d the f o r e s t e r i n managing Douglas f i r stands i n the P a c i f i c Northwest. By s i m u l a t i n g the growth of h i s stands from age ten t o age 100 years i n a few minutes he can study q u e s t i o n s t h a t would otherwise r e q u i r e s e v e r a l human gen e r a t i o n s t o e v a l u a t e . i v ACKNOWLEDGEMENTS The w r i t e r wishes t o acknowledge the help and u s e f u l c r i t i c i s m given t o him by Dr. J . H. G. Smith, P r o f e s s o r , F a c u l t y of F o r e s t r y , d u r i n g the two years he was working on t h i s p r o j e c t . The t h e s i s was reviewed by Drs. P. G. Haddock, J. H. G. Smith, and R. W. Wellwood of the F a c u l t y of F o r e s t r y and Dr. D. P. Ormrod of the D i v i s i o n of Plant Science i n the F a c u l t y of A g r i c u l t u r e . The w r i t e r i s g r a t e f u l f o r t h e i r c r i t i c a l review and a d v i c e . A s s i s t a n c e i n programming has been given by Messrs. R. J . Henderson and E. Froese of the U. B. C. Computing Centre. The f a c i l i t i e s of the Computing Centre have been f r e e l y p l a c e d at the d i s p o s a l of the w r i t e r . In the l a t e r stages of model development and t e s t i n g , most of the programs were run on the I. B. M. 7090 e l e c t r o n i c computer a t . t h e U n i v e r s i t y of Toronto. Much u s e f u l i n f o r m a t i o n about the Wind R i v e r Douglas f i r s p acing t r i a l s was obtained from Dr. D. L. Reukema, Research F o r e s t e r , Olympic Research Center, U. S. F o r e s t S e r v i c e . Mr. G. C. Warrack, Research F o r e s t e r , Research D i v i s i o n , B. C. F o r e s t S e r v i c e , s u p p l i e d the bulk of the data used t o c a l c u l a t e the hei g h t growth r e g r e s s i o n . Per-m i s s i o n t o use these data was given by the B. C. F o r e s t S e r v i c e and i s g r a t e f u l l y acknowledged. F i n a n c i a l support was giv e n by the F a c u l t y of F o r e s t r y , U n i v e r s i t y of B r i t i s h Columbia, i n the form of a U n i v e r s i t y F o r e s t F e l l o w s h i p and a Research A s s i s t a n t s h i p d u r i n g the 1962-3 s e s s i o n . The w r i t e r r e c e i v e d a N a t i o n a l Research C o u n c i l Studentship f o r the 1963-4 s e s s i o n . The r e s e a r c h p r o j e c t was supported by an Extra-Mural Research Grant from the Department of F o r e s t r y , Ottawa, from May, 1963. T h i s enabled the w r i t e r t o be employed f u l l - t i m e d u r i n g the two summers and a l s o gave a d d i t i o n a l f i n a n c i a l support d u r i n g the w i n t e r of 1963-4. The w r i t e r wishes t o thank a l l of these agencies f o r t h e i r support. v i TABLE OF CONTENTS Page ABSTRACT i ACKNOWLEDGEMENTS i v TABLE OF CONTENTS v i LIST OF TABLES . . . i x LIST OF FIGURES . . . . . x i LIST OF PLATES x v i l INTRODUCTION . . . 1 PART I. SUMMARY OF PAST WORK 7 The S p a t i a l P a t t e r n of F o r e s t Stands 8 The E f f e c t of the I n i t i a l Spacing on Growth . . . . 12 The D i s t r i b u t i o n of Trees i n a F o r e s t Stand . . . . 15 Growth . . 17 Diameter at Br e a s t Height and B a s a l Area 17 Height Growth 22 Volume Growth 24 Stand D e n s i t y 25 T h i n n i n g 27 N a t u r a l M o r t a l i t y 35 M o r t a l i t y due to Fungal A t t a c k . . . . .35 M o r t a l i t y due to Insect A t t a c k 39 N a t u r a l M o r t a l i t y through Suppression .42 C o n c l u s i o n s 43 PART I I . DEVELOPING A STAND MODEL FOR DOUGLAS FIR . . 44 Stand Model I 45 v i i Page Stand Model I I 46 The B a s i c P r i n c i p l e s and Assumptions . . . . . . . . 46 D e s c r i p t i o n of the Stand Model 56 R e s u l t s . . . . . 68 Number of Trees per Acre 72 Average Diameter at Breast Height 73 B a s a l Area per Acre 7^ 'REDPAC1 . 7^ Diameter Frequency D i s t r i b u t i o n s . . 76 D i s t r i b u t i o n s of Trees . 83 Stand Model IIA 87 C o n c l u s i o n s 89 PART I I I . TESTING STAND MODEL I I . 93 M o r t a l i t y F o l l o w i n g P l a n t i n g 93 Method of Generating D i s t r i b u t i o n s 94 R e s u l t s 96 S i t e Q u a l i t y 128 R e s u l t s 129 T h i n n i n g 129 Moderate Low Th i n n i n g 137 Severe Low T h i n n i n g . . . . 142 Crown T h i n n i n g or S e l e c t i o n F e l l i n g 142 Height Growth , 15^ C o n c l u s i o n s . . 156 SUMMARY AND SUGGESTIONS FOR FUTURE DEVELOPMENT . . . . . 160 v i i i Page REFERENCES 171 APPENDIX I. The D i s t r i b u t i o n s Encountered i n F o r e s t Research lQ& APPENDIX I I . D e s c r i p t i o n of the FORTRAN Program f o r Stand Model I I lB9 i x TABLES Table Page 1 The i n i t i a l d. b. h. matrix used i n the development of the model. Data from the U n i v e r s i t y Research F o r e s t , Haney 59 2 M o r t a l i t y by f i v e - y e a r p e r i o d s f o r D o u g l a s - f i r . Adapted from Barnes (U. B. C. F o r e s t Club, 1959) • Run IIA - 1 88 3 The growth of an unthinned stand. S i t e index: 140. Spacing: 6 .6 x 6.6 f t . Run I I - l . . . . 138 4 The e f f e c t of t h i n n i n g on growth and y i e l d . A l l t r e e s l e s s than (D-s) removed at i n t e r v a l s of 10 y e a r s . Spacing: 6 .6 x 6 .6 f t . Run I I - 9 . 1^ 1 5 The e f f e c t of t h i n n i n g on growth and y i e l d . A l l t r e e s l e s s than (D-s) removed at i n t e r v a l s of 20 y e a r s . Spacing: 6.6Sx 6.6 f t . Run 11-10 145 6 The e f f e c t of thinning_ on growth and y i e l d . A l l t r e e s between (D-s).and (D - 0 . 5 s ) j removed at i n t e r v a l s of 10 y e a r s . Spacing: 1 6.6 x 6.6 f t . Run 11-11 148 7 The e f f e c t of t h i n n i n g on growth and y i e l d . A l l t r e e s between (D-s) and (D - 0 . 5 s ) removed at i n t e r v a l s of 20 y e a r s . Spacing: 6.6 x 6 .6 f t . Run 11-12 . . . 151 8 The e f f e c t of thinning_ on growth . and_ y i e l d . A l l t r e e s between (D+0.75s) and (D+s) removed at i n t e r v a l s of 10 y e a r s . Spacing: 6.6 x 6.6 f t . Run 11-13 . . 153 9 Regressions of t r e e h e i g h t on d. b. h., number of t r e e s per a c r e , b a s a l area per acre, s i t e index and age . . . ' 157 10 The e f f e c t of amount and d i s t r i b u t i o n of m o r t a l i t y f o l l o w i n g p l a n t i n g on stand growth. S i t e index: 140. Mean d. b. h. o. b. at. age 10 years = 1.26 i n . Spacing: 6 .6 x 6 .6 f t . . . 164 11 The e f f e c t of s i t e q u a l i t y on stand growth. Amount of m o r t a l i t y f o l l o w i n g p l a n t i n g : n e g l i g i b l e . Spacing: 6 .6 x 6.6 f t . 165 X Table Page 12 The e f f e c t of t h i n n i n g on stand growth. S i t e index: 140. Mean d. b. h. o. b. at age 10 years = 1 . 2 6 , ,in. Amount of m o r t a l i t y f o l l o w i n g p l a n t i n g : n e g l i g i b l e . Spacing: 6 . 6 x 6 . 6 f t . . . . . . 166 13 The e f f e c t of i n i t i a l s p acing ( p l a n t i n g d i s t a n c e ) on stand growth. S i t e index: 1 4 0 . Mean d. b. h. o. b. at age 10 y e a r s : = 1 .26 i n . Amount of m o r t a l i t y f o l l o w i n g p l a n t i n g : n e g l i g i b l e . Run I I - l . . . . . . . . 167 x i ILLUSTRATIONS F i g u r e Page 1 (a) S t a e b l e r ' s method of measuring t r e e competition by the o v e r l a p method (From S t a e b l e r , 1951; F i g . 4) (b) The method of measuring t r e e competition i n the p r e s e n t study 8 2 The r e l a t i o n s h i p between crown width and d. b. h. o. b. 'of open-grown Douglas f i r . . . 48 3 The r e l a t i o n s h i p between d. b. h. i . b. and age. Open-grown Douglas f i r , Paul Lake 49 4 The r e l a t i o n s h i p between d. b. h. i . b. and age. Open-grown Douglas f i r , Saanich 51 5 The r e l a t i o n s h i p between d. b..hh i . b. and age. Open-grown Douglas f i r , Wind R i v e r 52 6 C a l c u l a t e d d. b. h. o. b./age curves (by i n c h c l a s s e s at age 10 years) 5^ 7 The r e l a t i o n s h i p between roo t growth and.crown width.of open-grown Douglas f i r , Paul Lake . . 55 8 Diameter frequency d i s t r i b u t i o n s of seven-year-o l d Douglas f i r p l a n t a t i o n s e s t a b l i s h e d at d i f f e r e n t i n i t i a l s p a c i n g s . U n i v e r s i t y Research F o r e s t , Haney . . . . . 57 9 The r e l a t i o n s h i p between number of t r e e s per acre and t o t a l age or years from p l a n t i n g (Duff, 1956) . . . . . 64 10 The r e l a t i o n s h i p between mean d. b. h. o. b. and t o t a l age or years from p l a n t i n g (Duff,. 1956) . 65 11 The r e l a t i o n s h i p between b a s a l area per acre a n d . t o t a l age or years from p l a n t i n g (Duff, 1956) . . . . . . . . . 66 12 The t r e e l o c a t i o n s t e s t e d f o r p o s s i b l e competitors of the t r e e b e i ng s t u d i e d ( l , J ) i n the model . 67 13 The r e l a t i o n s h i p between number, of t r e e s per' acre and age. Run II- 1 69 x i i F i g u r e Page 1 4 The r e l a t i o n s h i p between.mean d. b. h. o. b. and age. Run I I - l . . 7 0 1 5 The r e l a t i o n s h i p between b a s a l area per acre and age. Run I I - l 7 1 1 6 The r e l a t i o n s h i p between "REDFAC" and age. Run I I - l 7 5 1 7 Diameter frequency d i s t r i b u t i o n s . Spacing: 3 . 3 x 3 . 3 f t . Run I I - l 7 7 1 8 Diameter frequency d i s t r i b u t i o n s . Spacing: 6 . 6 x 6 . 6 f t . Run I I - l . . . . . 7 8 1 9 Diameter frequency d i s t r i b u t i o n s . Spacing: 9 . 9 x 9 - 9 f t . Run I I - l 7 9 2 0 Diameter frequency d i s t r i b u t i o n s . Spacing: 1 3 . 2 x 1 3 . 2 f t . Run I I - l 8 0 2 1 Cumulative d. b. h. o. b. frequency d i s t r i b u t i o n s . Spacing: (a) 3 . 3 x 3 . 3 f t . (b) 6 . 6 x 6 . 6 f t . Run I I - l 8 1 2 2 Cumulative d. b. h. o. b. frequency d i s t r i b u t i o n s . Spacing: (a) 9 . 9 f t . (b) 1 3 . 2 x 1 3 . 2 f t . Run I I - l , 8 2 . 2 3 I n i t i a l diameter matrix used f o r d e v e l o p i n g the model. Run I I - l 84 2 4 Stand s t r u c t u r e of the b a s i c model. Spacing: 6 . 6 x 6 . 6 f t . Run I I - l 8 5 2 5 Stand s t r u c t u r e of the b a s i c model at the d i f f e r e n t spacings at age 1 0 0 y e a r s . Run I I - l . 8 6 2 6 The r e l a t i o n s h i p between number of t r e e s per acre and age. Run I I A - 1 . 9 0 2 7 The r e l a t i o n s h i p between mean d. b. h. o. b. and age. Run I I A - 1 9 1 2 8 The r e l a t i o n s h i p between b a s a l area per acre and age. Run I I A - 1 9 2 2 9 The e f f e c t of amount and d i s t r i b u t i o n of m o r t a l i t y f o l l o w i n g p l a n t i n g on number of t r e e s per a c r e . Spacing: 3 . 3 x 3 . 3 f t . Runs I I - 2 t o I I - 6 9 7 x i i i F i g u r e Page 30 The e f f e c t of amount and d i s t r i b u t i o n of m o r t a l i t y . f o l l o w i n g p l a n t i n g on mean d. b. h. o . b . Spacing: 3.3 x 3.3 f t . Runs II-2 to II-6 . . . . . . . . . . . . . . 98 31 The e f f e c t of amount and d i s t r i b u t i o n of m o r t a l i t y f o l l o w i n g p l a n t i n g on b a s a l area per a c r e . Spacing: 3.3 x 3.3 f t . Runs II-2 t o II-6 . 99 32 The e f f e c t of amount and d i s t r i b u t i o n of m o r t a l i t y f o l l o w i n g ' p l a n t i n g on number of t r e e s per ac r e . Spacing: 6.6 x 6.6 f t . Runs II-2 t o II-6 . 100 33 The e f f e c t of amount and d i s t r i b u t i o n of m o r t a l i t y f o l l o w i n g p l a n t i n g on mean d. b. h. o. b. Spacing:"6.6'x'6.6 f t . Runs II-2 t o II-6 101 34 The e f f e c t of amount and d i s t r i b u t i o n of m o r t a l i t y f o l l o w i n g p l a n t i n g on b a s a l area per a c r e . Spacing: 6.6- x 6.6 f t . Runs II-2 t o II-6 102 35 The e f f e c t of amount and d i s t r i b u t i o n of m o r t a l i t y f o l l o w i n g p l a n t i n g on number of t r e e s per a c r e . Spading: 9.9 x 9.9 f t . Runs II-2 t o II-6 103 36 The e f f e c t of amount and d i s t r i b u t i o n of m o r t a l i t y f o l l o w i n g p l a n t i n g on" mean d. b. h. o. b. Spacing: 9-9 x 9-9 f t . Runs II-2 t o II-6 104 37 The e f f e c t of amount and d i s t r i b u t i o n of m o r t a l i t y f o l l o w i n g p l a n t i n g on b a s a l area per a c r e . Spacing: 9-9 x 9.9 f t . Runs II-2 to II-6 -. 105 38 The e f f e c t of amount and d i s t r i b u t i o n of m o r t a l i t y f o l l o w i n g p l a n t i n g on number of t r e e s per a c r e . Spacing: 13.2 x 13.2 ft-. Runs II-2 t o II-6 106 39 The e f f e c t of amount and d i s t r i b u t i o n of m o r t a l i t y f o l l o w i n g p l a n t i n g on mean d. b. h. o. b. Spacing: 13.2 x 13.2'ft. Runs II-2 to II-6 107 x i v F i g u r e Page 40 The e f f e c t of amount and d i s t r i b u t i o n of m o r t a l i t y f o l l o w i n g p l a n t i n g on b a s a l area per a c r e . Spacing: 13.2 x 13.2 f t . Runs I I -2 t o II-6 .108 .41 Cumulative d. b. h. o. b. frequency d i s t r i -b u t i o n s f o r (a) 10 and (b) 30 per cent d i s t r i b u t i o n s of m o r t a l i t y f o l l o w i n g p l a n t i n g . Spacing: 6.6 x 6.6 f t . Runs 11-2,3. 109 42 Cumulative d. b. h. o. b. frequency d i s t r i -b u t i o n s f o r (a) 50 per cent b i n o m i a l and (b) 50 per cent uniform ( r e c t a n g u l a r ) d i s t r i b u t i o n s of m o r t a l i t y f o l l o w i n g p l a n t -i n g . Spacing: 6.6 x .6.6 f t . Runs 11-4,5 . . H O 43 Cumulative d. b. h. o. b. frequency d i s t r i -b u t i o n s f o r two random i n f e c t i o n c e n t r e s (l4 per cent m o r t a l i t y ) . Run I I - 6 i l l 44 . I n i t i a l diameter matrix w i t h 10 per cent b i n o m i a l d i s t r i b u t i o n of mortality.. Run I.I-2 . 112 45 The e f f e c t of amount and d i s t r i b u t i o n of m o r t a l i t y f o l l o w i n g p l a n t i n g on stand s t r u c t u r e . B i n o m i a l d i s t r i b u t i o n (10 per cent m o r t a l i t y ) . ' Spacing: 6.6 x 6.6 f t . Run I I -2 113 46 • I n i t i a l diameter matrix w i t h 30 per cent b i n o m i a l d i s t r i b u t i o n of m o r t a l i t y . Run IX-3 . 115 47 The e f f e c t of amount and d i s t r i b u t i o n of m o r t a l i t y f o l l o w i n g p l a n t i n g on stand s t r u c t u r e . B i n o m i a l d i s t r i b u t i o n (30 per cent m o r t a l i t y ) . Spacing: 6.6 x 6.6 f t . Run II-3 H 6 48 I n i t i a l diameter matrix w i t h 50 per cent b i n o m i a l d i s t r i b u t i o n of m o r t a l i t y . Run I I -4 . 118 49 The e f f e c t of amount and d i s t r i b u t i o n of m o r t a l i t y f o l l o w i n g p l a n t i n g on stand s t r u c t u r e . B i n o m i a l d i s t r i b u t i o n (50 per cent m o r t a l i t y ) . Spacing:" 6 . 6 x 6.6 f t . Run II-4 . . .. . 119 50 I n i t i a l diameter matrix w i t h 50 per cent u n i f o r m ( r e c t a n g u l a r ) d i s t r i b u t i o n of m o r t a l i t y . Run II-5 121 XV F i g u r e Page 51 The e f f e c t of amount and d i s t r i b u t i o n of m o r t a l i t y f o l l o w i n g p l a n t i n g on stand s t r u c t u r e . Uniform ( r e c t a n g u l a r ) d i s t r i -b u t i o n (50 per cent m o r t a l i t y ) . Spacing: 6.6 x 6.6 f t . Run II-5 . . . . . . . . . . ...... 122 52 I n i t i a l diameter matrix w i t h two randomly l o c a t e d i n f e c t i o n c e n t r e s (1.4 per cent m o r t a l i t y ) . Run II-6 ' . . . . . '. • . . 124 53 The e f f e c t of amount -and d i s t r i b u t i o n of m o r t a l i t y f o l l o w i n g p l a n t i n g on stand s t r u c t u r e . Two random i n f e c t i o n c e n t r e s (l4 per cent m o r t a l i t y ) . Run II-6 . . . . . ... 125 54 The e f f e c t of s i t e q u a l i t y on number of t r e e s per a c r e . Spacing: 6.6 x 6.6 f t . Runs II-2, II-7 and IK-8 . 130 55 The e f f e c t of s i t e q u a l i t y on.mean d. b. h. o. b. Spacing: 6.6 x 6.6 f t . Runs 11-2,11-7 and .11-8 131 56 The e f f e c t of s i t e q u a l i t y on b a s a l area per a c r e . Spacing: 6.6 x 6.6 f t . Runs II-2, II-6 and I'I-7 . . . . . . . 132 57 Cumulative d. b. h. o. b. frequency d i s t r i b u t i o n s f o r (a) s i t e index 120 and (b) s i t e index 160. -Spacing: 6.6 x 6.6 f t . .Runs 11-7,8 . . . . . . . 133 58 The e f f e c t of s i t e q u a l i t y on stand s t r u c t u r e at age 100 y e a r s . Spacing:'6.6 x 6.6 f t . Runs 11-7,8 13^ 59 The e f f e c t of t h i n n i n g on number of t r e e s per a c r e . A l l t r e e s l e s s than (D - s) removed at 10-year i n t e r v a l s . Spacing: 6.6 x 6.6 f t . Run II-9 139 60 The e f f e c t of t h i n n i n g on gross, net (stand + t h i n n i n g s ) and stand b a s a l area y i e l d . A l l t r e e s l e s s than (D - s) removed at 10-year i n t e r v a l s . Spacing: 6.6 x 6.6 f t . Run II-9 • .. 14T> 61 The e f f e c t of t h i n n i n g on number of t r e e s per a c r e . A l l t r e e s l e s s than (D - s) removed at 20-year i n t e r v a l s . Spacing: 6.6 x 6.6 f t . Run 11-10 . . . . . . . . 143 x v i F i g u r e Page 62 The e f f e c t of t h i n n i n g on gross, net (stand + t h i n n i n g s ) and stand b a s a l area y i e l d . A l l t r e e s l e s s than (D - s) removed at 20-year i n t e r v a l s . Spacing: 6.6 x 6.6 f t . Run 11-10. . 144 63 The e f f e c t of t h i n n i n g on number_of t r e e s per a c r e . A l l t r e e s between (D - s) and (D - 0.5s) removed at 10-year i n t e r v a l s . Spacing: 6.6 x 6.6 f t . Run 11-11 '. 146 6 4 The e f f e c t of t h i n n i n g on gross, net (stand + t h i n n i n g s ) and stand b a s a l area y i e l d . A l l t r e e s between. (D - s) and (D - 0.5s) removed at 10-year i n t e r v a l s . Spacing: 6.6 x 6.6 f t . Run 11-11 . . . . '. . . .' , . . . . .... . . 147 65 The e f f e c t of t h i n n i n g on number of tree_s_ per. a c r e . A l l t r e e s between (D - s) and (D - 0.5s) removed at 20-year i n t e r v a l s . Spacing: 6.6 x 6.66ft. Run II-12 149 66 The e f f e c t of t h i n n i n g on gross, net (stand + t h i n n i n g s ) and stand b a s a l area y i e l d . A l l t r e e s between (D - s) and (D - 0.5s) removed at 20-year i n t e r v a l s . Spacing: 6.6 x 6.6 f t . Run 11-12 150 67 The e f f e c t of t h i n n i n g on number of t r e e s per ac r e . A l l t r e e s between (D + 0.75s) and . (D + s) removed at 10-year i n t e r v a l s . Spacing: 6.6 x 6.6 f t . Run 11-13 , 152 68 The e f f e c t of t h i n n i n g on gross, net (stand + t h i n n i n g s ) and stand b a s a l a r e a _ y i e l d . A l l t r e e s between (D + 0.75s) and (D + s), removed at 10-year i n t e r v a l s . Spacing: 6.6 x 6.6 f t . Run 11-13 155 69 C a l c u l a t e d t r e e h e i g h t / d . b. h. o. b. curves by stand b a s a l area i n sq. f t . per ac 158 70 C a l c u l a t i o n of © i n FORTRAN program . . . . . ... . 195 71 Example of program output . 196 x v i i PLATES P l a t e Page I Douglas f i r p l a n t a t i o n s on the U n i v e r s i t y Research F o r e s t , Haney, B. C. p l a n t e d -as two-year-old s e e d l i n g s i n the f a l l of 1957 at d i f f e r e n t spacings: (a) 3 x 3 f t . (b) 6 x 6 f t . (c) 9 x 9 f t . (d) 12 x 12 f t . Photographed i n September, 1963 . . . . 16 I I Group-dying of Douglas f i r caused by P o r i a w e l r i i Murr., Wind R i v e r , Washington. Photographed i n August, 1963 . . 36 I I I (a,b) Group-dying of Douglas f i r caused by Dendroctonus pseudotsugae Hopk., Paul Lake, Kamloops. Photographed i n J u l y , 1963 . 4 l I V Crown canopy photographs of 39-year-old Douglas f i r p l a n t a t i o n s e s t a b l i s h e d at (a) 4 x 4 f t . and (b) 8 x 8 f t . , Wind R i v e r , Washington. Photographed i n August, 1963 63 INTRODUCTION The present study was i n i t i a t e d i n 1962 to develop a stand model to describe the growth of Douglas f i r (Pseudotsuga menzies i i (Mirb. ) Franco) stands under the condit ions normally encountered i n the coastal region of B r i t i s h Columbia. A stand model should describe mathemati-c a l l y the growth of a stand by i n d i v i d u a l trees rather than by average height , average diameter at breast height , t o t a l basal area and volume (e i ther t o t a l or merchantable) as in published y i e l d tab les . Y i e l d tables general ly describe the "normal" or " fu l ly stocked" stand although i n Europe (e. g. Hummel and C h r i s t i e , 1953) and, more recent ly , i n North America (e. g. C l u t t e r , 1963) attempts have been made.to forecast growth under various th inning regimes. New Zealand tables for Douglas f i r (Duff, 1956) take the i n i t i a l spacing in the p lantat ions into account. These tables have the defect that once away from the normal or prescr ibed i t i s not poss ib le to forecast y i e l d on any sound bas i s . In the past ten years there has been a marked change i n th inning and p lant ing prac t i ce which has outdated many of the y i e l d tab les . In the perfect stand model the e f fect of the removal of any group of trees by th inning or natural morta l i ty can be described-by the model i n terms of increased or decreased growth and, what i s more important, the d i s t r i b u t i o n of that growth on the remaining trees can 1 2 a l s o b e e s t i m a t e d . I f s u c h a m o d e l c o u l d b e o b t a i n e d I t w o u l d b e o f g r e a t v a l u e t o t h e f o r e s t m a n a g e r i n d e t e r m i n i n g o p t i m u m I n i t i a l s p a c i n g i n p l a n t a t i o n s a n d t h e m o s t e c o n o m i c a l m e t h o d o f t h i n n i n g . P r o m p r e s e n t d a y k n o w l e d g e o f t h e g r o w t h o f s t a n d s a n d o f i n d i v i d u a l t r e e s i t s h o u l d b e p o s s i b l e t o o b t a i n a s a t i s f a c t o r y a p p r o x i m a t i o n t o t h e ' p e r f e c t ' m o d e l . F o r t h e p u r p o s e s o f t h e p r e s e n t s t u d y a s a t i s f a c t o r y m o d e l w a s d e f i n e d a s o n e w h i c h g a v e e s t i m a t e s o f m e a n d i a m e t e r g r o w t h , b a s a l a r e a g r o w t h . a n d n u m b e r o f t r e e s p e r a c r e w h i c h f e l l w i t h i n t h e b o u n d s o f t h e d a t a g i v e n i n s e v e r a l y i e l d t a b l e s f o r D o u g l a s f i r ( B a r n e s , (U. B . C . F o r e s t C l u b , 1959.), M c A r d l e e t a l , , 194-9, a n d D u f f , 1956) a n d w h i c h s h o w e d n o a b n o r m a l i t i e s i n d i a m e t e r d i s t r i b u t i o n . I n t h e m o d e l n o a t t e m p t h a s b e e n m a d e t o d e s c r i b e h e i g h t g r o w t h o f i n d i v i d u a l t r e e s i n d e t a i l , d u e t o t h e i m p r a c t i c a l i t y o f m e a s u r i n g t h e l a r g e n u m b e r o f t r e e h e i g h t s r e q u i r e d , o r v o l u m e g r o w t h , a l t h o u g h t h e l a t t e r i s u s u a l l y c o n s i d e r e d t o b e c l o s e l y c o r r e l a t e d w i t h b a s a l a r e a g r o w t h . 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 c a n b e c a l c u l a t e d f o r i n d i v i d u a l s t a n d s f r o m e m p i r i c a l d a t a . I t s h o u l d a l s o b e n o t e d t h a t , a l t h o u g h t h e c a l c u l -a t i o n s h a v e b e e n m a d e o n i n d i v i d u a l t r e e s , t h e m o d e l s h o u l d n o t b e u s e d t o d e s c r i b e t h e g r o w t h o f a n y p a r t i c u l a r t r e e b u t r a t h e r t h e s t a n d a s a w h o l e . I t I s a n t i c i p a t e d t h a t t h e g r e a t e s t v a l u e o f t h e m o d e l w i l l b e f o r c o m p a r i n g d i f f e r e n t m e t h o d s o f s t a n d m a n a g e m e n t . M o s t o f t h e i n f o r m a t i o n o n t h e g r o w t h o f D o u g l a s f i r u s e d i n t h i s s t u d y i s b a s e d o n d a t a c o l l e c t e d a t t h e 3 U n i v e r s i t y Research F o r e s t , Haney, B r i t i s h Columbia s i n c e 19^9. Much of t h i s i n f o r m a t i o n has been p u b l i s h e d by Smith et a l , (1961) and by G r i f f i t h (i960). F u r t h e r data were c o l l e c t e d i n the summer of 1963 i n the i n t e r i o r of B r i t i s h Columbia (Paul Lake, near Kamloops), i n the d r i e r p a r t s of the c o a s t a l r e g i o n of B r i t i s h Columbia (Saanich P e n i n s u l a , Vancouver I s l a n d ) , and on the h i g h e r r a i n f a l l , western slopes of the Cascade Mountains i n Washington State (Wind R i v e r ) . The growth of a f o r e s t stand i s a very complex s u b j e c t and one which has probably been s t u d i e d more i n t e n -s i v e l y than any other t o p i c i n f o r e s t r y . The p r i n c i p l e s of the mensurational a s p e c t s of growth - h e i g h t and diameter growth, s t o c k i n g , d e n s i t y , y i e l d , t h i n n i n g and c o m p e t i t i o n -are g e n e r a l l y understood but attempts to f i t mathematical models t o the b a s i c theory have e i t h e r been complicated and l i m i t e d i n t h e i r use ( C l u t t e r , 1962 and 1963; Crane, 1962; Czarnowski, 196l, and Meyer, 1930) or e l s e d i d not g i v e the s a t i s f a c t o r y r e s u l t s t h a t had been d e s i r e d ( S t a e b l e r , 1951). Due t o the complexity of the growth p a t t e r n s which i n v o l v e e x t e n s i v e computations, r e s e a r c h e r s i n the past have a l s o been p h y s i c a l l y l i m i t e d i n the amount of work they c o u l d undertake. With the advent of modern, high-speed, e l e c t r o n i c computers and t h e i r widespread use i n f o r e s t r y , t h i s problem has been reduced and the f o r e s t r e s e a r c h e r ' s h o r i z o n has been g r e a t l y widened (see Csizmazia, 1963). The advantages of these advances have been summarized by J e f f e r s (1962): 4 For the f i r s t time in.human h i s t o r y , not only i s i t p o s s i b l e f o r us t o undertake l a r g e s c a l e f o r e s t enumerations showing the s t a t e of the f o r e s t at a p a r t i c u l a r time, i t i s p o s s i b l e t o p r o j e c t these enumerations forward w i t h a f a i r degree of r e l i a b i -l i t y , and.to b u i l d from t h i s i n f o r m a t i o n , mathematical models of the f o r e s t . a n d i t s p r o d u c t i o n . The e f f e c t of management d e c i s i o n s on t h i s model may then be t e s t e d , so t h a t the f o r e s t e r can have a f a i r idea, of the con-sequence of h i s a c t i o n s , before treatments are ever a p p l i e d . T h i s•paragraph emphasizes the importance of stand models i n f o r e s t management and a l s o the need t o e s t a b l i s h a model, or models, which w i l l s a t i s f a c t o r i l y p r e d i c t growth under a wide range of stand c o n d i t i o n s . The growth of an i n d i v i d u a l t r e e i s dependent on a number of f a c t o r s . Baker ( 1 9 5 0 ) o u t l i n e d these as: 1 . The s i t e f a c t o r s , which are v i r t u a l l y f i x e d and can be but s l i g h t l y m o d i f i e d by the f o r e s t e r ' s a r t . 2. The. inhe r e n t c a p a c i t y of the lea v e s t o c a r r y on p h o t o s y n t h e s i s ( t o l e r a n c e i s i n v o l v e d h e r e ) . 3. The input of l i g h t energy, water and n u t r i e n t s . 4. The p h o t o s y n t h e t i c a r e a . U n t i l r e c e n t l y , the f o r e s t e r could c o n t r o l , or i n f l u e n c e , only the t h i r d and f o u r t h f a c t o r s . By f e r t i l i z i n g , i r r i -g a t i n g or d r a i n i n g , the f o r e s t e r can now improve the q u a l i t y of the s i t e and, i n t r e e - b r e e d i n g , i t i s p o s s i b l e . t o s e l e c t those t r e e s having the g r e a t e s t growth c a p a c i t y . The f a c t o r s i n f l u e n c i n g s i t e are m u l t i t u d i n o u s and, because of t h i s , no two f o r e s t s i t e s are ever e x a c t l y a l i k e . In a d d i t i o n t o the s o i l c h a r a c t e r i s t i c s (depth, p e r m e a b i l i t y , o r g a n i c mat-e r i a l content and n u t r i e n t c o n t e n t ) , s i t e i s a f f e c t e d by cl i m a t e ( p r e c i p i t a t i o n , temperature, hours of sunshine, f r o s t and winds), topography (type of slope, steepness, aspect 5 and e l e v a t i o n ) , and the a r t i f i c i a l e f f e c t s of man and domestic animals. In n a t u r a l stands growth may vary from year t o year due to c l i m a t i c changes so t h a t growth s t u d i e s should always be averaged over a number of years t o e l i m i n a t e , s u c h v a r i a t i o n as f a r as i s p o s s i b l e . Not only does growth vary from si.te to s i t e but growth w i t h i n a stand on a p a r t i c u l a r s i t e i s a l s o v a r i a b l e . I f we c o n s i d e r a stand where a l l the t r e e s are the same age and s p e c i e s , and have s u f f i c i e n t room t h a t they are not i n competition w i t h each other, there w i l l be s t i l l c o n s i d e r a b l e v a r i a t i o n i n growth (Baker, 1923). These d i f f e r e n c e s w i l l be due t o two sources: f i r s t , the i n h e r i t e d g e n e t i c a l d i f f e r e n c e s between t r e e s , and second, l o c a l d i f f e r e n c e s i n s i t e due t o changes i n m i c r o c l i m a t e , microtopography, the presence of l o c a l pockets of f e r t i l i t y , the presence of r o o t systems of a former stand and many other causes. T h i s p a r t i a l l y e x p l a i n s the l a c k of success i n . p r e d i c t i n g the growth of i n d i v i d u a l t r e e s ( S t a e b l e r , 1951). In p r e d i c t i n g growth, even i f i t i s on an i n d i v i d u a l t r e e b a s i s , the o v e r a l l e f f e c t must be con-s i d e r e d and not the i n d i v i d u a l e f f e c t . Smith (1964) d e s c r i b e d progress t o date and the problems i n v o l v e d i n the p r e p a r a t i o n of a model of stand development from stem a n a l y s i s . In order to apply h i s data secured by stem a n a l y s i s i n the t e s t i n g of v a r i o u s a l t e r n a -t i v e s i n f o r e s t management, i t was necessary t o develop a comprehensive, mathematical model of the growth of Douglas f i r stands which can be manipulated q u i c k l y and e a s i l y . The 6 w r i t e r has been concerned f o r two years w i t h the p r e p a r a t i o n and t e s t i n g of the r e q u i r e d mathematical model. In t h i s t h e s i s p r e p a r a t i o n of the model w i l l be d e s c r i b e d . The developed model w i l l be used t o t e s t the e f f e c t s of v a r i o u s amounts and d i s t r i b u t i o n s of m o r t a l i t y f o l l o w i n g p l a n t i n g on b a s a l area growth. Four d i f f e r e n t spacings r a n g i n g from 3 . 3 x 3 . 3 f t . t o 13.2 x 13.2 f t . w i l l be t e s t e d on three d i f f e r e n t s i t e q u a l i t i e s - poor ( s i t e index 120), medium ( s i t e Index 140) and good ( s i t e index l 6 o ) . Two wider spacings, 1 6 . 5 x 1 6 . 5 f t . and 19.8 x 19.8 f t . , w i l l be t e s t e d on the medium s i t e q u a l i t y . D i f f e r e n t types and i n t e n s i t i e s of t h i n n i n g w i l l be t e s t e d i n an attempt to determine the best t h i n n i n g schedule f o r the management of Douglas f i r p l a n t a t i o n s i n the c o a s t a l r e g i o n of B r i t i s h Columbia. PART I SUMMARY OP PAST WORK As stated prev ious ly , very l i t t l e a t tent ion has been paid to the development of comprehensive stand models, due to the enormity of the mathematical computations involved. Staebler (1951) suggested a number of hypotheses about the growth of Douglas f i r . His f i r s t hypothesis was that a t ree ' s growth var ied inverse ly to the competition which i t received from neighbouring trees . It was assumed that competition was d i r e c t l y proport iona l to some funct ion of the competing trees (e .g . diameter at breast height or crown c lass) and inverse ly proport iona l to t h e i r distance apart . A l l the competing trees taken together explain the growth of the study tree . Staebler ' s "area of overlap" hypothesis i s probably of most in teres t i n r e l a t i o n to the present study. According to t h i s hypothesis , trees require a c i r c u l a r area, which var ies with the d. b . h . of the t ree , in which to grow. Two trees which are growing so close together that t h e i r c i r c l e s overlap are considered to be competing, with each other. The growth of any tree i s inverse ly proport iona l to the amount of t h i s overlap, d' ( F i g . l a ) . Staebler suggested that competition might be better measured by the area of overlap rather than the l i n e a l measurement. In the present study an angular measure, Q, has been used to measure tree competition ( F i g . l b ) . Stae.bler's t h i r d hypothesis was to 7 8 P i g . l . ( a ) S t a e b l e r ' s method of. measuring t r e e competition by the o v e r l a p method (from S t a e b l e r , 1 9 5 1 : F i g . *0 . (b) The method of measuring t r e e c o m p e t i t i o n i n the present study. d e f i n e c ompetition as a f r a c t i o n of " f u l l growth". " F u l l growth" was d e f i n e d as two standard d e v i a t i o n s above the b a s i c diameter growth/d. b. h. curve f o r the stand. A curve of " f u l l growth" could then be used as u n i f o r m r e f e r e n c e p o i n t f o r any stand. U n f o r t u n a t e l y , the r e g r e s s i o n equations t h a t S t a e b l e r developed t o d e s c r i b e the diameter growth of i n d i -v i d u a l t r e e s d i d not remove much of the v a r i a t i o n i n growth between t r e e s , p o s s i b l y due to the small s i z e of the sample he was able to use (40 t r e e s ) . S t a e b l e r ' s r e s e a r c h o u t l i n e d above i s probably the only attempt t h a t has been made to develop a mathematical model t o d e s c r i b e the growth of a f o r e s t stand on an i n d i v i -d ual t r e e b a s i s . Other methods have used stand averages; these are d e s c r i b e d below- to g e t h e r w i t h other f a c t o r s which a f f e c t the growth of f o r e s t stands. The S p a t i a l P a t t e r n of F o r e s t Stands " S p a t i a l p a t t e r n " i s the arrangement of t r e e stems w i t h i n the stand. T h i s arrangement i s u s u a l l y c l e a r l y d e f i n a b l e 9 i n the e a r l y l i f e of a p l a n t a t i o n as the t r e e s w i l l normally have been p l a n t e d i n some g e o m e t r i c a l arrangement at roughly equal d i s t a n c e s . A square p a t t e r n i s u s u a l l y adopted as t h i s g i v e s a balanced, equal space t o each t r e e and i s easy to apply i n p r a c t i c e . Although there i s always a c e r t a i n amount of m o r t a l i t y , and sometimes ingrowth of n a t u r a l l y regenerated t r e e s , the p l a n t i n g p a t t e r n can u s u a l l y be seen f o r some time. P l a n t i n g the t r e e s c l o s e r w i t h i n the rows than the d i s t a n c e between the rows i s a f a i r l y common p r a c t i c e i n p o p l a r c u l t i -v a t i o n and may become more important i n the establishment of Douglas f i r p l a n t a t i o n s i n the P a c i f i c Northwest. Van Slyke (1964 a,b) has adopted the systematic designs of Nelder (1962) f o r t e s t i n g the e f f e c t s of s p a c i n g and r e c t a n g u l a r i t y i n f o r e s t stands. The t r e e s i n a n a t u r a l stand are never e q u a l l y spaced and t h e i r s p a t i a l p a t t e r n i s o f t e n i n d e s c r i b a b l e by mathematical formulae. Where r e g e n e r a t i o n i s dense and com-p e t i t i o n i s i n t e n s e , or when the stand becomes o l d e r , the s p a t i a l p a t t e r n does become more r e g u l a r . Development of p a t t e r n can be i l l u s t r a t e d simply. I f a number of pennies are p l a c e d i n a f l a t t r a y so t h a t they do not o v e r l a p and the t r a y then t i l t e d so t h a t the c o i n s s l i d e t o one edge i t w i l l be seen t h a t , except f o r the edge pennies, each penny i s surrounded by s i x other pennies. I f the c o i n s could now be compressed so t h a t the spaces between them were e l i m i n a t e d , the c o i n s would change t h e i r shape to s i x - s i d e d polygons whose c e n t r e s would be l o c a t e d at the 10 corners of e q u i l a t e r a l t r i a n g l e s . Such a phenomenon i s common in nature when the u n i t s , or c e l l s , are occupying a l l the ava i l ab l e space as, for example, in a honeycomb. Dice (1952) adopted t r iangu lar spacing in descr ib ing the s p a t i a l pattern of p l a n t s . For these reasons i t may be argued that the s p a t i a l pattern of natural stands should be t r i a n g u l a r since the stand w i l l tend to u t i l i z e the s i t e completely. However, few stands have 100 per cent crown closure and by adopting.an a r b i t r a r y square spacing, allowance can be made for t h i s . Hummel (195^) suggested that square spacing was at least as j u s t i f i a b l e as t r i a n g u l a r spacing because the d i s t r i b u t i o n of trees i n a stand was never quite regu lar . It i s a lso simpler i n prac t i ce and fores ters are more accustomed to th ink i n terms of i t . Bright (1914), Crane (1962), Lemmon and Schumacher (1963), Smith (1958 and 1963) and Wiley (1959) have a lso adopted the square spacing concept to allow for the spaces between crowns in the canopy. The argument that a stand i s only f u l l y u t i l i z i n g a s i t e when the crown closure i s more or less complete i s f a l l a c i o u s i n some instances . In South A f r i c a i t was found that root competition set in long before the crowns of plantation-grown, exot ic pines came into competition (Hi ley , 19^8 and 195^)• When water i s scarce, growth i s more inf luenced by root r e s t r i c t i o n than by crown r e s t r i c t i o n (Hi ley , 1959)• Part of the root system of trees normally extends beyond the perimeters of the crowns and may i n t e r l o c k with the roots of the surrounding trees (Spurr, 1952). This has been found s p e c i f i c a l l y to be so i n Douglas f i r (Hengst, 1938; and McMinn, 1955 and 1963).. McMinn (1963), however, found that the greatest concentration of feeding roots was confined to an area considerably smaller than the area occupied by the crown and that the spread of the root sys-tems was r e s t r i c t e d by the presence of other root systems in the s o i l . Bright (1914), who studied yellow pine (Pinus  ponderosa Laws.) , claimed that the roots rarely .extended beyond the crown. It appears probable that he did not measure the f u l l extent of the root systems as he confined h i s mea-surements to the roots of wind-blown trees . The various methods of descr ib ing pattern in plant ecology have been described i n d e t a i l by Greig-Smith (1957.). Foster and Johnson (1963c) have described the pattern and frequency d i s t r i b u t i o n of forest disease In Douglas f i r p lantat ions on Vancouver I s land. Although the i n i t i a l pat-tern of spacing must have been regu lar , or nearly so, and the p lantat ions had only been establ i shed th i r teen years , the d i s t r i b u t i o n of the l i v i n g trees was described as regular in only one of the f ive areas sampled. • In the two areas with the heaviest morta l i ty the d i s t r i b u t i o n was aggregated whi l s t i n the remaining two areas i t was i r r e g u l a r . Root-rot affected trees were a lso aggregated whereas trees affected by f r o s t -l e s i o n and sunscald conformed to an i r r e g u l a r pat tern . The negative binomial d i s t r i b u t i o n was found to adequately des-cr ibe the d i s t r i b u t i o n of r o o t - r o t affected trees . The actual s p a t i a l pattern of a fores t canopy can best be determined from crown canopy photographs taken from 12 below or from stand maps. Stand maps have been drawn by some European f o r e s t e r s but these have g e n e r a l l y been of uneven-aged, stands (Miegroet, 1950) or e l s e are not e x t e n s i v e enough to draw d e f i n i t e c o n c l u s i o n s ( K o s t l e r , 1953). The E f f e c t of I n i t i a l Spacing on Growth The " i n i t i a l s p a c i n g " i n p l a n t a t i o n s i s the p l a n t i n g d i s t a n c e or, i n n a t u r a l l y regenerated stands, i t i s the average d i s t a n c e between ' e s t a b l i s h e d ' s e e d l i n g s . What i s meant by ' e s t a b l i s h e d ' w i l l depend t o a . l a r g e extent on the s i t e and s p e c i e s . D i c k (1963) d e f i n e d an e s t a b l i s h e d ponderosa pine s e e d l i n g as "one at l e a s t 1 f o o t i n height or growing i n h e i g h t at a r a t e of at l e a s t 1/10-foot per year". These f i g u r e s should probably be doubled f o r Douglas f i r i n the c o a s t a l r e g i o n of B r i t i s h Columbia because of i n c r e a s e d competition from more v i g o r o u s ground v e g e t a t i o n . Many spa c i n g t r i a l s have been c a r r i e d out to t e s t the e f f e c t s on growth of d i f f e r e n t p l a n t i n g d i s t a n c e s . In the P a c i f i c Northwest a s e r i e s of Douglas f i r p l a n t a t i o n s was e s t -a b l i s h e d at Wind R i v e r , Washington, i n 1925 at spacings vary-i n g from 4 x 4 to 12 x 12 f t . The establishment and the r e s u l t s of t h i s experiment have been d e s c r i b e d i n d e t a i l i n v a r i o u s unpublished r e p o r t s of the P a c i f i c Northwest F o r e s t and Range Experiment S t a t i o n of the United S t a t e s F o r e s t S e r v i c e (Isaac, 1926; Isaac and Meagher, 1936; Isaac and Petersen, 1940; and Reukema, 1961.) . Less d e t a i l e d r e p o r t s have been.published by E v e r s o l e (1955) and by Reukema (1959). Morse (1962) has c a r r i e d out an economic a n a l y s i s u s i n g data from the Wind R i v e r experiment to determine the optimum i n i t i a l s p a c i n g i n f o r e s t p l a n t a t i o n s . The i960 measurements from t h i s experiment (Reukema, 196l) showed t h a t b a s a l area decreased w i t h i n c r e a s e i n spacing while c u b i c volume ( t r e e s 1.5 i n . d. b. h. and l a r g e r ) i n c r e a s e d with.wider s p a c i n g . T h i s i n c r e a s e i n volume was more pronounced when only t r e e s 6.5 i n . i n d. b. h. and g r e a t e r were c o n s i d e r e d . The d. b. h. of the average t r e e i n the 12 x 12 f t . , s p a c i n g was more than twice t h a t i n the 4 x 4 f t . s p a c i n g . Merchantable cu b i c volume i n the 12 x. 12 f t . spacing, was three times t h a t i n the 4 x 4 f t . s p a c i n g . Net p e r i o d i c annual growth (1957-1960) by a l l measures was p r o g r e s s i v e l y b e t t e r w i t h wider s p a c i n g . M o r t a l i t y was c o n f i n e d t o t r e e s s i x inches d. b. h. and under. The l a r g e s t t r e e s per acre contained 57 per cent of the t o t a l stand c u b i c volume i n the 12 x 12 f t . spacing but only nine-teen per cent i n the 4 x 4 f t . s p a c i n g . T o t a l h e i g h t of the dominant and codominant t r e e s appeared to be g r e a t e s t i n the widest spacings (Reukema, 1959)• In Great B r i t a i n , Mackenzie (1951) found t h a t i n i t i a l s p a cing (4 x 4, 6 x 6 or 8 x 8 f t . ) had no e f f e c t on the height growth of Douglas f i r or the other c o n i f e r s t e s t e d . He a l s o found t h a t there was c o n s i d e r a b l e i n t e r l a c i n g of the branches i n the c l o s e r p l a n t i n g s . The dominant h e i g h t growth of south-ern p i n e s , e s t a b l i s h e d at spacings from 4 x 4 to 16 x 16 f t . , was not a f f e c t e d by s p a c i n g d u r i n g the f i r s t f o u r t e e n growing seasons (Ware and S t a h e l i n , 19^8). S i m i l a r r e s u l t s were 14 obtained w i t h Norway spruce ( P i c e a a b i e s Karst.) i n B a v a r i a ( G u i l l e b a u d , 1951). I t seems reasonable t o suppose t h a t , p r o v i d i n g the average h e i g h t of the dominant t r e e s or of the 100 l a r g e s t t r e e s per acre, i s used as the parameter, spacing does not a f f e c t the f i r s t 30 to 40 y e a r s ' h e i g h t growth, except at the extremes of open sp a c i n g or very c l o s e s p a c i n g . On the other hand, diameter growth i s g r e a t l y improved w i t h wider s p a c i n g because of the e x t r a space a v a i l a b l e f o r crown development. U n t i l r e c e n t l y , e s p e c i a l l y i n Europe, the p r a c t i c e has been to e s t a b l i s h p l a n t a t i o n s at r e l a t i v e l y c l o s e spacings (4 x 4 t o 6 x 6 f t . ) and then t o t h i n e a r l y i n the r o t a t i o n . T h i s r e s u l t s i n h i g h . i n i t i a l c o s t s of establishment and, where e a r l y t h i n n i n g s are u n s a l e a b l e , f u r t h e r f i n a n c i a l l o s s . I f t h i n n i n g i s ignored the crop's r o t a t i o n may have to be lengthened. Smith (l958),and Smith _et j a l . ( l 9 6 l ) have sug-gested e s t a b l i s h i n g stands at wide spacings which w i l l permit the r a p i d growth r a t e c h a r a c t e r i s t i c of open-growth t r e e s d u r i n g the e a r l y p a r t of the r o t a t i o n but which, by the time the t r e e s have reached h a r v e s t a b l e s i z e , would be normally stocked. The i n i t i a l spacings could be as much as twelve f e e t or g r e a t e r . Such an idea i s not e n t i r e l y new. S t a f f o r d (1931) advocated " s k e l e t o n p l a n t i n g " , t h a t i s , p l a n t i n g the u l t i m a t e stand, on the Swann F o r e s t i n Massachusetts. In South A f r i c a p l a n t a t i o n s have been e s t a b l i s h e d at spacings of 9 x 9 f t . or g r e a t e r f o r a number of years ( H i l e y , 1959). Cromer and Pawsey (1957) found f o r r a d i a t a pine (Pinus r a d i a t a D. Don.) t h a t the optimum spa c i n g f o r maximum merchantable 15 volume growth at age f i f t e e n years was between 9 x 9 and 10 x 10 f t . Cook (1963) observed t h a t the t r a d i t i o n a l 6 x 6 f t . s p a c i n g i n the n o r t h e a s t e r n United States was c o s t l y to e s t a b l i s h , produced much unusable wood and r e q u i r e d c u l t u r a l o p e r a t i o n s t h a t were d i f f i c u l t t o perform w i t h modern equip-ment and l a b o u r . He advocated a 6 x 10 f t . . p l a n t i n g program which.would produce adequate s t o c k i n g and maximum merchantable volume on w e l l formed stems. I t i s almost c e r t a i n t h a t wide i n i t i a l s p a c i n g w i l l be more e x t e n s i v e l y used i n the establishment of Douglas f i r i n the P a c i f i c Northwest i n the f u t u r e . To t e s t the e f f e c t of spacings v a r y i n g from 3 x 3 f t . to 15 x 15 f t . , experimental p l a n t a t i o n s of Douglas f i r were e s t a b l i s h e d i n 1957 on the U n i v e r s i t y Research F o r e s t at Haney (U. B. C. F a c u l t y of F o r e s t r y , 1959)- Four of these p l a n t a t i o n s are shown as of October, 1963, i n P l a t e s Ia-d. The poorer growth i n the p l a n t a t i o n e s t a b l i s h e d at 12 x 12 f t . i s probably due to s o i l compaction as a r e s u l t of l o g g i n g the p r e v i o u s stand. The D i s t r i b u t i o n of Trees In a F o r e s t Stand The d i s t r i b u t i o n or "manner of occurrence" of t r e e s i n a f o r e s t stand g i v e s i n f o r m a t i o n on the r e g u l a r i t y , or i r r e g u l a r i t y , of the s c a t t e r of the t r e e s . I t i s important t h a t the d i s t r i b u t i o n i s known as, f o r i n s t a n c e , the volume growth behavior of t i g h t l y c l u s t e r e d groups of t r e e s of a l l s i z e s w i l l be d i f f e r e n t from the behavior of evenly spaced t r e e s of more or l e s s u n i f o r m s i z e growing under s i m i l a r s i t e PLATE I: Douglas f i r p l a n t a t i o n s on the U n i v e r s i t y Research F o r e s t , Haney, B. C , p l a n t e d as two-year-old s e e d l i n g s i n the f a l l of 1957 at d i f f e r e n t s p a c i n gs: (a) 3 x 3 f t . (b) 6 x 6 f t . (c) 9 x 9 f t . (d) 12 x 12 f t . Photographed i n September, 1963. 17 c o n d i t i o n s (Grosenbaugh, 1948). The h i g h e s t y i e l d s should be obtained when the t r e e s are spaced i n a r e g u l a r p a t t e r n (Poster and Johnson, 1963b). A summary of the frequency d i s t r i b u t i o n s t h a t have been found t o have a p p l i c a t i o n s i n f o r e s t r y i s given i n Appendix I. A d e s c r i p t i o n of some of the d i s t r i b u t i o n s found i n sampling f o r e s t stands has been given by Smith and Ker (1958). The d i s t r i b u t i o n of the t r e e s i s obtained by t a k i n g a number of sample p l o t s and r e c o r d i n g the number of t r e e s i n each p l o t or, a l t e r n a t i v e l y , d i v i d i n g the p l o t i n t o a number of s u b - p l o t s and counting the number of stocked s u b - p l o t s , a sub-plot being c l a s s i f i e d as stocked I f i t c o n t a i n s an e s t -a b l i s h e d s e e d l i n g of the d e s i r e d s p e c i e s . The parameters of most d i s t r i b u t i o n s are estimated from the mean and/or the v a r i a n c e of the sample (see Appendix I ) . The d i s t r i b u t i o n w i l l vary a c c o r d i n g t o the s i z e of quadrat used (Smith and Ker, 1957). Growth Diameter at Breast Height and B a s a l Area B r e a s t h e i g h t i s taken at 4 f t . 6 i n . i n North America and i n New Zealand, 4 f t . 3 i n . i n Great B r i t a i n and 1.3 metres (4.2650 on the c o n t i n e n t of Europe. The d i f f e r -ences i n diameter growth i n these t h r e e systems can be con-s i d e r e d to be n e g l i g i b l e . The age of a p l a n t a t i o n i n Great 18 B r i t a i n and the r e s t of Europe and i n New Zealand, i s u s u a l l y reckoned from the date of the p l a n t a t i o n ' s establishment and not from the date of germination of the seed as i n North Ameriica. These f a c t s should be remembered when c o n s u l t i n g y i e l d t a b l e s . The methods of p r e d i c t i n g diameter growth t h a t are commonly used i n f o r e s t r y have been d e s c r i b e d by Spurr (1952) and by Husch ( 1 9 6 3 ) . For Douglas f i r i n Washington and Oregon Spurr (1952) gave the f o l l o w i n g formulae f o r b a s a l area growth: G B = 8 5 . 3 1 - 2.995B; or G f i = 87 .30 - 2.039B - 0 . 3 9 6 A ; where Gg i s the b a s a l area growth In square f e e t per acre f o r a p e r i o d of f i f t e e n y ears, B i s the b a s a l area at the begin-ning of the p e r i o d , and A i s the age of the stand. Spurr l a t e r s t u d i e d the growth of Douglas f i r i n New Zealand and found t h a t , r e g a r d l e s s of the type of t h i n n i n g c a r r i e d out,, the r e l a t i o n s h i p between mean b a s a l area of the 100 l a r g e s t t r e e s per acre and age was l i n e a r .(Spurr, 1 9 6 3 ). For p r e d i c t i n g the mean diameter i n inches of the 100 l a r g e s t t r e e s per acre (D) Spurr gave the equation: D = y i 3 . 5 A - 160 where A i s the age of the p l a n t a t i o n In years from the date of es t a b l i s h m e n t . Warrack (1959a) s t u d i e d the diameter growth f o l l o w -i n g t h i n n i n g of an e i g h t e e n - y e a r - o l d stand of Douglas f i r and found t h a t 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 increment. The i n c l u s i o n of other v a r i a b l e s , such as crown width or crown index (crown width x crown l e n g t h ) , d i d not s i g n i f i c a n t l y improve the r e s u l t s . Rouse (1962) has p u b l i s h e d estimates of r a d i a l growth of Douglas f i r f o r f i v e - and ten-year p e r i o d s f o r d i f f e r e n t i n i t i a l diameters. His estimates were based on the B r i t i s h y i e l d t a b l e s .for Douglas f i r (Hummel and C h r i s t i e , 1953). Ker (1953) and Smith et a l . (1961) d e s c r i b e d the growth of i n d i v i d u a l Douglas f i r t r e e s . A d e t a i l e d , study of the r a d i a l growth of Douglas f i r and i t s r e l a t i o n t o c l i m a t e and s o i l has been.made by G r i f f i t h (i960) on the U n i v e r s i t y Research F o r e s t over a . p e r i o d of f i v e y e a r s . Crane (1962) has given formulae f o r p r e d i c t i n g the average b a s a l area increment per t r e e f o r r a d i a t a pine depending on whether the stand i s open-grown, dense, or i n the t r a n s i t i o n p e r i o d between open-grown and dense. Not only i s i t necessary to p r e d i c t diameter and b a s a l area growth of both stands and i n d i v i d u a l t r e e s , i t i s a l s o necessary t o be able to d e s c r i b e the manner i n which the d i s t r i b u t i o n of diameter c l a s s e s v a r i e s w i t h age. One of the e a r l i e s t , and probably the most e x t e n s i v e , s t u d i e s In t h i s f i e l d was c a r r i e d out by Meyer (1930). Although diameter d i s -t r i b u t i o n s may approximate the normal d i s t r i b u t i o n , f o r c i n g them to do so o f t e n leads to s e r i o u s e r r o r s i n stand d i s t r i b u t i o n t a b l e s . For t h i s 1 reason g r a p h i c a l methods are c o n s i d e r e d more a c c e p t a b l e than mathematical methods i n f i t t i n g frequency d i s t r i b u t i o n curves (Spurr, 1952). To be analysed mathematically the frequency d i s t r i b u t i o n s have t o 20 be defined by.-parameters which take into account departures from the symmetrical, normal d i s t r i b u t i o n . These parameters are the coe f f i c i en t of asymmetry (a measure of skewness) and the coe f f i c i en t of excess (which represents approximately the extent to which the actual d i s t r i b u t i o n d i f f e r s i n height from the corresponding normal curve) . The C h a r l i e r Type A curve deviates around the normal frequency curve and i s derived from: where: , L N 2 | j T T faty=Jzn&e. = &0(x) 5 004 being the normal p r o b a b i l i t y funct ion 0j(x)= S40m(^ _) , where i s the t h i r d der ivat ive of (^ (x)= SS0'E(^) , where i s the fourth der iva t ive of 0(x) j^t = number of trees , /$3 = coe f f i c i en t of asymmetry, = coe f f i c i en t of excess, £ - s tandard.dev iat ion , b = average diameter of the stand. The G h a r l i e r Type B curve progresses from extreme negative symmetry to the normal curve and i s derived from: 21 where: /,/ \ e sinflx[~"i h. _ A ^ _ A . . . . yHx) = | _ £ - x 17^5 2^7 2) 3$""$ A / ( x > |x-Ksh)|J A2y(^)= T^TJ - 1 ^ 2 ) 5 etc.,etc. w, c, A and the Bs are parameters which are defined by the co n d i t i o n s of the s o l u t i o n . Meyer (1930) f i t t e d Type A curves to 113 d i s t r i b u t i o n s of Douglas f i r . He found that the c o e f f i -c i e n t of asymmetry, /Sg, decreased r a p i d l y at f i r s t w i t h increase i n age' but then more or l e s s l e v e l l e d o f f . The c o e f f i c i e n t of excess, p , was p o s i t i v e ( i . e. c e n t r a l c l a s s frequencies r a i s e d ) but decreased w i t h age and became negative. Type B curve f i t t i n g s were su p e r i o r t o Type A when the average d i a -meter' was small ( l e s s than seven i n c h e s ) . Prodan (1953) a l s o has f i t t e d C h a r l i e r Type A curves to diameter frequency d i s t r i b u t i o n s of even-aged stands. Besides using the c o e f f i -c i e n t s of asymmetry and excess i n f i t t i n g diameter d i s t r i b u -t i o n s of l o b l o l l y pine (Pinus taeda L . ) , Nelson (196*0 a l s o used the gamma d i s t r i b u t i o n (see Appendix I) and- obtained s a t i s f a c t o r y r e s u l t s w i t h the same d i s t r i b u t i o n . Anderson (1937) has described the a p p l i c a t i o n of F o u r i e r ' s s e r i e s i n f o r e s t mensuration. A simple method of o b t a i n i n g the range of a diameter d i s t r i b u t i o n , given the mean, has been suggested by Smith and Ker ( i960) . They found that the minimum diameter i s seldom l e s s than h a l f the mean and the maximum u s u a l l y never more than twice the mean. L a t e r , Smith et a l . (1961) noted that i n 22 plantation-grown Douglas f i r i n New Zealand the upper l i m i t was only 1.6 times the mean. Vezina (1963) found t h a t i n dense, n a t u r a l stands of balsam f i r (Abies balsamea (L.) M i l l . ) , the most vig o r o u s t r e e s had diameters of about twice the average but a l s o observed t h a t the r a t i o of l a r g e s t - t o -average d. b. h. tended t o decrease w i t h age. T h i s would i n d i c a t e t h a t the diameter d i s t r i b u t i o n was probably becoming more normal. Height Growth The r e l a t i o n s h i p of h e i g h t t o age i s the best guide t o s i t e q u a l i t y (Spurr, 1952). In s p i t e of t h i s important f a c t h e i g h t growth and d i s t r i b u t i o n have been l e s s exten-s i v e l y s t u d i e d than diameter growth and d i s t r i b u t i o n . T h i s i s p a r t l y due t o the f a c t t h a t the h e i g h t of s t a n d i n g t r e e s i s more d i f f i c u l t t o measure and takes longer than the measurement of diameter at b r e a s t h e i g h t . Spurr (1952) has d e s c r i b e d the form of t y p i c a l height-over-age curves. For a sh o r t p e r i o d i n the e a r l y l i f e of the t r e e h e i g h t growth i s e x p o n e n t i a l . There i s then a long p e r i o d when the growth curve i s l i n e a r . The curve then g r a d u a l l y becomes h o r i z o n t a l as i t reaches the maximum hei g h t f o r the s i t e . Smith et a l . (i960) have d i s c u s s e d the r e l a t i v e m e r i t s of n a t u r a l and c o n v e n t i o n a l height/age curves and Smith (1962) has g i v e n f a c t o r s f o r c o n v e r t i n g h e i g h t at any age between 10 and 100 years to h e i g h t at age 50 years ( s i t e i n d e x ) . 23 Mathematical e x p r e s s i o n s f o r d e s c r i b i n g height growth have been produced by Meyer (lgko) and C o i l e (Schumacher, 1962). Meyer's formula i s : Y c = H ( l - e - a x ) where Y Q = h e i g h t , x = age, and H = maximum h e i g h t . The constant, a, v a r i e d between 0.04 and 0.12. C o i l e ' s equation i s : l o g H = a - 6.528(1/A) where H i s the average h e i g h t of 40 t r e e s per acre i n the dominant and codominant c l a s s e s and A i s the age i n y e a r s . The best p r a c t i c e i s to p r e d i c t mean h e i g h t growth f o r the 100 l a r g e s t diameter t r e e s per acre r a t h e r than f o r the average h e i g h t of dominants and codominants or the aver-age stand h e i g h t as has been done i n the p a s t . Mean h e i g h t of the 100 l a r g e s t t r e e s i s not so g r e a t l y a f f e c t e d by the d e n s i t y of s t o c k i n g as i s average h e i g h t and t h e r e f o r e g i v e s more c o n s i s t e n t growth t r e n d s . Because age or t h i n n i n g may reduce the number of stems per acre t o below 100, Spurr (1963) t e n t a t i v e l y suggested t h a t the mean hei g h t of the 40 l a r g e s t diameter stems per acre would be a b e t t e r measure of stand h e i g h t . Stoate and C r o s s i n (1959) claimed t h a t the r e l a t i o n between he i g h t and g i r t h (or diameter) at b r e a s t h e i g h t can be used as an index of s i t e where age i s unknown or i n d e t e r -minable. H e i g h t - o v e r - g i r t h curves were drawn f o r codominant t r e e s on d i f f e r e n t s i t e s and i t was found t h a t the g i r t h at which the curve l e v e l l e d out i n c r e a s e d w i t h improvement i n 24 s i t e q u a l i t y . Volume Growth The t r u e , or cubic f o o t , volume curve f o l l o w s the same sigmoid p a t t e r n as the height and diameter curves except t h a t the p e r i o d of e x p o n e n t i a l growth at the b e g i n n i n g of the l i f e of the t r e e s i s more prolonged (Spurr, 1952). An equation has been developed t o d e s c r i b e the growth curve of l o b l o l l y pine by C l u t t e r (1962 and 1963.) . T h i s i s of the form: l o g V = a + b_S + b„log B + b 0 A _ 1  & e 1 2 toe 3 where V i s the volume i n s i d e bark i n c u b i c f e e t per acre, A i s the stand age i n years, S i s the s i t e index i n f e e t , and B i s the b a s a l area i n square f e e t . C l u t t e r ' s equation f o r volume increment i s obtained by d i f f e r e n t i a t i n g the above equation w i t h r e s p e c t to age: ||.= b 2VB _ 1(dB/dA) - b 3VA" 2 where — = - B ( l o g B ) A _ 1 + C A" 1 B + C-.BSA"1. Thus: dA x °e ' o 1 | [ j = b 2 V ( l o g e B ) A _ 1 + b 2 C o V A _ 1 - b ^ V S A - 1 - b ^ A " 2 . The a c t u a l equations obtained f o r l o b l o l l y pine were: l o g V = 2.8076 + 0.015108S + 0.9^931(log B) - 2 1 . 8 6 3 A " 1 and: |^ = V ( 5 . 7 9 0 7 A _ 1 - 0.78l66(log eB) + 3.6562 + 0.0174lS)A S i m i l a r equations have been developed by Buckman (1962) f o r red pine (Pinus resonosa A i t . ) . 25 Y i e l d t a b l e s f o r Douglas f i r have been produced by Barnes and by Alexander (both i n U. B. C. F o r e s t Club, 1959) and by McArdle e_t al_. (19^9) f o r the c o a s t a l r e g i o n of the P a c i f i c Northwest. Y i e l d t a b l e s f o r Douglas f i r i n C a l i f o r n i a have been developed by Schumacher (1930). Hummel and C h r i s t i e (1953) p u b l i s h e d y i e l d t a b l e s f o r plantation-grown Douglas f i r i n Great B r i t a i n . These p l a n t a t i o n s have been thinned a c c o r d i n g t o standard F o r e s t r y Commission p r a c t i c e and the y i e l d t a b l e s g i v e the y i e l d from t h i n n i n g s as w e l l as the main crop. Barnes (1956) d i s c u s s e d the a p p l i c a t i o n of the B r i t i s h t a b l e s i n the P a c i f i c Northwest. Grandjean and Van Soest (1953) p u b l i s h e d s i m i l a r t a b l e s f o r the Netherlands and Parde (1956) adapted the B r i t i s h t a b l e s f o r French p l a n t a -t i o n s . Y i e l d t a b l e s have been p u b l i s h e d f o r unthinned Douglas f i r i n New Zealand (Duff, 1956.) . Stand D e n s i t y The three important f a c t o r s d e s c r i b i n g stand d e n s i t y are t r e e s per ac r e , mean hei g h t and mean diameter (Hummel, 1954). The product of these three v a r i a b l e s i s p r o p o r t i o n a l t o the "bble a r e a " (Lexen, 19^3; and Hummel, 1954) or the "cambial area" (Anucin, i960). The bole area remains more or l e s s constant from the time of canopy c l o s u r e u n t i l the l i m i t of he i g h t growth i s reached. Although i t i s cumbersome t o apply, i t has the advantage t h a t i t i s d i r e c t l y p r o p o r t i o n a l t o the a c t u a l growing s u r f a c e of the stem. The product of number of t r e e s and mean diameter ats r e l a t e d t o b a s a l area. A c c o r d i n g to Hummel (1954) a more p r a c t i c a l index of d e n s i t y , e s p e c i a l l y f o r t h i n n i n g , i s Hart's stand d e n s i t y index which i s the average d i s t a n c e between t r e e s expressed as a per-centage of the mean top h e i g h t of the 100 l a r g e s t t r e e s per he c t a r e . Hummel suggested t h a t i t would be b e t t e r t o work wi t h the l a r g e s t 250 t r e e s per h e c t a r e . ( o r , approximately, the l a r g e s t 100 t r e e s per a c r e ) . An index of twenty per cent should be found a f t e r t h i n n i n g t o the B r i t i s h C/D grade. Czarnowski (1961) put forward the h y p o t h e s i s : In pure, even-aged stands of a given s p e c i e s growing on land of i d e n t i c a l s i t e q u a l i t y and under c o n d i t i o n s of comparable competition f o r growing space, the number of t r e e s per u n i t of land area i s i n v e r s e l y p r o p o r t i o n a l t o the square of the mean hei g h t of the stand. For a measure of d e n s i t y Czarnowski d e f i n e d the "crowding f a c t o r " which was the r a t i o of the a c t u a l number of t r e e s per u n i t of land area t o the normal number. The average d. b. h. of the stand i s p r o p o r t i o n a l t o t h i s crowding f a c -t o r . Another measure of d e n s i t y used by Czarnowski was the "compactness f a c t o r " , the r a t i o between the a c t u a l volume per acre and the maximum volume a t t a i n a b l e f o r the s i t e . The use of "number of t r e e s per a c r e " has the d i s -advantage t h a t t h i s number may vary widely without a f f e c t i n g d e n s i t y (Spurr, 1952). B a s a l area i s not so a f f e c t e d and i s simple, o b j e c t i v e and easy t o use but has the disadvantage t h a t i t g i v e s equal weight t o n o n - f u n c t i o n i n g heartwood and f u n c t i o n i n g sapwood and a l s o t h a t cambial (or bole) area i s the f i r s t exponent of diameter and not the second (Nelson and Brender, 1963). Reineke's stand d e n s i t y index (Spurr, 1952) g i v e s the number of t r e e s per acre, N, when the t r e e of average b a s a l area has a d . b. h. of ten i n c h e s . T h i s index can be obtained from the r e f e r e n c e curve: l o g 1 Q N = -1.6051og 1 0D + K where K i s a constant v a r y i n g w i t h the s p e c i e s . The crown width/d. b. h. r a t i o s of Smith et a l . (l96l) can a l s o be used as an index of d e n s i t y . Stands w i l l then vary i n d e n s i t y from a CW/D r a t i o of 0.7 f o r dense, to about one f o r w e l l - s t o c k e d or "normal" stands, to two or more f o r n e a r l y open-grown t r e e s . B r i e g l e b (195 2) obtained r e g r e s -s i o n s f o r crown width and crown l e n g t h on t r e e h e i g h t and d. b. h. f o r Douglas f i r . Having-done t h i s he was able to c a l c u l a t e the "crown p r o j e c t i o n " and the crown s u r f a c e area and suggested t h a t , as these i n d i c e s remained more or l e s s constant r e g a r d l e s s of s t o c k i n g , they c o u l d be used as mea-sures of d e n s i t y . Other methods of measuring d e n s i t y have been d i s -cussed by Vezina (1962). T h i n n i n g Once the f o r e s t stand has been e s t a b l i s h e d there i s probably no way i n which the f o r e s t e r can a l t e r i t s develop-ment so much as by t h i n n i n g . In Europe and other c o u n t r i e s where f o r e s t management i s i n t e n s i v e , t h i n n i n g i s c a r r i e d out at r e g u l a r i n t e r v a l s from the time of canopy c l o s u r e to the time of f i n a l h a r v e s t i n g . The h e a v i e s t t h i n n i n g s are probably undertaken i n South A f r i c a where the t r e e s are p r a c t i c a l l y i n 28 a s t a t e of f r e e growth f o r a l a r g e p a r t of the r o t a t i o n . Based on South A f r i c a n experience w i t h e x o t i c p i n e s , H i l e y (1948) drew up a. schedule f o r t h i n n i n g Douglas f i r . T h i n n i n g i n B r i t a i n i s not heavy but the t h i n n i n g i n t e r v a l s are sh o r t , v a r y i n g between three years f o r young stands to f i v e or s i x years f o r more mature stands (Hummel and C h r i s t i e , 1953)• In New Zealand, Douglas f i r p l a n t a t i o n s are thinned on a ten-year c y c l e from age 30 years, removing o n e - t h i r d of the b a s a l area at each t h i n n i n g (Spurr, 1963). In the P a c i f i c Northwest t h i n n i n g has not been c a r r i e d much f a r t h e r than the e x p e r i -mental stage. I t i s g e n e r a l l y accepted that the i n c r e a s e d growing space caused by t h i n n i n g r e s u l t s i n the improved diameter growth of the remaining t r e e s , w i t h the r e s u l t that the merchantable volume i s o f t e n i n c r e a s e d (Mulloy, 1946). The immediate response may at f i r s t be a r e d u c t i o n i n diameter growth of the remaining t r e e s compared w i t h unthinned t r e e s due t o 'shock' ( s t a e b l e r , 1956a). Growth r e d u c t i o n may be due to one of three reasons. F i r s t l y , i f the stand remains unthinned u n t i l the crowns of the t r e e s become very s m a l l , then the t r e e s w i l l o f t e n not respond t o t h i n n i n g because t h e i r r e l e a s e i n c r e a s e s t o t a l r e s p i r a t i o n more than t o t a l p h o t o s y n t h e s i s (Kramer and Kozlowslki, i960) . Large amounts of carbohydrate are removed i n the r e s p i r a t i o n of cambial t i s s u e and the exposure of the stems t o the d i r e c t h e a t i n g e f f e c t of the sun r e s u l t s i n g r e a t l y i n c r e a s e d r e s p i r a t i o n i n the stem t i s s u e s . Trees i n 29 overcrowded stands may possess i n s u f f i c i e n t food s u p p l i e s t o q u i c k l y develop enlarged crowns capable of i n c r e a s e d photo-s y n t h e s i s . I t i s t h e r e f o r e important t o t h i n e a r l y while the crowns are s u f f i c i e n t l y l a r g e t o s h i f t the carbohydrate balance i n favour of p h o t o s y n t h e s i s over r e s p i r a t i o n . The second p o s s i b l e reason i s t h a t on r e l e a s e the stem puts on i n c r e a s e d growth at the base at the expense of growth higher up (Larson, 1963). F i n a l l y , r e d u c t i o n i n diameter growth may be due to damage of the r e s i d u a l t r e e s d u r i n g the f e l l i n g and e x t r a c t i o n of the thinned t r e e s . 'Shock' f o l l o w i n g t h i n n i n g i s probably c o n f i n e d t o dense stands t h a t are h e a v i l y t h i n n e d . Stephens and Spurr (1948) d e t e c t e d an immediate r e s -ponse t o t h i n n i n g i n a twenty-year-old stand of red pine, r a d i a l growth being i n c r e a s e d by 4 l per cent w i t h i n 24 hours. In t h i s p a r t i c u l a r stand r o o t competition was probably the l i m i t i n g f a c t o r r a t h e r than crown competition as the s o i l was sandy. Part of t h i s i n c r e a s e may have been due t o s w e l l i n g and not a c t u a l growth. Height growth of t r e e s i s l e s s i n f l u e n c e d by t h i n n i n g than i s diameter growth ( H a l l , 1954). K i t t r e d g e (1927) and Mulloy (1946) found t h a t t h i n n i n g young stands of red pine d i d not a f f e c t h e i g h t growth. In South A f r i c a , thinning,was found t o a f f e c t the mean hei g h t growth but d i d not a f f e c t top he i g h t ( H i l e y , 1959). S t a e b l e r (1956) observed a r e d u c t i o n i n h e i g h t growth i n the f i r s t season f o l l o w i n g t h i n n i n g of Douglas f i r i n an experiment a t Wind R i v e r , Washington. At Snow Creek, however, Worthington (1961) found t h a t the mean 30 annual h e i g h t growth was i n c r e a s e d from 2.2 t o 2.4 f t . i n the s i x years f o l l o w i n g a crown t h i n n i n g i n s i m i l a r stands. Adams (1936) found t h a t i n c r e a s e d h e i g h t growth occurred only i n e x c e p t i o n a l circumstances and t h a t growth could be r e t a r d e d f o r a few years f o l l o w i n g t h i n n i n g . Spurr (1952) s t a t e d t h a t at l e a s t some open-grown c o n i f e r s do not a t t a i n the h e i g h t of forest-grown t r e e s and t h a t a very h i g h d e n s i t y w i l l a d v e r s e l y a f f e c t h e i g h t growth through s t a g n a t i o n . Gen-e r a l l y , top hei g h t growth, or the height growth of dominant and codominant t r e e s , i s not a f f e c t e d by t h i n n i n g . Any d i f f -erences i n hei g h t growth noted f o l l o w i n g t h i n n i n g are probably due t o the f a c t t h a t mean h e i g h t growth was measured. The e f f e c t of t h i n n i n g on volume growth v a r i e s . Unless the t h i n n i n g removes such a l a r g e p r o p o r t i o n of the t r e e s t h a t the s i t e i s not f u l l y occupied f o r many years, the o v e r a l l r e s u l t i s a ga i n i n gross volume ( i n c l u d i n g volume from t h i n n i n g s ) as a r e s u l t of t h i n n i n g . H a n z l i k (1924), i n d e s c r i b i n g a Norway spruce t h i n n i n g experiment i n Sweden, showed t h a t the volume of an unthinned stand i n c r e a s e d by 91 per cent i n f i f t e e n years as opposed t o f i g u r e s of 72, 63 and 27 per cent f o r . s t a n d s t r e a t e d w i t h v a r i o u s t h i n n i n g grades. T o t a l volume p r o d u c t i o n was f a r g r e a t e r i n the h e a v i l y t h i n n e d p l o t s . K i t t r e d g e (1927) found t h a t the p e r i o d i c volume increment of red pine could be i n c r e a s e d by. 15 to 23 per cent by t h i n n i n g . L i (1923) found t h i s i n c r e a s e t o be as h i g h as 23 to 34 per cent f o r white pine (Pinus strobus L . ) . Of t h i s i n c r e a s e , 43 to 57 per cent was a t t r i b u t e d t o u t i l i z a t i o n of 31 m a t e r i a l l o s t by m o r t a l i t y i n the unthinned stand and the remainder due to a c c e l e r a t e d p r o d u c t i o n i n the thinned stand. M o l l e r ( 1 9 ^ 7 ) r e f u t e d the h y p o t h e s i s t h a t t h i n n i n g i n c r e a s e s gross volume increment. Genman experiments i n d i c a t e d t h a t the degree of t h i n n i n g , even.within very wide l i m i t s , had no i n f l u e n c e on gross increment over an extended p e r i o d of time. The i n c r e a s e d increment caused by heavy t h i n n i n g l a s t e d only ten t o twenty y e a r s . G e n e r a l l y , one would expect t h i n n i n g , p r o v i d i n g that i t i s not e x c e s s i v e l y heavy, to i n c r e a s e the merchantable volume of the stand and the gross volume y i e l d , but to have l i t t l e e f f e c t on the t o t a l c u b i c - f o o t volume of the stand. • V a r i o u s methods of t h i n n i n g and t h i n n i n g schedules have been d e r i v e d to s u i t d i f f e r e n t c o n d i t i o n s . Of i n t e r e s t t o f o r e s t e r s i n the P a c i f i c Northwest i s the work of Heiberg and Haddock ( 1 9 5 5 ) , Warrack ( 1 9 5 9 b ) and S t a e b l e r ( i 9 6 0 ) . Heiberg and Haddock suggested t h a t s i t e index 150 Douglas f i r should have t h i r t e e n t h i n n i n g s between the 2 8 t h and the 8 5 t h years t o reduce the number of stems from 7 3 5 t o 4 7 per a c r e . Under such a schedule volume p r o d u c t i o n i s 3 9 and value 54 per cent g r e a t e r than i n unmanaged stands. T h i s number of t h i n n i n g s i s c o n s i d e r a b l y g r e a t e r than that- u s u a l l y advocated i n t h i s r e g i o n and, apart from the delay i n t h e i r commence-ment, resembles European p r a c t i c e . Unless the stands were r e l a t i v e l y f r e e growing t o t h e i r 2 8 t h year i t would probably be more economical i f a " c l e a n i n g " had been done much e a r l i e r i n the r o t a t i o n so t h a t the t r e e s were at the spacing d e s i r e d at age 28 or, i f p l a n t i n g was c a r r i e d out, to p l a n t the t r e e s at t h a t s p a c i n g . Warrack (l959h). p r e s c r i b e d t h i n n i n g to the d/D r a t i o , where d i s the average d. b. h. of the t h i n n i n g s and D the average d. b. h. of the crop before t h i n n i n g . A r a t i o of O.65 i n d i c a t e s a c l e a n i n g , O.65 to 0.75 i s a low t h i n n i n g , 0.75 to 0.90 i s a severe low, or a l i g h t crown, t h i n n i n g ; and a r a t i o g r e a t e r than 1.00 i s a s e l e c t i o n t h i n n i n g . T h i n n i n g schedules were given f o r a n a t u r a l stand and f o r a p l a n t a t i o n showing the number of t r e e s , average d. b. h. and h e i g h t , b a s a l area and volume before and a f t e r t h i n n i n g . S t a e b l e r .(i960) based h i s t h i n n i n g schedules on the assumption t h a t the t o t a l p r o d u c t i o n of c u b i c volume by a stand of a given composition on a given s i t e was constant and optimum f o r a.wide range of s t o c k i n g d e n s i t i e s . The l e n g t h of h i s t h i n n i n g c y c l e corresponded to a t e n - f t . i n c r e a s e i n the t o t a l h e i g h t of dominants and codominants. For s i t e index 170 i t was assumed t h a t the average d. b. h. at age twenty years was e i g h t i n . The schedule was drawn up so t h a t the diameter growth i n the 21st year was one-half i n c h and the growth r a t e decreased t h e r e a f t e r at a r a t e of 0.0035 i n . each year. The main disadvantage w i t h S t a e b l e r ' s model i s t h a t he assumed a l l the t r e e s i n the stand, i n c l u d -i n g those removed i n the t h i n n i n g , were the same s i z e . These three papers show an advance i n stand manage-ment i n the P a c i f i c Northwest over the past twelve years f o r , i n 1952, Warrack was advocating not t h i n n i n g u n t i l t h e r e were 33 s u f f i c i e n t t r e e s (without removing a l l the l a r g e s t ) to make i t p r o f i t a b l e . T h i s would have been at an age of 35 years and might i n p r a c t i c e have l e d to e x p l o i t a t i o n f e l l i n g s r a t h e r than t h i n n i n g s . A l t e r n a t i v e l y , a t h i n n i n g could have been c a r r i e d out before the crop was twenty years o l d , removing a l l but the dominants and best codominants. In other r e g i o n s , Hummel (195*0- h a s d e f i n e d t h i n n i n g treatments by means of Hart's stand d e n s i t y index and Johnston and Waters (1961) have suggested c o n t r o l l i n g t h i n n i n g s by means of a b a s a l area/top h e i g h t curve. Spurr (19*1-8) advo-cated row t h i n n i n g s as they are cheaper and p r a c t i c a l l y as e f f i c i e n t as s e l e c t i v e t h i n n i n g s . They are good f o r improving r o o t and s o i l c o n d i t i o n s but l e a d to uneven crown development. Row t h i n n i n g s may be u s e f u l i n young p l a n t a t i o n s i n the f i r s t and second t h i n n i n g s or u n t i l the thinned produce becomes merchantable. L i t t l e and Mohr (1963) recommended the removal of every t h i r d row when t h i n n i n g l o b l o l l y p i n e . The e f f e c t of t h i n n i n g on the growth of i n d i v i d u a l Douglas f i r t r e e s has been s t u d i e d by S t a e b l e r (1956b) and Krueger (1959). In a 4 l - y e a r - o l d n a t u r a l stand, S t a e b l e r found t h a t , a f t e r three y e a r s , the e f f e c t of r e l e a s e produced a g r e a t e r d. b. h. growth i n dominant t r e e s than i n codomlnant or i n t e r m e d i a t e t r e e s . There was a p r o g r e s s i v e i n c r e a s e i n diameter growth as from zero to three competitors were cut from around each t r e e . T h i s i n c r e a s e was most marked between zero and one and l e a s t marked between two and three competi-t o r s cut. Dominant t r e e s which were thought t o be growing at 34 the maximum r a t e p o s s i b l e f o r the s i t e responded w e l l t o t h i n n i n g . The amount of r e l e a s e d i d not a f f e c t stem form (Yerkes, i960). Krueger (1959) r e p o r t e d on a : s i m i l a r e x p e r i -ment c a r r i e d out i n a 30-year-old Douglas f i r p l a n t a t i o n . The r e s u l t s were s i m i l a r t o those obtained i n the n a t u r a l stand except t h a t there were no s i g n i f i c a n t d i f f e r e n c e s i n diameter growth between the d i f f e r e n t amounts of r e l e a s e . G u i l l e b a u d and Hummel (1949) made ob s e r v a t i o n s on the movement of t r e e c l a s s e s i n Douglas f i r (and other c o n i -f e r o u s ) p l a n t a t i o n s i n Great B r i t a i n s ubjected t o d i f f e r e n t grades o f t h i n n i n g . In n e a r l y every case, dominants showed a net l o s s and subdominants ( i n t e r m e d i a t e s ) a net g a i n . I t was p o s s i b l e f o r codominants to move up to the dominant crown c l a s s . T h i n n i n g may a l s o a f f e c t the even t u a l y i e l d of a stand i n i n d i r e c t ways. Rish b e t h (1951) has r e p o r t e d t h a t t h i n n i n g i n c r e a s e d the r i s k of a t t a c k by b u t t - r o t (Forties  annosus Fr.) i n Douglas f i r p l a n t a t i o n s i n East A n g l i a . Conversely, Weir and Hubert (1919) claimed t h a t the t h i n n i n g of western hemlock (Tsuga h e t e r o p h y l l a (Raf.) Sarg.) and grand f i r (Abies g r a n d i s L i n d l . ) made c o n d i t i o n s l e s s f a v o u r a b l e f o r fu n g a l a t t a c k . Mulloy (1946) found t h a t t h i n n i n g reduced storm damage i n red p i n e . I t has been suggested t h a t t h i n n i n g may reduce the volume of st a n d i n g timber destroyed by f i r e by as much as 50 per cent ( S t a e b l e r , 1955a). A l l these f a c t o r s , a lthough t o a l a r g e extent u n p r e d i c t a b l e , a f f e c t the growth of the f o r e s t stand. 35 N a t u r a l M o r t a l i t y N a t u r a l m o r t a l i t y can be caused by the suppression of the weaker t r e e s by surrounding, more vig o r o u s t r e e s , by I n s e c t s , by f u n g i , by extremes of c l i m a t e , by f i r e or by a combination of any of these e f f e c t s . Often the cause of death cannot be a t t r i b u t e d t o any p a r t i c u l a r one of these. U s u a l l y one agent causes a d e c l i n e i n a t r e e ' s v i g o r , a second car-r i e s i t f u r t h e r and, o c c a s i o n a l l y , a t h i r d agent completes the k i l l i n g p r o c e s s . The l o s s t o the f o r e s t can take two forms, complete l o s s of i n d i v i d u a l t r e e s due t o death or a weakening of the t r e e s c a using a d e c l i n e i n the growth pro-cesses w i t h subsequent l o s s i n volume increment. Spurr (1962) found t h a t r a d i a t a pine t r e e s growing'on pumice i n New Zealand began t o e x h i b i t a marked decrease i n growth from at l e a s t s i x t o e l e v e n years before death. Trees showing a (d. b. h.) increment of l e s s than f o u r (inches) had a l i f e expectancy of l e s s than e i g h t y e a r s . M o r t a l i t y due to Fungal A t t a c k The most important fungus causing death and decay i n Douglas f i r stands i n the P a c i f i c Northwest i s the laminated r o o t - r o t , P o r i a w e i r i i Murr. I t i s present i n most Douglas f i r stands but i s not always s e r i o u s ( C h i l d s , 1955). Trees of a l l ages and s i z e s are a t t a c k e d but more e s p e c i a l l y those t r e e s from about 40 to 125 years o l d (.Childs, i960). The d i s e a s e occurs i n patches or c e n t r e s of i n f e c t i o n ( P l a t e II) PLATE I I ; Group-dying of Douglas f i r caused by P o r i a w e i r i i Murr., Wind R i v e r , Washington. Photographed i n August, 1963. 37 from a few hundred square feet to an acre or more in extent. The d i s t r i b u t i o n of the disease has been described as " e r r a t i c " (Anon., 1955). Within a t y p i c a l centre there w i l l be several trees, standing or down, that have been dead for d i f f e r i n g lengths of time. A few of the l i v i n g trees i n the centre may be showing signs of attack - leaning, t h i n crowns or poor colour. In a 112-year-old stand of Douglas f i r the percentage of l i v i n g trees having i n f e c t i o n v i s i b l e on the stump was found to vary inversely with the distance to the nearest tree k i l l e d by the disease (Anon., 1955). Within ten feet, 88 per cent of the trees were affected but at a distance greater than 50 feet only four per cent.were affected. Infection was usually on the portion of the stump nearest the dead tree but the extent of decay within the tree showed no c o r r e l a t i o n with the distance from the k i l l e d tree. The rate of damage usually doubles every ten to twenty years (Childs, 1955). The fungus spreads when spores i n f e c t wounds at or near the base of l i v i n g trees (Childs, i960). The fungus .can l i v e i n dead roots for 50 years or more. In young stands, damage increases i n geometric proportion to age as the i n f e c t i o n centres enlarge. Infected trees seldom respond well to release from competition and are often wind-thrown within a few years i f not k i l l e d , by the fungus (Childs, 1955). The wind-thrown trees often provide breeding material for bark beetles (Wright and Lauterbach, 1958). Poster and Johnson (1959a, 1959b and i960).have carried out a series of disease-sampling, studies i n young 38 ( t h i r t e e n t o seventeen years) Douglas f i r p l a n t a t i o n s on Vancouver I s l a n d . In the f i r s t t h i r t e e n years a f t e r p l a n t i n g the Douglas f i r had s u f f e r e d a 57 per cent m o r t a l i t y . There was a h i g h i n c i d e n c e of r o o t r o t . The s h o e - s t r i n g fungus, A r m i l l a r i a mellea ( V a h l . ex. Fr.) Quel., was most p r e v a l e n t but P o r i a w e i r i i and Fomes annosus (recorded f o r the f i r s t time i n a p l a n t a t i o n i n B r i t i s h Columbia) were a l s o p r e s e n t . The r o o t d i s o r d e r s were aggregated or contagious and f o l l o w e d the negative b i n o m i a l d i s t r i b u t i o n when the stands were sampled w i t h p l o t s v a r y i n g i n s i z e from 4/400 to 36/400 a c r e s . Terminal l e a d e r dieback, f r o s t - l e s i o n s and s u n s c a l d were a l s o observed. These i n j u r i e s were u s u a l l y randomly d i s t r i b u t e d and f o l l o w e d the Poisson model. In the two l a t e r papers ( F o s t e r and Johnson, 1959t> and i960) the red h e a r t - r o t , Stereum sanguinolentum A. & S., was found t o be present and i t was thought that t h i s fungus would l e a d to m o r t a l i t y i n the f u t u r e . Most of the openings caused by r o o t - r o t were l e s s than 9.9 f t . i n diameter and of i n s u f f i -c i e n t s i z e t o support a t r e e more than 22 years o l d . Only 8.2 per cent of the 574 i n f e c t i o n c e n t r e s were g r e a t e r than 9.9 f t . i n diameter and 1.6 per cent were g r e a t e r than 15.2 f t . ( F o s t e r and Johnson, 1963a). F u r t h e r d e t a i l s of the assessment of p a t t e r n , frequency d i s t r i b u t i o n and sampling of f o r e s t d i s e a s e i n Douglas f i r p l a n t a t i o n s have been given i n F o s t e r and Johnson (1963c.) . Fomes annosus i s very common i n Douglas f i r p l a n t -a t i o n s i n Great B r i t a i n . R i s h b e t h (1951) found t h a t t r e e s 39 were k i l l e d near i n f e c t e d stumps soon a f t e r p l a n t i n g . ' Stumps caused by t h i n n i n g are c o l o n i z e d by the fungus and act'as c e n t r e s o f i n f e c t i o n f o r the surrounding .trees...... I t ..may be a d v i s a b l e t o d e l a y t h i n n i n g u n t i l - t h e t r e e s are over 25 y e a r s , at which'age they are m o r e ' r e s i s t a n t to a t t a c k . ' ; Other f u n g i t h a t may cause damage i n Douglas f i r stands i n the P a c i f i c Northwest have been d e s c r i b e d by Harvey (1962). M o r t a l i t y due to I n s e c t A t t a c k The only i n s e c t c a u s i n g m o r t a l i t y and volume l o s s of economic importance i n ' t h e P a c i f i c Northwest i s the Douglas f i r bark b e e t l e , Dendroctonus pseudotsugae Hopk. Ac c o r d i n g t o Evenden and Wright (1955) i t ' i s ' present at a l l times, k i l l i n g s c a t t e r e d t r e e s and small groups (probably i n a s s o c i a t i o n w i t h r o o t - r o t ) . When i t throws o f f the c o n t r o l l i n g e f f e c t s of i t s n a t u r a l enemies i t becomes epidemic, k i l l i n g much of the Douglas f i r over, l a r g e areas i n a. few y e a r s . I t .can .apparently k i l l h e a l t h y t r e e s but p r e f e r s windfalls', damaged or- d e f o l i a t e d t r e e s and l o g g i n g s l a s h . Mathers (1951) found t h a t i n a- 700-acre stand near Quesnel, B r i t i s h Columbia, t h a t the b e e t l e s spread in. a n o r t h e a s t e r l y d i r e c t i o n . - There were f o u r t e e n pockets of a t t a c k r a n g i n g i n s i z e from two to twenty t r e e s . There was no c o r r e l a t i o n between the t r e e s k i l l e d and crown \ c l a s s , shape or .size of crown or d.^ b. h. but there was w i t h t r e e . v i g o u r , those t r e e s w i t h a slower growth ( p o s s i b l y due t o f u n g a l i n f e c t i o n ) being most l i a b l e t o a t t a c k . Thomas and C r a i g (1958) found.that w i n t e r i n j u r y due to f r o s t -weakened t r e e s , p a r t i c u l a r l y dominants and codominants, had made them more s u s c e p t i b l e t o a t t a c k by Dendroctonus. In the M i l l i c o m a F o r e s t , Oregon, which has 150,000 a c r e s of predomin-a n t l y Douglas f i r , Dendroctonus was r e s p o n s i b l e f o r 59 per cent of the average annual m o r t a l i t y . Walters (1954) c l a s s i -f i e d t r e e s as t o s u s c e p t i b i l i t y t o a t t a c k by age and v i g o u r . Older, slower-growing t r e e s were most l i a b l e t o i n f e c t i o n . A t y p i c a l group of b e e t l e - k i l l e d t r e e s i s shown i n P l a t e s I l i a and b. M o r t a l i t y caused by i n s e c t s , u n l e s s i t i s a s s o c i a t e d w i t h f u n g a l i n f e c t i o n , i s g e n e r a l l y s c a t t e r e d or randomly d i s t r i b u t e d i n the f o r e s t stand. L i t t l e harm i s caused by small " k i l l s " as the surrounding t r e e s b e n e f i t from the r e l e a s e (Hoffman and Anderson, 194-5) • F o s t e r and Johnson (1963a) suggested t h a t such m o r t a l i t y may be b e n e f i c i a l i n overstocked stands although i t i s d e t r i m e n t a l i n understocked stands. I t can be seen t h a t the u s u a l p a t t e r n of m o r t a l i t y i n Douglas f i r stands f o l l o w s the p a t t e r n of, f i r s t , i n f e c -t i o n by r o o t - r o t , which makes the t r e e more l i a b l e t o wind-throw which i n t u r n p r o v i d e s s u i t a b l e breeding m a t e r i a l f o r bark b e e t l e s (Mathers, 1951). When c o n d i t i o n s are s u i t a b l e , the bark b e e t l e p o p u l a t i o n r i s e s and an epidemic may occur. 41 PLATE I I l ( a , b ) : G r o u p - d y i n g o f D o u g l a s f i r c a u s e d b y D e n d r o c t o n u s p s e u d o t s u g a e Hopk., P a u l L a k e , K a m l o o p s . P h o t o g r a p h e d i n J u l y , 1 9 6 3 . 42 N a t u r a l M o r t a l i t y t h r o u g h S u p p r e s s i o n W h e reas t h e number o f t r e e s k i l l e d by i n s e c t s , f u n g i o r c l i m a t i c e x t r e m e s and t h e i r d i s t r i b u t i o n c a n n o t be a c c u r -a t e l y f o r e c a s t , an a t t e m p t c a n be made t o f o r e c a s t n a t u r a l m o r t a l i t y t h r o u g h s u p p r e s s i o n . I t i s known t h a t , p r o v i d i n g e x t e r n a l f o r c e s do n o t come i n t o p l a y , t h e s m a l l e s t , o r s u p -p r e s s e d t r e e s i n a s t a n d w i l l be t h e f i r s t t o d i e a s t h e s t a n d becomes o l d e r . F r o m p e r m a n e n t s a m p l e p l o t s i t i s p o s s i b l e t o t e l l , f o r a n y s i t e q u a l i t y , how many t r e e s p e r a c r e t h e r e w i l l be i n a n o r m a l l y s t o c k e d s t a n d a t any g i v e n a g e . F r o m t h i s i t i s p o s s i b l e t o e s t i m a t e t h e number o f t r e e s t h a t s h o u l d d i e i n a g i v e n p e r i o d . B e c a u s e t h e l a r g e s t t r e e s a n d t h e s m a l l e s t t r e e s a r e n e v e r e v e n l y s p a c e d o v e r t h e a r e a , t h e m o r t a l i t y w i l l n o t remove e n t i r e l y t h e s m a l l e s t d i a m e t e r c l a s s e s b u t w i l l be g e n e r a l l y c o n f i n e d t o them. S t a e b l e r (1953) h a s e s t i -m a ted t h e m o r t a l i t y i n f u l l y - s t o c k e d s t a n d s o f y o u n g - g r o w t h D o u g l a s f i r ( a g e d 26 t o 93 y e a r s ) on 36 p e r m a n e n t s a m p l e p l o t s i n W a s h i n g t o n and O r e g o n . P l o t s w h e r e " i r r e g u l a r " m o r t a l i t y h a d o c c u r r e d w e r e r e j e c t e d . He d e r i v e d two e q u a -t i o n s f o r d e t e r m i n i n g t h e p e r c e n t a g e o f t r e e s t h a t w i l l d i e i n a t e n - y e a r p e r i o d . T h e s e a r e : f o r d o m i n a n t s and c o d o m i n a n t s , % m o r t a l i t y = 4.96 + 0.08(age) - 0.4l(d.b.h.) (R = 0.266) f o r i n t e r m e d i a t e s a n d s u p p r e s s e d , % m o r t a l i t y = 13.01 + 0.5M'S.I.) + 0 . 6 l ( a g e ) - 7B3(d.b.h.) (R = 0.715) Gross y i e l d and m o r t a l i t y t a b l e s were given i n a l a t e r paper ( S t a e b l e r , 1 9 5 5 b ) . F u r t h e r data on m o r t a l i t y i n Douglas f i r have been g i v e n by E v e r s o l e (1955) and G r i f f i t h ( i 9 6 0 ) . C o n c l u s i o n s From t h i s review of the l i t e r a t u r e i t can be seen t h a t much i s known q u a l i t a t i v e l y but l i t t l e i s known q u a n t i t a t i v e l y about the growth of Douglas f i r . Our q u a n t i t a t i v e knowledge i s co n f i n e d t o the p u b l i s h e d y i e l d t a b l e s based on "normal" or f u l l y - s t o c k e d stands and to more or l e s s l o c a l i z e d r e s e a r c h p r o j e c t s which, although y i e l d i n g much v a l u a b l e i n f o r m a t i o n , cannot be a p p l i e d on a gen e r a l b a s i s i n the Douglas f i r r e g i o n w i t h any degree of r e l i a b i l i t y . In the development of a mathematical stand model assumptions w i l l t h e r e f o r e have t o be made on the b a s i s of the a v a i l a b l e i n f o r m a t i o n . PART I I DEVELOPING A STAND MODEL FOR DOUGLAS FIR The object of the present study i s to develop a mathematical model that w i l l describe the growth of a Douglas f i r stand from an age of ten years, by which time i t i s assumed that a l l the t r e e s w i l l have reached breast height (4.5 f t . ) , to an age when the stand might normally be expected to be har-vested. This i s assumed to be a t , or before, 1 0 0 years. A.sound mathematical model should have as i t s b a s i s sound b i o l o g i c a l theory. Unfortunately our knowledge of the growth of f o r e s t stands i s not by any means complete, p a r t i -c u l a r l y i n the pe r i o d between the onset of competition between t r e e s i n the stand and the time when m o r t a l i t y , through sup-p r e s s i o n of the weaker t r e e s , occurs and the stand becomes normally stocked ( c f . the open-to-normal concept of Smith et a l . , 1 9 6 1 ) . Table 111 of Smithoet_ al. . (1961) i n d i c a t e d that the le n g t h of time taken t o grow Douglas f i r stands t o an average d. b. h. of twelve inches can be reduced by 3 0 to 40 per cent i f the stands are e s t a b l i s h e d at such an open-spacing that they become "normal" when the average d. b. h. i s twelve i n . Because of the gaps i n our knowledge of tre e growth, c e r t a i n assumptions have been made i n developing the model which i t i s hoped w i l l be j u s t i f i e d when the model i s compared w i t h f i e l d c o n d i t i o n s . A l l assumptions made w i l l be discussed f u l l y . 44 45 Stand Model I T h i s model has been d e s c r i b e d i n d e t a i l i n an e a r l i e r r e p o r t (Newnham, 1963). In t h i s , as i n l a t e r models, i n i t i a l square spacing was assumed, t h a t i s , t r e e s could be l o c a t e d only at the i n t e r s e c t i o n s of a square l a t t i c e . The p a t t e r n could be m o d i f i e d by o m i t t i n g t r e e s from c e r t a i n l o c a t i o n s . The reasons f o r adopting square sp a c i n g have been d i s c u s s e d i n Part I of t h i s t h e s i s . I t s main advantage i n model development i s t h a t i t f a c i l i t a t e s computations. S t o c k i n g was assumed to be 1,000 t r e e s per acre at age ten y e a r s , d e c r e a s i n g to 250 t r e e s at age 50 y e a r s . A b a s i c matrix of 100 t r e e s was used w i t h each t r e e being given a rank number depending on the magnitude of i t s d. b. h., rank No. 1 being the l a r g e s t t r e e and rank No. 100 the s m a l l e s t . Competition was e v a l u a t e d by comparing the rank number of each t r e e with.those of the surrounding t r e e s . I f a ''competitor" had a lower rank number than the t r e e being s t u d i e d , the rank number of the t r e e was i n c r e a s e d by an amount i n v e r s e l y p r o p o r t i o n a l to the d i s t a n c e of the "competi-t o r " from the t r e e . At the end of each f i v e - y e a r p e r i o d those t r e e s having the g r e a t e s t i n c r e a s e i n rank number were con-s i d e r e d t o have " d i e d " u n t i l the d e s i r e d l e v e l of s t o c k i n g was obtained. The process was repeated at f i v e - y e a r i n t e r v a l s t o age 50 y e a r s . Diameter growth was p r e d i c t e d by assuming a constant r a t e of b a s a l area growth f o r open-grown or f r e e -growing, t r e e s (Spurr, 1952). The f i v e - y e a r d. b. h. growth 46 of each t r e e was then reduced by an amount p r o p o r t i o n a l to the rank p o s i t i o n of the t r e e and a l s o i t s i n c r e a s e i n rank dur-i n g the f i v e - y e a r p e r i o d . Although the method used i n th i s , model was based on a r b i t r a r y assumptions, the r e s u l t s (see Newnham, 1963) con-formed t o a p a t t e r n t h a t might w e l l be assumed t o occur i n nature. I t had the advantage t h a t the c a l c u l a t i o n s were s t r a i g h t f o r w a r d and could be r a p i d l y c a r r i e d out. The main disadvantage was the d i f f i c u l t y of adapting i t f o r d i f f e r e n t i n i t i a l spacings and d i s t r i b u t i o n s of t r e e s as i t was designed t o c o n s i d e r only those competitors w i t h i n 13.2 f e e t of the t r e e being s t u d i e d . For t h i s reason work on t h i s model was d i s c o n t i n u e d . Stand Model I I The e a r l y work on the development of t h i s model has been p r e v i o u s l y d e s c r i b e d (Newnham, 1963 and 1964). Most of t h i s work c o n s i s t e d of v a r y i n g the v a l u e s of the parameters used i n the model i n order t o make i t gi v e r e s u l t s which com-pared f a v o u r a b l y w i t h the p u b l i s h e d y i e l d t a b l e data. The B a s i c P r i n c i p l e s and Assumptions Data were c o l l e c t e d In the i n t e r i o r of B r i t i s h Colum-b i a (Paul Lake, near Kamloops), i n the c o a s t a l r e g i o n of B r i t i s h Columbia (Saanich P e n i n s u l a , Vancouver I s l a n d and the U n i v e r s i t y Research F o r e s t , Haney) and on the western slopes of the Cascade Mountains i n Washington State (Wind R i v e r ) . 47 T h e r e l a t i o n s h i p b e t w e e n c r o w n w i d t h ( t h e s u m o f t h e m e a s u r e -m e n t s o f t h e l o n g e s t b r a n c h o n t w o s i d e s o f t h e t r e e ) a n d d i a m e t e r a t b r e a s t h e i g h t o u t s i d e b a r k w a s c a l c u l a t e d f r o m t h e s e . T h i s r e l a t i o n s h i p c o u l d b e s t b e d e s c r i b e d b y t w o s t r a i g h t - l i n e r e g r e s s i o n s , o n e f o r t r e e s l e s s t h a n t h r e e i n . d . b . h . , b a s e d o n d a t a c o l l e c t e d f r o m a s e v e n - y e a r - o l d p l a n t a t i o n e s t a b l i s h e d a t H a n e y a t a s p a c i n g o f 9 x 9 f t . i n w h i c h t h e c r o w n s w e r e n o t o v e r l a p p i n g ( P l a t e I c ) , a n d o n e f o r t r e e s t h r e e i n . i n d . b . h . o r g r e a t e r , b a s e d o n t h e d a t a c o l -l e c t e d f r o m t h e r e m a i n i n g s i t e s ( F i g . 2 ) . T h e s e r e g r e s s i o n s a r e : t r e e s < 3 i n ; ; , d . b . h . CW = 2.270 + 2.399D r = . 8 2 0 , N = 274, S = +0.765 f t . t r e e s >: 3 i n s . d . b . h . CW = 5.031 + 1 . 4 2 3 D r = .917 N = 152, S = +4.517 f t . T h e s e r e s u l t s a r e n e a r l y i d e n t i c a l t o t h o s e p u b -l i s h e d b y S m i t h _e_t a l . (1961) a n d t h e m o r e r e c e n t r e s u l t s o b t a i n e d b y S m i t h a n d J a k o y (1963, u n p u b l i s h e d d a t a ) f r o m m e a s u r e m e n t s o b t a i n e d o n t h e U n i v e r s i t y R e s e a r c h F o r e s t a t H a n e y . T h e r e l a t i o n s h i p b e t w e e n d i a m e t e r a t b r e a s t h e i g h t a n d a g e w a s a l s o s t u d i e d f o r t h e s e o p e n - g r o w n t r e e s b y t a k i n g i n c r e m e n t b o r i n g s . A g e w a s d e t e r m i n e d b y c o u n t i n g t h e n u m b e r o f a n n u a l b r a n c h - w h o r l s b e l o w b r e a s t h e i g h t a n d a d d i n g t h i s n u m b e r t o t h e n u m b e r o f r i n g s o n t h e i n c r e m e n t b o r i n g . T h e r e l a t i o n s h i p w a s f o u n d t o b e p e r s i s t e n t l y l i n e a r f o r t h e K a m l o o p s d a t a ( F i g . 3) a n d a l s o f o r t h e p o o r e r s i t e s o n t h e « h k k h is h » « M , - , D I A M E T E R A T B R E A S T H E I G H T IN I N C H E S FIG. 2: The rela t i o n s h i p between crown width and d. b. h. o. b. of open-grown Douglas f i r . 49 20,-O 20 40 60 80 lOO A G E IN Y E A R S -PIG. 3: The rel a t i o n s h i p between d. b. h. i . b. and age. Open-grown Douglas f i r , Paul Lake. Saanich Peninsula ( F i g . 4) and at Wind R i v e r ( F i g . 5). On the b e t t e r s i t e s , the r a t e of diameter growth tended t o decrease w i t h i n c r e a s e i n age. For the Saanich data a r e g r e s s i o n was c a l c u l a t e d f o r f i v e - y e a r r a d i a l increment ((-y rRg) on age, diameter at age 25 years ( D 2 c ; - i n t e r p o l a t e d where necessary from the d. b. h./age curves of the i n d i v i d u a l t r e e s ) and the i n i t i a l diameter at the beginning of each f i v e - y e a r p e r i o d ( D ^ ) . Trees were r e j e c t e d i f th e r e were any apparent e r r o r s i n age e s t i m a t i o n . The r e g r e s s i o n was: 5 y r R g = 0.9920 - 0.07223~D± + 0 . 1 5 3 8 8 D 2 5 + 0 . 0 1 9 6 9 A g e -0.00005029Age2 N = l 4 l ( f r o m 18 t r e e s ) , R = 0.784, s = +O.270 i n . To f a c i l i t a t e programming, the r e g r e s s i o n of LV>^  on D ^ Q , the diameter at age ten year s , was l a t e r c a l c u l a t e d ( D 2 5 = 1.706 + 2.754D 1 q, r = 0.950, s = 1.128 i n . ) and the f i n a l r e g r e s s i o n became: 5 y r R g = 0.06338 - 0.07223D1 + 0.4237D1Q + 0.01969Age -0.00005029Age2 In the model t h i s , l a s t r e g r e s s i o n has been a p p l i e d t o o u t s i d e bark measurements whereas the r e g r e s s i o n i s based on i n s i d e bark measurements. The e r r o r i n v o l v e d was not great enough t o cause any s e r i o u s d e f e c t i n the model because i t c o uld only underestimate growth i n d. b. h. by e i g h t t o twelve per cent. The Saanich data were s e l e c t e d as they were rep-r e s e n t a t i v e of s i t e c o n d i t i o n s i n d r i e r p a r t s of the c o a s t a l 51 PIG. 5'- The r e l a t i o n s h i p between d. b. h. i . b. and age. Open-grown Douglas f i r , Wind River. 53 reg ion . The Kamloops data were rejected on the grounds that t h e i r i n c l u s i o n would have given too much weight to the poorer s i t e s not often encountered i n the coastal region; the Wind River data were not included as the trees were not s u f f i c i e n t l y o l d . Diameter/age curves were constructed using the above regress ion and are shown in P i g . 6. It can be seen that the ca lcu lated diameter/age r e l a t i o n s h i p i s more or less l i n e a r on the poorer s i tes but shows some c u r v l l i n e a r i t y on the better ones. For the model i t was necessary to estimate the approximate age at which trees , i n i t i a l l y es tabl i shed as open-grown, came into competition with one another. As the determination of root spread i s d i f f i c u l t i n p r a c t i c e , crown spread has often been used as an ind ica tor of root spread. This r e l a t i o n s h i p was studied i n d e t a i l on the open-grown Douglas f i r trees at Paul Lake, Kamlopps. The annual p r e c i -p i t a t i o n in t h i s region i s low ( f i f t een to twenty inches) and the s o i l s are therefore dry . The general mass of roots was found not to spread much beyond the extent of the crown ( F i g . 7 ) although occasional roots spread to greater d i s -tances. When studied i n the other areas sampled t h i s r e l a t i o n -ship a lso appeared to ho ld . Crown spread was therefore assumed to be a good ind ica tor of root spread when considering the competitive status of t rees . In the seven-year-old p lanta t ion establ ished at Haney at an i n i t i a l spacing of 3 x 3 f t . i t was found that , although the branches of the trees overlapped by as much as 54 AGE IN YEARS P I G . 6 : C a l c u l a t e d d. b. h. o. b./age c u r v e s ( b y i n c h d. b. h. o. b. c l a s s e s a t age 10 y e a r s ) . RR — 1-975 + 0-4132CW S-3-37U. r - 73 Basis no. of data m 84 25f-o 50 per cent, there was no apparent r e d u c t i o n i n diameter growth compared w i t h the p l a n t a t i o n s e s t a b l i s h e d at wider spacings which were s t i l l open-grown (see F i g . 8 and P l a t e s I a - d ) . To determine the c o m p e t i t i v e s t a t u s of each t r e e i n the e a r l y stages of the model, the crown widths (determined from the d. b. h. of each t r e e u s i n g the crown width/d. b. h. r e g r e s s i o n ) were t h e r e f o r e m u l t i p l i e d by a r e d u c t i o n f a c t o r (the v a r i a b l e "REDPAC" i n the FORTRAN program) which was chosen a f t e r t e s t i n g s e v e r a l d i f f e r e n t v a l u e s u s i n g the b a s i c matrix of diameters. The new measurement thus obtained was c a l l e d the c o m p e t i t i v e crown width. For a s t a r t i n g p o i n t i n the model, the diameters of a matrix of 15 x 15 t r e e s were measured i n the 6 x 6 f t . p l a n t a t i o n (aged seven years) at Haney. These t r e e s were growing on.a good s i t e (probably s i t e index 170 f e e t at 100 y e a r s ) . To adapt these measurements t o a poorer s i t e more commonly encountered i n the c o a s t a l r e g i o n of B r i t i s h Columbia, i t was assumed t h a t the same diameter d i s t r i b u t i o n would be found on s i t e index 140 but at an age of ten y e a r s . D e s c r i p t i o n of the Stand Model As s t a t e d , the model s t a r t e d w i t h a matrix of 15 x 15 t r e e s at age ten y e a r s . The number of t r e e s per acre t h e r e f o r e v a r i e d w i t h the i n i t i a l s p a c i n g r a n g i n g from 4000 t r e e s per acre at a- 3.3 x 3.'3 f t . spacing to 250 t r e e s per acre at a 13.2 x 13.2 f t . s p a c i n g . To a v o i d "edge e f f e c t s " , i t was assumed t h a t t h i s matrix was repeated every f i f t e e n 57 30r 2 0 -IO-9x9feet Spacing 3 0 Mean- IH6in. S.D.»iO-455in. N » 274 20 L-n l-O 1-5 2-0 D. B. H. CLASS IO-25 in I ng 6x61 ttt Spacing Mean = l-284in. S.0.= ±O 454in. N - 2 I 8 Ul I O 3 D D. B. H. CLASS 30r-20-Ln 1 0 3x3feet Spacing Mean - I 123in. S.D.= ±0-772in. N =304 Ln j _ I 5 l-O l-S 2-0 D.B.H. CLASS 2-5 in. PIG. 8 : Diameter frequency d i s t r i b u t i o n s of seven-year-old Douglas f i r p l a n t a t i o n s e s t a b l i s h e d at d i f f e r e n t i n i t i a l s p a c i n g s . U n i v e r s i t y Research F o r e s t , Haney. 58 rows and every f i f t e e n columns. Thus the t r e e s In row No. 15 were next t o the t r e e s i n row No. 14 on one s i d e and row No. 1 on the other. The diameters at b r e a s t height of these t r e e s were known (Table 1) and from these the crown widths of the t r e e s c o u l d be c a l c u l a t e d (see P i g . l ) and the com p e t i t i v e crown widths obtained u s i n g the r e d u c t i o n f a c t o r , "REDPAC". Taking one t r e e i n the matrix at a time the model was t e s t e d t o see i f any of the surrounding t r e e s were com-p e t i n g by determining whether the competitive crowns over-lapped. I f o v e r l a p occurred, the angle subtended a t the centre of the crown by the two p o i n t s of i n t e r s e c t i o n of the com-p e t i t i v e crown perimeters f o r each competitor was measured ( i n r a d i a n s : 2fl r a d i a n s = 3 6 0 ° . See P i g . l b ) . T h i s measure-ment was weighted i n each case by the r a t i o of the crown width of the competitor t o the crown width of the t r e e being s t u d i e d , thus r e c o g n i z i n g t h a t the t r e e s w i t h the l a r g e r crowns u s u a l l y had the added advantage of being t a l l e r . The method of c a l c u l a t i n g t h i s angle i s given i n the d e s c r i p t i o n of the FORTRAN, program i n Appendix I I (see SUBROUTINE CROWN). For each t r e e the sum of these angles was d i v i d e d by 2TV to gi v e the p r o p o r t i o n of the circumference of the " c o m p e t i t i v e " crown of the t r e e occupied by the crowns of i t s competitors (the FORTRAN v a r i a b l e "SOC"). Thus a value, which v a r i e d between zero and one (or more i f the circumference of the crown was occupied by s e v e r a l o v e r l a p p i n g crowns), was obtained f o r the com p e t i t i v e s t a t u s of each t r e e . The sum of the angles subtended.at the centre of a t r e e by the i n t e r s e c t i o n s of the crowns of the competing TABLE 1: The i n i t i a l d. b. h. matrix used i n the development of the model. Data from the U n i v e r s i t y Research F o r e s t , Haney, B. C. Diameter at breast height (inches) Row No. 1 2 3 4 5 6 Column 7 8 No. 9 1 0 11 12 13 14 15 1 1 . 4 1 . 4 1 . 5 1 . 0 1 . 5 2 . 0 1 . 6 1 . 3 1 . 2 : < N 0 . 7 1 . 1 1 . 4 1 . 8 1 . 7 2 1 . 5 1 . 1 0 . 2 1 , 7 0 . 9 0 . 9 1 . 4 1 . 5 1 . 5 0 . 8 1 . 3 1 . 2 1 . 4 1 . 6 0 . 8 3 1 . 3 0 . 9 1 . 0 0 . 8 0 . 9 1 . 0 1 . 9 1 . 7 1 . 7 0 . 7 1 . 5 1 . 4 1 . 3 1 . 6 1 . 2 4 0 . 9 1 . 2 1 . 7 0 . 5 1 . 9 1 . 9 l . l 2 . 0 1 . 8 1 . 3 1 . 0 1 . 7 0 . 8 1 . 0 0 . 6 5 1 . 2 1 . 1 0 . 6 0 . 8 1 . 4 1 . 3 1 . 8 1 . 5 2 . 0 1 . 2 l . l 1 . 4 0 . 6 l . l 1 . 4 6 0 . 8 0 . 4 1 . 0 0 . 9 0 . 2 . 1 . 6 1 . 6 1 . 4 0 . 8 1 . 3 1 . 5 0 . 6 0 . 9 0 . 9 0 . 2 7 0 . 7 0 . 2 1 . 6 0 . 9 1 . 3 0 . 9 2 . 0 1 . 5 1 . 2 1 . 9 1 . 6 1 . 5 1 . 2 0 . 6 1 . 1 8 0 . 8 0 . 8 - 1 . 4 1 . 0 - 2 . 1 1 . 4 0 . 6 1 . 3 1 . 6 - 0 . 6 0 . 9 1 . 5 9 0 . 9 1 . 4 1 . 3 1 . 2 . 1 . 2 1 . 5 1 . 1 1 . 6 1 . 1 1 . 5 l . l 0 . 9 1 . 3 0 . 9 l . l 10 1 . 7 1 . 4 1 . 6 2 . 2 1 . 5 1 . 3 • ' 0 . 7 0 . 7 1 . 0 1 . 0 1 . 3 1 . 3 0 . 5 1 . 2 0 . 9 11 1 . 4 1 . 9 1 . 8 1 . 5 1 . 0 0 . 8 - 1-3 1 . 4 1 . 7 1 . 5 1 . 9 1 . 4 0 . 7 0 . 7 12 1 . 0 2 . 0 1 . 8 1 . 5 2 . 1 1 . 2 - 0 . 9 1 . 3 1 . 2 1 . 6 1 . 0 1 . 7 0 . 9 1 . 5 13 1 . 2 1 . 0 1 . 9 1 . 8 2 . 0 1 . 3 1 . 4 0 . 7 1 . 7 l . l 1 . 7 0 . 6 1 . 5 2 . 1 1 . 3 14 1 . 5 0 . 6 1 . 2 1 . 0 2 . 0 1 . 3 1 . 2 2 . 1 1 . 4 l . l - 0 . 9 - 1 . 3 l . l 15 1 . 4 1 . 5 1 . 5 2 . 4 1 . 4 1 . 5 1 . 6 0 . 9 - - 1 . 7 0 . 8 l . l 1 . 4 1 . 7 60 t r e e s , as an index of competition, i s r e l a t e d t o the "number of s i d e s f r e e " index of competition used by Tinney and. Malmberg .(1948) and by Ker (1953). Ker- found t h a t f o r 65-year-o l d Douglas f i r t r e e s , r a d i a l growth at b r e a s t h e i g h t i n c r e a s e d w i t h i n i t i a l d. b. h., crown c l a s s (the w e i g h t i n g f a c t o r des-c r i b e d above would have a s i m i l a r e f f e c t i n the model) and number of s i d e s f r e e . The v a r i a b l e "SOC" used i n the present model d i f f e r e d only i n t h a t i t measured " s i d e s occupied" r a t h e r than " s i d e s f r e e " . S t a e b l e r (1951) used the amount of crown o v e r l a p as a measure of competition (see F i g . l a ) . Using the r e g r e s s i o n d e s c r i b e d above f o r f i v e - y e a r r a d i a l growth on age, d. b. h. at age ten y e a r s , d. b. h. at the b e g i n n i n g of the f i v e - y e a r p e r i o d and age, the model next c a l c u l a t e d the f i v e - y e a r diameter increment f o r each t r e e , assuming t h a t the t r e e was open-grown r e g a r d l e s s of whether i t was or was not so. T h i s f i v e - y e a r increment was then reduced by an amount which depended on the c o m p e t i t i v e s t a t u s of each t r e e (see l i n e s 370-371 of the FORTRAN prog-ram i n Appendix I I ) . The amount of r e d u c t i o n would vary from zero f o r t r e e s t h a t were free-growing ( i . e . "SOC" = 0) to -100 per cent f o r t r e e s whose crowns were completely over-lapped by surrounding competitors ("SOC" =. l ) . . I f t h i s increment was not g r e a t e r than a c e r t a i n percentage ("DINC") of the d. b.h. at the b e g i n n i n g of the f i v e - y e a r p e r i o d , the t r e e was c o n s i d e r e d t o have d i e d . I f i t was g r e a t e r , the new diameter of the t r e e ("DAPJj") was c a l -c u l a t e d . These new diameters were then used as a b a s i s f o r 61 c a l c u l a t i n g the next f i v e y e a r s ' growth of the stand and the process repeated to age 100 y e a r s . The v a l u e s of "DINC" used t o d e f i n e m o r t a l i t y were chosen a r b i t r a r i l y . Attempts t o o b t a i n r e a l i s t i c v a l u e s by sampling dead t r e e s i n the f i e l d were of l i t t l e use as the v a r i a t i o n i n the l a s t f i v e y e a r s ' d. b. h. growth of Douglas f i r was very g r e a t . Some t r e e s were able t o s u r v i v e and grow at the r a t e of 100 r i n g s per i n c h f o r 25 years or more while o t h e r s , a p p a r e n t l y k i l l e d by i n s e c t s or f u n g i , d i e d w i t h very l i t t l e decrease i n growth r a t e d u r i n g the l a s t f i v e years of t h e i r l i v e s . The v a l u e s of "DINC" chosen range from f i v e per cent at age ten years to 0 .1 per cent at age 45 years or above. The i n c l u s i o n of "DINC" v a l u e s i n the model was found t o be necessary f o r , i f i t was assumed t h a t death occurred only when the p e r i o d i c diameter increment became zero, the onset of m o r t a l i t y was delayed and, when i t d i d occur, was very heavy c a u s i n g widespread d e p l e t i o n of the stand. T h i s was because, i n nature, m o r t a l i t y i s a- continuous process o c c u r r i n g every year, whereas i n the model, m o r t a l i t y was a d i s c r e t e p rocess o c c u r r i n g only at the end of each f i v e -year p e r i o d . Thus t r e e s t h a t would have d i e d at the begin- . n i n g of the p e r i o d remained " a l i v e " t o the end and thus reduced the growth of neighbouring t r e e s , which would nor-mally have been r e l e a s e d , t o a l e v e l where death might occur too. As soon- as m o r t a l i t y had s t a r t e d i n the model the value of "REDPAC" was i n c r e a s e d each f i v e - y e a r p e r i o d by an amount "REDINC", again chosen a r b i t r a r i l y . Competition i n the model was based on the crown-dimensions of open-grown . t r e e s . Thus w i t h the r e d u c t i o n i n diameter growth as competi-t i o n set i n , the c a l c u l a t e d competitive crown widths were l e s s than they would have been had the t r e e s remained c o m p e t i t i o n - f r e e . I n c r e a s i n g the value of "REDPAC" compen-sated f o r t h i s . I t should be noted t h a t , i n c l o s e d stands of Douglas f i r a f t e r the i n i t i a l p e r i o d . o f i n t e n s e competition, there may be c o n s i d e r a b l e gaps i n the crown canopy (see P l a t e s IVa & b) whereas the r o o t s may occupy these gaps and ov e r l a p t o a c o n s i d e r a b l e extent (McMinn,.1963). Much of the e a r l y work i n the development of the model i n v o l v e d s e l e c t i n g a combination of va l u e s of "REDFAC", "REDINC" and "DINC" which, when the model was run, would .i giv e r e s u l t s t h a t f i t t e d p u b l i s h e d y i e l d t a b l e data (Fig;;. 9-H) s a t i s f a c t o r i l y . I t was found (Newnham, 1964) t h a t "REDFAC" c o n t r o l l e d the age a t which m o r t a l i t y f i r s t o c c urred and "REDINC" the amount of m o r t a l i t y t h e r e a f t e r . Reducing "REDFAC" delayed the onset of m o r t a l i t y ; r e d u c i n g "REDINC" decreased the amount of m o r t a l i t y . When s e a r c h i n g f o r p o s s i b l e competitors at each f i v e -year r e - a p p r a i s a l the program was designed t o c o n s i d e r a l l t r e e s or, more c o r r e c t l y , a l l p o s s i b l e t r e e l o c a t i o n s , w i t h i n a d i s t a n c e of e i g h t times the i n i t i a l s p a c i n g of the t r e e b e i n g s t u d i e d . The area around each t r e e was d i v i d e d i n t o o c t a n t s ( F i g . 12) and the c l o s e s t t r e e i n each octant was t e s t e d t o see whether i t was a competitor. I f a competitor 63 a b i ..If " ' iJBiIFI i1r|i r, | ^  B^0 idB .. v*-'*^i&fliKit PLATE I V : Crown canopy photographs of 3 9 - y e a r - o l d Douglas f i r p l a n t a t i o n s e s t a b l i s h e d at (a) 4 x 4 f t . (b) 10 x 10 f t . , Wind R i v e r , Washington. Photographed i n August, 1 9 6 3 . 64 40OOr-YIELD TABLES S.I. McArdl c, Meyer & Bruce, 1949 140 Barney 1936 140 Duff, 1956: 6x6ft. cl20 — — 8x8ft. cl20 35CO-3000-UJ ct o < Ui a in Hi ui a u. o a Ui ID Z 2500-2000-1500-IOOO-500-\ \ \ \ \ \ \ \ \ A ~ 60 BO Sty "4*5" AGE IN YEARS FIG. 9 : The r e l a t i o n s h i p between number of t r e e s per acre and t o t a l age or years from p l a n t i n g (Duff, 1956). 6 5 YIELD TABLES S.I. McArdle. Meyer & Bruce, 1949 140 20r~ Bornei, 1936 140 Duff, 1956: 6x6ft. cl20 • — — — 8 x8ft. CI20 , I8r- / / / I4r- / / Z Z lOr < ui a m a UJ 9 // 2 / I I / / / ' ,7/ / / / / / / / / / /// ' / / / / / / J L 2 0 4 0 6 0 S O I C O AGE IN YEARS PIG. 10: The r e l a t i o n s h i p between mean d. b. h. o. b. and t o t a l age or years from p l a n t i n g (Duff, 1956).. 6 6 400|-350 300 Ui OC g250|-a. ui It! Ui 5200L-o < Ui 5 I50|-- l < < 0 100-50r-YIELD TABLES S.I. McArdle, Meyer t Bruce, 1949 140 Barncc, 1936 140 Duff, 1956: 6 x6ft. cl20 ^ _ 6x8ft. cl20 1 i i I : £ t t / 1 / 11 / • / ' / !// ii/ <'// " J O " T6" AGE IN YEARS PIG. 1 1 : The r e l a t i o n s h i p between b a s a l area per acre and t o t a l age or ye a r s from p l a n t i n g (Duff, 1 9 5 6 ) . 67 J - 8 J -7 J - 6 J-S J-4 J -3 J -2 J - l J J + l J+2 J + 3 J+4J+5 J+6 J+7 J+8 FIG. 12: The t r e e l o c a t i o n s t e s t e d f o r p o s s i b l e competitors of the t r e e b e i ng s t u d i e d ( I , J) i n the model. Octants are numbered 1 t o 8, l o c a t i o n s w i t h i n each octant 1 to 31 i h the order of i n c r e a s i n g d i s t a n c e from t r e e ( I , j ) . 68 was only h a l f i n an octant, the competitive s t a t u s of the t r e e was halv e d . A l a r g e s e c t i o n of the FORTRAN program (L i n e s 53-369) i s r e q u i r e d t o c a l c u l a t e the d i s t a n c e s t o each l o c a -t i o n i n the octant and t o make the necessary t e s t s i n v o l v e d . The model assumed t h a t i f a t r e e was r e l e a s e d from competition at any stage, i t s p a t t e r n of diameter growth u n t i l c ompetition again set i n would be t h a t of an open-grown t r e e of the same d. b. h. and age. T h i s Ignored the p o s s i b l e shock e f f e c t ( S t a e b l e r , 1956) d i s c u s s e d i n Part I of t h i s t h e s i s . I t was thought t h a t the e r r o r s caused by making t h i s assumption would be s m a l l . R e s u l t s As s t a t e d e a r l i e r , the model was developed u s i n g a b a s i c matrix of diameters from a 6 x 6 f t . p l a n t a t i o n of Douglas f i r on the U n i v e r s i t y Research F o r e s t at Haney. Ten (4.4 per cent) of the 225 t r e e s i n t h i s matrix were dead or mi s s i n g . There was no evidence t o show t h a t t h i s m o r t a l i t y was not randomly d i s t r i b u t e d . S i t e index was approximately 140 a c c o r d i n g t o Barnes (-U. B. C. F o r e s t Club, 1959). Four i n i t i a l spacings ( p l a n t i n g d i s t a n c e s ) were t e s t e d : 3.3 x 3.3> 6.6 x 6.6, 9.9 x 9.9 and 13.2. x 13.2 f t . L a t e r the two wider spacings, 16.5 x 1.6.5 f t . and 19.-8 x 19.8 f t . , were t e s t e d . They are i n c l u d e d i n F i g . 13-15 i n order t h a t the e f f e c t of i n i t i a l s p a c i n g on stand growth can be s t u d i e d over very wide range of spa c i n g s . The program was run on the I. B. M. 7090 e l e c t r o n i c computer at the U n i v e r s i t y of Toronto which 69 4000|-35CO-3000-UJ OC O 2500-OC UJ a IA UJ S! , H 2OOO — * OC UJ 0 z I5CO lOOO-soo-INITIAL SPACING 3 3 x 33ft. 6-6x 66ft. 99 x 9.9ft. 13.2 x l3-2ft. 16-5 x I6-Sft. -19-Sx 198ft. \ \ X _L 20 BO ^ O S O AGE IN YEARS FIG. 1 3 : The r e l a t i o n s h i p between number of trees per acre and age. Run I I - 1 . lOO 201-18 / /// 14 «/> Ul X o z ? 12 I-z o Hi z -i- »° < 11 cc $ a ce ui ui 2 * 6 7 0 INITIAL SPACING 3 3 x3-3ft. /. 6 6 x 6-6ft. / 9.9 x 9-9ft. 13-2 x 13-2ft. / ' 16 5 x 16 5ft. 19 8 x 19 8ft. / // / / // // /// ,4 / / / / / / x / / / / / // / / / / // 2h // J L O 20 40 6O SO IGO AGE IN YEARS FIG.- 14: The re l a t i o n s h i p between mean d. b. h. o. b. and age. Run II-1. 71 if 400r-35C+-INITIAL SPACING 3-3 X 3-3ft. 6-6 X 6-6U. 9-9 X 99ft. 13-2 x 132ft. 16-5 X 165ft. 19-8 X 19-8ft. 30C+-ui CC O < CC UJ a 2 5 C 4 -H ui UJ u. UJ oc < § 2 0 0 r -< UJ a < i I50H < ID lOO— 50r-FIG. 15: / / / / / y / / - 7 / / / / / / / 20 _L 40 60 80 AGE IN YEARS The r e l a t i o n s h i p between basal area per acre and age. Run I I - 1 . fob 72 i s 300 times fas ter than an I . B. M. 1620. The time taken for program compilat ion, input , processing and output (but not l i s t i n g the resu l t s ) was about f ive minutes. The number, of trees per acre , mean d. b. h. (with i t s variance and stand-ard dev ia t ion) , basal area per acre , mean and per iod i c basal area increments, diameter of the tree of mean.basal area, gross basal area y i e l d per acre and morta l i ty were obtained at the end of each f ive -year p e r i o d . In add i t i on , the diameter frequency d i s t r i b u t i o n table and the diameter matrix could be pr inted out as r e q u i r e d . An example of the program output i s given in Appendix II (P ig . 70). Number of Trees per Acre (P ig . 13) The number of trees per acre remained constant u n t i l morta l i ty from competition set i n at age twenty years (3.3 x 3.3 f t . ) , 30 years (6.6 x 6.6 f t . ) , 50 years (9.9 x 9.9 f t . ) , 70 years (13.2 x 13.2 f t . ) or 95 years (16.5 x 16.5 f t . ) . M o r t a l i t y , as defined here, d id not occur at a l l i n the 19.8 x 19.8 f t . spacing during the f i r s t 100 years of the l i f e of the stand. M o r t a l i t y d id not usua l ly occur u n t i l the number of trees at any age was greater than that given i n the y i e l d tables of McArdle et a l . (1949). In the c losest spac-ing (3.3 x 3.3 f t . ) the stocking was considerably higher than the y i e l d table data . Part of t h i s d i f ference was probably a t t r i b u t a b l e to the fact that t h i s model only takes account of natura l morta l i ty through suppression of the weaker trees by the more vigorous trees and not by 73 agents t h a t cause i r r e g u l a r m o r t a l i t y such as f u n g i , i n s e c t s or wind. By. age 80 years the d i f f e r e n c e s i n s t o c k i n g due t o the i n i t i a l s p a c i n g have g r e a t l y diminished except f o r the 3.3 ,x 3.3 f t . s p a c i n g . Average Diameter at B r e a s t Height The average diameter at b r e a s t h e i g h t o u t s i d e bark given i n P i g . 14 i s the simple a r i t h m e t i c mean of a l l t r e e s . Y i e l d t a b l e s are based on those t r e e s above a c e r t a i n minimum d. b. h. ( u s u a l l y 1.5 i n c h e s ) . The New Zealand t a b l e s (Duff, 1956) d i f f e r from the y i e l d t a b l e s of the P a c i f i c Northwest i n t h a t the average d. b. h. given i s the d. b. h. of the t r e e of average b a s a l area. Age i s c a l c u l a t e d from the date, of •establishment o f . t h e p l a n t a t i o n and not from the date of ger-mination of the seed. These d i f f e r e n c e s i n nomenclature should be remembered when comparing the d i f f e r e n t y i e l d t a b l e s and the stand model. Prom P i g . 14 i t can.be seen.that, i n the e a r l y stages , of the model, the average d. b. h. was g r e a t e r than t h a t of the y i e l d t a b l e s of the P a c i f i c Northwest due t o the open--grown nature of the stands compared wi t h the normal stands given i n the y i e l d t a b l e s . As competition set i n the average d. b. h. growth f a l l s o f f . A f t e r the f i r s t t r e e s were removed as m o r t a l i t y , d.. b. h. growth.picked up again and remained more or l e s s l i n e a r . The g e n e r a l t r e n d of the d. b. h. growth curves f o l l o w e d those given by McArdle e_t a l . (1949) and Barnes (U. B. C. F o r e s t Club, 1959), except as 74 o u t l i n e d above. The average diameter of the 3.3 x 3.3 f t . spacing appeared.to be h i g h e r than expected above age 80 y e a r s . T h i s was because the model f a i l e d t o determine the t o t a l amount of competition f o r each t r e e . B a s a l Area per Acre The r e l a t i o n s h i p between b a s a l area and age ( P i g . 15) proved t o be the most d i f f i c u l t t o harmonize w i t h the y i e l d t a b l e r e l a t i o n s h i p s (Newnham, 1964) and the f i n a l model could probably s t i l l be improved. The b a s a l area of the 3.3 x 3-3 f t . s p a c i n g was e x c e s s i v e l y high'above 40 years, due t o a combination of a h i g h s t o c k i n g of t r e e s ( F i g . 13) and a h i g h average d. b. h. ( F i g . 14). I t appears that the model w i l l be of l i m i t e d use i n performing t e s t s on t h i s s p a c i n g above 40 y e a r s , or when the number of t r e e s i n the matrix i s reduced t o l e s s than 25. With the other three spacings, b a s a l area growth was most r a p i d i n the 6.6 x 6.6 f t . spacing, and l e a s t at the 19.8 x 19.8 f t . s p a c i n g . B a s a l area growth appears to l e v e l o f f at between 250 and,300 sq. f t . per a c r e . B a s a l area y i e l d appeared c l o s e r t o the y i e l d t a b l e s of Barnes (U. B. C. F o r e s t Club, 1959) than to those of McArdle et_ a l . (1949). "REDFAC" The v a r i a t i o n of the value of the r e d u c t i o n f a c t o r , "REDFAC", w i t h age i s shown i n F i g . 16. I t can be seen t h a t i t s value remained constant at each sp a c i n g u n t i l m o r t a l i t y 75 INITIAL SPACING l-2i IOr-0-8 O < u. O O 6 ui CC 0-4 0-2 3 3 x 3-3ft. 6 6 X6 6ft. 9 . 9 x 9 . 9 a . . 13-2 x 132ft. _L 20 40 60 AGE IN YEARS 80 ICO FIG. 16: The r e l a t i o n s h i p between "REDFAC" and age. Run I I - l . 76 o c c u r r e d . Prom then on i t i n c r e a s e d , the i n c r e a s e being more r a p i d w i t h i n c r e a s e i n age. Above a value of 1.0, "REDFAC" Is no longer a " r e d u c t i o n " f a c t o r . The reasons f o r modifying the value of "REDFAC" have been e x p l a i n e d above. Diameter Frequency D i s t r i b u t i o n s Frequency polygons have been drawn at ten-year i n t e r v a l s on a t r e e s - p e r - a c r e b a s i s ( F i g . 17-20). They show no s i g n i f i -cant departure from t h a t which might be expected from small samples taken from p l a n t a t i o n s . The i r r e g u l a r i t i e s i n the d i s t r i b u t i o n s of the l a s t three or f o u r decades of the c l o s e r spacings were probably due to the f a c t t h a t , although the number of t r e e s per acre i n each age c l a s s was r e l a t i v e l y l a r g e , the numbers i n the matrix on which the d i s t r i b u t i o n was based were r e l a t i v e l y s m a l l . For example, one t r e e pre-sent i n the matrix was e q u i v a l e n t to 17.8 t r e e s per acre at 3.3 x 3.3 f t . spacing, but only 1.1 t r e e s per acre at 13.2 x 13.2 f t . s p a c i n g . The small range i n diameters probably was a l s o due to the small number of t r e e s . The cumulative frequency d i s t r i b u t i o n s ( F i g . 21 and 22) have the c h a r a c t e r i s t i c sigmoid shape of n a t u r a l d i s t r i b u t i o n s . The range of diameters f o r any g i v e n mean i n c r e a s e s w i t h i n c r e a s e i n i n i t i a l s p a c i n g although t h i s i s probably due, i n p a r t , t o the g r e a t e r number of t r e e s p r e s e n t i n the matrix at the wider s p a c i n g s . l O O O r 400r 20 yrs. [ L a o yr». SOO 200-77 u, 200I-oc O < C C J3 ioc4-UJ cc u. o OC U) CO Z 40 yri. f 3 IO 50 yn. 1 60 y n . I it 100 - 70 y n . LTL II 3 S O yri. „ JL in lOOr 90 yn. Jl 16 £ ICO yri. T lb 24 „ T n , „ , _ DIAMETER, CLASS-ONE INCH INTERVALS FIG. 17: Diameter frequency d i s t r i b u t i o n s . Spacing: 3.3 3.3 f t . Run II-1 . x 7 8 400 n 20 yri. 30 yri. 200-£ I 2 7 1 _L UJ cc U 2COr cc UJ a UJ UJ ? lOO-u. O cc U l CO Z 40 yri. 50 yri. II 1 13 60 yri. i 15 ICOr 50-70 yri. r so yri. 9 19 IOQ-50-90 lOO yri. I DIAMETER CLASS - ONE INCH INTERVALS FIG. 1 8 : Diameter frequency d i s t r i b u t i o n s . Spacing: 6 . 6 x 6 . 6 f t . Run I I - 1 . ICOr 50 90 IOO yrt. li R DIAMETER CLASS - ONE INCH INTERVALS TT T 4 FIG. 19: Diameter frequency d i s t r i b u t i o n s . Spacing: 919 x 9.9 f t . Run I I - 1 . 80 DIAMETER CLASS-ONE INCH INTERVALS PIG. 20: Diameter frequency d i s t r i b u t i o n s . Spacing: 1 3 . 2 x 1 3 . 2 f t . Run I I - 1 . DIAMETER AT BREAST HEIGHT IN INCHES FIG. 21: Cumulative d. b. h. o. b. frequency d i s t r i b u t i o n s . Spacing: (A) 3.3 x 3.3 f t . ; (B) 6.6 x 6.6 f t . Run I I - l . DIAMETER AT BREAST HEIGHT IN INCHES PIG. 22: Cumulative d. b. h. o. b. frequency d i s t r i b u t i o n s . Spacing: (A) 9.9 x 9.9 f t . ; (B) 13.2 X 13.2 f t . Run I I -1. 83 D i s t r i b u t i o n s , of Trees The stand s t r u c t u r e of the b a s i c matrix i s shown i n P i g . 23 and the development of the stand f o r the 6.6 x 6.6 f t . s p a c i n g i s shown in. P i g . 24. The . f i n a l stand s t r u c t u r e f o r each sp a c i n g i s shown i n P i g . 25. The stand development between age ten and age 100 years f o r the spacings not shown i s s i m i l a r t o t h a t of the 6.6 x 6.6 f t . spacing,' the o n l y major d i f f e r e n c e b e i ng the ages at which the v a r i o u s stages shown i n Pig.. 24 occur. For the 3.3 x 3.3 f t . . s p a c i n g s the stages are reached e a r l i e r and f o r the wider spacings they are reached l a t e r . In order t h a t the s t r u c t u r e s of the stands may be understood, the t r e e s have been d i v i d e d i n t o f o u r c l a s s e s by d. b. h. (see F i g . 23). A f t e r c o mpetition has set i n , these c l a s s e s probably correspond t o the f o u r crown c l a s s e s : dominant, codominant, i n t e r m e d i a t e and suppressed. However, where there i s a c o n c e n t r a t i o n of t r e e s i n the upper d. b. h. c l a s s , i t i s u n l i k e l y t h a t they would a l l be dominant t r e e s , as would be i n d i c a t e d by t h i s method. Conversely, a group of t r e e s i n the s m a l l e s t d. b. h. c l a s s would c o n t a i n t r e e s i n the h i g h e r crown c l a s s e s . Using t h i s diagrammatic r e p r e s e n t a t i o n of the stand, i t i s p o s s i b l e t o t r a c e move-ment of t r e e s between crown c l a s s e s and to see how the v a r i o u s d i s t r i b u t i o n s of t r e e s , t o be t e s t e d l a t e r i n t h i s t h e s i s , are m o d i f i e d as the stand develops. Although the p l a n t a t i o n , on which the b a s i c matrix of diameters i s based, appeared remarkedly u n i f o r m i n growth, 84 INITIAL DIAMETER MATRIX AGE IO YEARS 5 - I 26 in. • - tO-42in. Suppressed Q D ^ B - t Intermediate ^ D - K O S B Codominant @ 5 < 0 s 5 + i Dominant eft D>D + » SCALE I 3 3x 33 ft. I I 6-6x6-6 ft. | I 99 X 9-9 ft. | , 13-2 x 13-2 ft. , PIG. 23: I n i t i a l diameter matrix used f o r d e v e l o p i n g the model. Data c o l l e c t e d from a Douglas f i r p l a n t a t i o n at Haney. (The c i r c l e s r e p r e s e n t i n g t r e e s are not drawn t o s c a l e . ) Run I I - 1 . The s c a l e s r e p r e s e n t 66 f t . at each spacing. 85 D I CD B - 4 04in. s - ± 0 9 7 i n . D-6-92in. s-±l-32in. AGE 60 YEARS > o o om ® © © m o 1 o # o o o _ _ _ . o • © o ® © o o ® o ® o © o © o I ® o ® ®# B-I0 09in. s=»± l-94in. ® o o m AGE BO YEARS o oo ® ® o mo o © © ® L ® B - 14 26 in. s-±2-49in. FIG. 2 4 : Stand s tructure of the basic model. Spacing: 6 . 6 x 6 . 6 f t . Run I I - 1 . 86 3 3x3-3ft. 6-6x6-6ft. r CD-~ l o © o L j £ B-I9'90in. »=.± 2-1 tin. o _ B-I897in. s-±2-37in. 9-9 x 9-9 ft. o o • ® o © © o o D O 0 O ® O O 5-l7-94in. s-±2-50in. o # o © o< 13 2 x 13 2 ft. ® ® mm® © o ® • o © @ ® ® @ • 3 A ® - • • ® ® 5—»8-39in. s-±2-2lin. Q FIG. 2 5 : Stand s t r u c t u r e of the b a s i c model at the d i f f e r e n t spacings at age 100 y e a r s . Run I I - 1 . 87 i t can be seen t h a t there i s some s i t e v a r i a t i o n w i t h i n the stand at age ten y e a r s . M o r t a l i t y occurs f i r s t i n the l o c a l patches of h i g h e s t s i t e q u a l i t y ( P i g . 24, age 40 years) as the competition i s more i n t e n s e . By age 60 y e a r s , m o r t a l i t y has occurred among the lower s i t e groups and the t r e e s have become more u n i f o r m l y d i s t r i b u t e d . . Before the number of t r e e s i s reduced t o the l e v e l shown f o r the 3.3 x 3.3 f t . s p a c i n g at age 100 years ( F i g . 25), the model ceases t o f u n c t i o n p r o p e r l y . The model only searches t o a d i s t a n c e of e i g h t times the i n i t i a l s p acing (26.4 f t . i n t h i s i n s t a n c e ) f o r c o m p e t i t o r s . T h e r e f o r e , at c l o s e spacings, some of the t r e e s are c o n s i d e r e d t o be f r e e of competition i n one or more oc t a n t s , diameter growth i s not reduced s u f f i c i e n t l y and b a s a l area consequently r i s e s t o the h i g h l e v e l p r e v i o u s l y d e s c r i b e d (see P i g . 15). Stand Model I I A As i t f a i l e d t o g i v e s a t i s f a c t o r y r e s u l t s a t the 3.3 x 3.3 f t . s p a c i n g above age 40 y e a r s , stand Model I I was modi-f i e d by p r e s c r i b i n g a f i x e d amount of m o r t a l i t y each f i v e -year p e r i o d . T h i s c o n s i s t e d of 0.5 per cent random mort-a l i t y , which was a p p l i e d t o the stand each f i v e - y e a r p e r i o d from the s t a r t of the model. A " c o m p e t i t i o n " m o r t a l i t y was a p p l i e d at the end of each f i v e - y e a r p e r i o d as soon as f i v e per cent of the stand had shown no i n c r e a s e i n diameter growth d u r i n g a f i v e - y e a r p e r i o d . The v a l u e s of t h i s mort-a l i t y were, i n the f i r s t i n s t a n c e , i n t e r p o l a t e d from the 8 8 y i e l d t a b l e f o r Douglas f i r of Barnes (U. B. C. F o r e s t Club, 1 9 5 9 ) but were l a t e r m o d i f i e d t o s u i t the requirements of the model (Table 2 ) . T h i s competition m o r t a l i t y was not a l l o c a t e d TABLE 2 : M o r t a l i t y by f i v e - y e a r p e r i o d s f o r Douglas f i r . Adapted from Barnes (U. B. C. F o r e s t Club, 1 9 5 9 ) . Run I I A - 1 . Age Previous 5 y e a r s ' (yr . ) m o r t a l i t y {% of t o t a l no. of t r e e s ) 15 2 5 2 0 2 3 2 5 21 30 1 9 3 5 17 40 14 45 12 50 10 5 5 8 6 0 7 6 5 7 7 0 6 7 5 6 8 0 5 8 5 5 90 4 9 5 4 1 0 0 3 u n i f o r m l y t o a l l diameter c l a s s e s but w i t h decreased probab-i l i t y i n c l a s s e s g r e a t e r than the mean. Within each c l a s s t r e e s were s e l e c t e d at random f o r m o r t a l i t y . A d e t a i l e d d e s c r i p t i o n of t h i s method of a p p l y i n g m o r t a l i t y was given i n an e a r l i e r r e p o r t (Newnham, 1 9 6 4 ) . T h i s m o d i f i e d model was run u s i n g the same matrix of diameters as was used i n stand Model I I d e s c r i b e d above. 89 R e s u l t s ( P i g . 26-28) show a reasonable correspondence w i t h the p u b l i s h e d y i e l d t a b l e data except t h a t the b a s a l area was again too h i g h i n the. 3.3 x 3.3 f t . s p a c i n g . Diameter d i s t r i -b u t i o n s u s u a l l y covered a g r e a t e r range of valu e s than i n stand model I I . As there was no improvement over stand model I I , the development of t h i s model was not pursued f u r t h e r . I t was a l s o thought more d e s i r a b l e t h a t the f i n a l model should generate i t s own co m p e t i t i o n m o r t a l i t y r a t h e r than to have i t predetermined. C o n c l u s i o n s Of the three stand models t e s t e d , stand model I can be d i s r e g a r d e d f o r f u r t h e r use owing t o i t s severe l i m i t a t i o n s i n t e s t i n g d i f f e r e n t s p a c i n g s . Stand model I I meets the requirements of t h i s t h e s i s i n t h a t m o r t a l i t y i s s e l f -p r e s c r i b e d and i t can be used t o t e s t a wide range of spacings. Stand model IIA could probably a l s o be developed i n t o a s a t i s -f a c t o r y model w i t h f u r t h e r m o d i f i c a t i o n . I t has the advantage, which could a l s o be b u i l t i n t o stand model I I i f r e q u i r e d , t h a t a small amount of random m o r t a l i t y i s a l l o c a t e d at the end of each f i v e - y e a r p e r i o d as w e l l as co m p e t i t i o n m o r t a l i t y . Both stand models I I and IIA are of only l i m i t e d use w i t h spacings as c l o s e as 3.3 x 3.3 f t . 90 4000r-3 S O C T -INITIAL SPACING 3.3 x 3 3 ft. 6-6x 6-6 ft. 99 x 9 9 ft. 13-2 x 13-2 ft. 3000 r ui ce U 2500F ce U i a. ui ce 2 O O O L -a: ui co I I500h z lOOOr V SOO}-PIG. 26: 2 0 eo 40 60 AGE IN YEARS The re l a t i o n s h i p between number of trees per acre and age. Run IIA-1. 1 0 0 91 2Q I8H I6L-14 ui X o z ? 1 2 1 X o Ui X H ICf-< Ul cc CO < 8 OC Ul I-Ul < 6 INITIAL SPACING 3 3 x 3-3 ft. 6-6 X 6-6 ft. — 9-9 X 9-9 ft. 13-2 x 13-2 ft. .y / / / / y y y y y / / 'y' / 1/ ' 7 y / / / 20 80 40 60 AGE IN YEARS PIG. 27: The rel a t i o n s h i p between mean d. b. h. o. b. and age. Run IIA - 1 . IOO 92 400|— INITIAL SPACING • - 3-3 x 3-3 ft. 6-6 x 6-6 U. 9-9 x 9-9 ft. 13-2 x 13-2 ft. 35C+-300— ui CC u < cc Ul 0- 250h-I-Ul Ul u. Ul cc I 200t~~ O < Ul cc < I50I-< < CO IOC4-/ / / SOL-/ / / / / / / / / / I / 1 / / 1 FIG. 28: 20~ 40 60 AGE IN YEARS BO lOO The r e l a t i o n s h i p between basal area per acre and age. Run IIA - 1 . PART I I I TESTING STAND MODEL I I Having developed a s a t i s f a c t o r y model f o r a normal or f u l l y - s t o c k e d stand of s i t e index 140 f e e t at 100 years, i t was next necessary to t e s t t h i s model f o r stands having v a r y i n g amounts and d i s t r i b u t i o n s of m o r t a l i t y f o l l o w i n g planting,, f o r v a r i o u s s i t e s and v a r i o u s t h i n n i n g schedules. For a l l t e s t s , except those used i n t e s t i n g s i t e d i f f e r e n c e s , the same diameter d i s t r i b u t i o n as that used i n d e v e l o p i n g the model, i . e . N( jl = 1.26, cr = 0.177)* was assumed. In the t e s t runs, however, the t h e o r e t i c a l normal d i s t r i b u t i o n was used as opposed t o the e m p i r i c a l d i s t r i b u t i o n , obtained from the Douglas f i r p l a n t a t i o n at Haney, which was used i n d e v e l o p i n g the model. For t e s t i n g s i t e d i f f e r e n c e s each diameter i n the b a s i c matrix was m u l t i p l i e d by a constant, which v a r i e d w i t h the s i t e index being t e s t e d . The d i s t r i -b u t i o n of l o c a t i o n s occupied by t r e e s t h e r e f o r e remained constant over the range of s i t e s t e s t e d w h i l e the d i s t r i -b u t i o n of diameters v a r i e d . An advantage of these methods of t e s t i n g over f i e l d experiments i s t h a t a l l the s i t e f a c t o r s , except the f a c t o r b e i n g t e s t e d , are h e l d constant. M o r t a l i t y F o l l o w i n g P l a n t i n g M o r t a l i t y f o l l o w i n g p l a n t i n g d e s c r i b e s the number of t r e e s m i s s i n g , or dead, i n the i n i t i a l diameter matrix at age ten y e a r s . In the p l a n t a t i o n which was used, i n d e v e l o p i n g the model t h i s was 4.4 per cent which, i n p r a c t i c e , 93 9h even at spacings as wide as 1 3 . 2 x 1 3 . 2 f t . , would be i g n o r e d . The purpose of these t e s t s was to determine the amoun.t of m o r t a l i t y t h a t may occur at each spacing before the f i n a l y i e l d would be a f f e c t e d . The d i s t r i b u t i o n bf the dead t r e e s i s important. I f the dead t r e e s occur i n clumps, as opposed to a random d i s t r i b u t i o n , the stand w i l l take longer t o r e t u r n to f u l l s t o c k i n g (see Smith et_ al_., 1 9 6 l , Table 1 6 ) . Method of Generating D i s t r i b u t i o n s The d i s t r i b u t i o n s t e s t e d were: bi n o m i a l ( 1 0 , 30 and 50 per cent m o r t a l i t y ) , u n i f o r m or r e c t a n g u l a r , ( 5 0 per cent m o r t a l i t y ) and an a r t i f i c i a l d i s t r i b u t i o n c o n s i s t i n g of two random i n f e c t i o n c e n t r e s ( f o u r t e e n per cent m o r t a l i t y ) . To a l l o c a t e each d i s t r i b u t i o n of m o r t a l i t y , the b a s i c matrix of 15 x 15 t r e e l o c a t i o n s was d i v i d e d i n t o 9 - l o c a t i o n square p l o t s . The number of p l o t s w i t h 0 , 1 , 2 , 9 t r e e s was c a l c u l a t e d from the d e n s i t y f u n c t i o n f o r the a p p r o p r i a t e d i s t r i b u t i o n (see Appendix I ) ; each p l o t was a l l o c a t e d one of these numbers at random. The l o c a t i o n s of the t r e e s w i t h i n each p l o t were chosen at random and, f i n a l l y , a diameter was a l l o c a t e d at random to each t r e e from a normal d i s t r i b u t i o n (JL= 1 . 2 6 , cr 2 = 0 . 1 7 7 ) . The two random i n f e c t i o n c e n t r e s were chosen to r e p r e s e n t the clumped m o r t a l i t y a s s o c i a t e d with, f o r example, P o r i a w e i r i l (see P l a t e I I ) . Two l o c a t i o n s i n the matrix were chosen at random as c e n t r e s of i n f e c t i o n . The i n f e c t i o n was then assumed to have spread outwards from each c e n t r e , k i l l i n g t r e e s w i t h p r o b a b i l i t y i n v e r s e l y p r o p o r t i o n a l to the d i s t a n c e from the centre l o c a t i o n . To a l l o c a t e t h i s mortal-i t y to the i n i t i a l , matrix, i t was assumed t h a t a l l e i g h t t r e e s i n the f i r s t square " s h e l l " surrounding the centre l o c a t i o n , f o u r of the s i x t e e n t r e e s i n the second " s h e l l " , two of the 24 t r e e s i n the t h i r d " s h e l l " and one of the 32 t r e e s i n the f o u r t h " s h e l l " , were dead. The a l l o c a t i o n of the dead t r e e s i n the second, t h i r d and f o u r t h s h e l l s was c a r r i e d out at random. Apart from the two random i n f e c t i o n c e n t r e s no other m o r t a l i t y was i n t r o d u c e d . I t was assumed t h a t the l i v i n g t r e e s surrounding the two dentres were not " i n f e c t e d " and t h e r e f o r e showed no d e c l i n e i n v i g o u r . T h i s i s u s u a l l y not the case i n p r a c t i c e but the problem of determining the amount of r e d u c t i o n i n growth due to i n f e c t i o n was o u t s i d e the scope of the present p r o j e c t . The d i s t r i b u t i o n s of m o r t a l i t y t e s t e d t h e r e f o r e cover random m o r t a l i t y , d e s c r i b e d by the b i n o m i a l d i s t r i b u -t i o n s , and clumped m o r t a l i t y , d e s c r i b e d by the u n i f o r m d i s t r i b u t i o n . .The random i n f e c t i o n centres d e s c r i b e an o extreme of a g g r e g a t i o n . The b i n o m i a l d i s t r i b u t i o n was used t o give a random d i s t r i b u t i o n of m o r t a l i t y , i n s t e a d of the Poisson d i s t r i b u t i o n , which i s more u s u a l l y a s s o c i a t e d w i t h randomness, as the g r e a t e s t number of t r e e s t h a t could occur i n a p l o t was l i m i t e d to n i n e . The Poisson d i s t r i b u t i o n r e q u i r e s t h a t t h e r e be no upper bound. 96 R e s u l t s For comparative purposes i t has been assumed i n these t e s t s t h a t the stand w i t h only ten per cent m o r t a l i t y , binom-i a l l y d i s t r i b u t e d , i s f u l l y stocked f o r each spacing-at age ten y e a r s . A m o r t a l i t y of l e s s than ten per cent f o l l o w i n g p l a n t i n g i s unusual i n p r a c t i c e . The Douglas f i r p l a n t a t i o n at Haney, which was used i n the development of the model, was s p e c i a l l y chosen as r e p r e s e n t a t i v e of the maximum s t o c k i n g that could be expected i n p r a c t i c e . The r e s u l t s of the t e s t s are summarised, g r a p h i c a l l y , i n F i g . 29-40. Cumulative frequency d i s t r i b u t i o n s are given i n F i g . 41-43 and.the development of the stand s t r u c t u r e under the d i f f e r e n t types of m o r t a l i t y i s given f o r the 6.6 x 6.6 f t . s p a c i n g i n F i g . 44-53. As would be expected, the i n i t i a l p l a n t i n g d i s t a n c e p l a y s an important p a r t i n determining the l e n g t h of time r e q u i r e d by the stand to again reach f u l l s t o c k i n g , by b a s a l area. At the c l o s e s t spacing ( F i g . 31)* d i f f e r e n c e s i n b a s a l area have disappeared by age twenty, r e g a r d l e s s of the amount or d i s t r i b u t i o n of m o r t a l i t y at age ten y e a r s . Doubling the p l a n t i n g d i s t a n c e i n c r e a s e s the age at which r e c o v e r y takes p l a c e t o between 60 and JO y e a r s , depending on the amount and d i s t r i b u t i o n of m o r t a l i t y ( F i g . 3^). At a spacing of 9.9 x 9.9 f t . , the stands w i t h 30 and 50 per cent b i n o m i a l m o r t a l i t i e s have recovered by age 85 to 95 years but the clumped, uniform (or r e c t a n g u l a r ) 50 per cent m o r t a l i t y has not f u l l y r ecovered at age 100 years ( F i g . 37). At the widest 97 4000r-3500U 3000r ui ce ^ 2500|-ce (/> Ul Ul * 2000r u. O cc ui co 3 I50C+ Z I c o o k SOOT-DISTRIBUTION Binomial •-Uniform 2 Rand. MORTAL I IO 30 50 50 T Y - % Centres _L ± FIG. 29: 20 40 oO AGE IN YEARS 80 100 The e f f e c t of amount and d i s t r i b u t i o n of m o r t a l i t y f o l l o w i n g p l a n t i n g on number of t r e e s per a c r e . Spacing: 3.3 x.3.3 f t . Runs II -2 to II-6. 20r -2 -98 , / DISTRIBUTION MORTALITY-% / ..^  Binomial 10 / ffij 30 / / / / / SO / , / / - — II — Uniform 50 2 Rand. Inf. Centre* 14 /// iff/ ff/ <*// /I// •t f t . //// //// /// /ft// J I L loo 20 40 60 SO AGE IN YEARS EG. 30: The e f f e c t of amount and d i s t r i b u t i o n of m o r t a l i t y f o l l o w i n g p l a n t i n g on mean d. b. h. o. b. Spacing: 3.3 x 3.3 f t . Runs II-2 to II-6. 400r-350r-300U 2COh ISOh-lOCf-50h-DISTRIBUTION MORTALITY - % — Binomial IO 30 - so Uniform SO 2 Rand. Inf. Ccntrtt 20 40 60 SO ICO AGE IN YEARS PIG. 31 : The ef fect of amount and d i s t r i b u t i o n of morta l i ty fo l lowing p lant ing on basal area per acre . Spacing: 3.3 x 3.3 f t . Runs I I - 2 to I I - 6 . 100 4 0 O Q - DISTRIBUTION Binomial Uniform MORTALITY IO 30 SO SO 35O0-2 Rand. Inf. Centres 14 3OO0-UJ CC a ID a «n ui UJ CC 25O0-2OO0-cc UJ ID 5 Z I500-IOOO-5 0 0 -— • — ^•^Br 20 40 60 AGE IN YEARS SO IOO PIG. 32: The e f f e c t of amount and d i s t r i b u t i o n of m o r t a l i t y f o l l o w i n g p l a n t i n g on number of tr e e s per acre. Spacing: 6 .6 x 6 .6 f t . Runs I I - 2 to I I - 6 . 1 0 1 20r-DISTRIBUTION MORTALITY ~% 18 16 14 (A Ui X O z X x io < Ui oc ID oc UJ »-UJ 2 < 6 O Binomial Uniform 2 Rand. Inf. Centres IO 30 SO 50 14 20 40 60 AGE IN YEARS BO FIG. 33: The e f f e c t of amount and d i s t r i b u t i o n . o f m o r t a l i t y f o l l o w i n g p l a n t i n g on mean d. b. h. o. b. Spacing: 6.6 x 6.6 f t . Runs II-2 to 11-6. IOO 102 400P-350 r DISTRIBUTION Binomial MORTALITY - % 10 . 30 . 50 Uniform 50 2 Rand. Inf. Centre* 14 300r-14 cc 5c ce ui a ui ui ui ce < o IA < Ul ce < < < CO 2SOh 200\-\SO\-IOOr SO\-/ — v-y / / / / / ! / // // / / / /// / If/// / ll l // li l // a // /// /// 20 40 60 SO lOO AGE IN YEARS PIG. 34: The e f f e c t of amount and d i s t r i b u t i o n of m o r t a l i t y f o l l o w i n g p l a n t i n g on b a s a l area per a c r e . Spacing: 6.6 x 6.6 f t . Runs II-2 to II-6. 1 0 3 4000r-3SOO 3000 -UJ a 2SOO oe ui a (A Ui g 2 0 0 0 -oe Ui a | ISOO z IOOO SOO-01 ST RI BUT ION Binomial MORTALITY"* IO 30 50 Uniform 50 2 Rand. Inf. Centres 14 "20 JL "45 60 AGE IN YEARS ret PIG. 3 5 : The e f f e c t of amount and d i s t r i b u t i o n of m o r t a l i t y f o l l o w i n g p l a n t i n g on number of t r e e s per ac r e . Spacing: 9 . 9 x 9 - 9 f t . Runs I I - 2 t o I I - 6 . 104 20r-18 16 14 vt u i X o z X o tn < u i ce CO H < ce Ui Ui 2 < 6 O 12 IC4-DISTRIBUTION MORTALITY-% Binomial IO . 30 . SO • / / 2 Rand. Inf. Centre* 14 Uniform SO / ' //// //// / / sty / //// .4'/ // 20 40 oO SO IOO AGE IN YEARS FIG. 3 6 : The e f f e c t of amount and d i s t r i b u t i o n ' o f m o r t a l i t y f o l l o w i n g p l a n t i n g on mean d. b . h . o'. b. Spacing: 9 . 9 x 9 . 9 f t . Runs I I - 2 to I I - 6 . 105 400r-350h DISTRIBUTION Binomial Uniform • 2 Rand. Inf. Centres MORTALITY - % IO 30 50 SO 14 300r-U J cc cc ui a ui U J U J 2 SOL 200f-< U J cc < < I50r ID lOOh 50r / / // / 7 / / / / s I / II I / / / / / III //• / /// / f If /// /// ^ JL _L 20 40 60 AGE IN YEARS 60 IOO P I G . 37: The e f f e c t o f amount a n d d i s t r i b u t i o n o f m o r t a l i t y f o l l o w i n g p l a n t i n g on b a s a l . a r e a p e r a c r e . S p a c i n g : 9.9 x 9.9 f t . R u n s . I I - 2 t o I I - 6 . 106 4000r-3500-DISTRIBUTION MORTALITY - % Binomial IO . 30 N SO Uniform 50 2 Rand. Inf. Centres 14 3000 -u 2SOO. O < ec ui a (A ui 2 OOO a i -ec ui 5 ISOOr-3 IOOO-500 — 20 40 60 AGE IN YEARS BO too FIG. 38: The ef fect of amount and d i s t r i b u t i o n of morta l i ty fo l lowing p lant ing on number of trees per acre . Spacing:. 13.2 x 13.2 f t . Runs I I - 2 to I I - 6 . 107 DISTRIBUTION MORTALITY - X Binomial 1 0 y — « 30 / , " *° A / A Uniform 50 y / •// 2 Rand. Inf. C*ntr«« 14 / / / A ///// /As Ay AA / / / / / / / / / / / / / / / / _L 20 40 6 O 80 lOO AGE IN YEARS 39: The e f f e c t o f amount a n d d i s t r i b u t i o n ' o f ' m o r t a l i t y f o l l o w i n g p l a n t i n g on mean d. b . ' h . o. b. S p a c i n g : 13.2 x 13.2 f t . Runs I I - 2 t o I I - 6 . i o 8 400r-350 300 250 Ui CC o < CE Ui Ck fr-il l Ui U. Ui cc < 8 200| < u i oc < < I50| < co IOO DISTRIBUTION Binomial MORTALITY - % IO .. 30 ,, SO uniform 50 2 Rand. Inf. Centres 14 / X . / / / / // / // , / / / // / / i s / / // /// / // / s< if / / // / / /// 20 40 60 AGE IN YEARS SO IOO FIG. 40: The e f f e c t of amount and d i s t r i b u t i o n of m o r t a l i t y f o l l o w i n g p l a n t i n g on b a s a l area per a c r e . Spacing: 1 3 . 2 x 1 3 . 2 f t . Runs I I - 2 to I I - 6 . 4 8 12 16 DIAMETER AT BREAST HEIGHT IN INCHES FIG. 41: Cumulative d. b. h. o. b. frequency d i s t r i b u t i o n s for (A) ; 10 a (B) 30. per cent binomial d i s t r i b u t i o n s of morta l i ty fo l lowing p lant ing . Spacing: 6 . 6 x 6 . 6 f t . Runs 1 1 - 2 , 3 . Ul u 8 12 16 20 DIAMETER AT BREAST HEIGHT IN INCHES 24 28 F I G . 42: Cumulative d. b. h. o. b. frequency d i s t r i b u t i o n s for (A) 50 per cent binomial and (B) 50 per cent uniform.(rectangular) d i s t r i b u t i o n s of mortal i ty fol lowing p lant ing . Spacing: 6 .6 x 6 .6 f t . Runs 11-4,5-o DIAMETER AT BREAST HEIGHT IN INCHES FIG. 4 3 : Cumulative d. b. h. o. b. frequency d i s t r i b u t i o n s for two random in fec t ion centres ( l 4 per cent morta l i ty fol lowing p lant ing ) . Spacing: 6 . 6 x 6 . 6 f t . Run II-6. 112 INITIAL DIAMETER MATRIX AGE 10 YEARS o<§ o © © o @ © o @ o o#o QQ©Q D- I-26in. s - ±0-42in. Suppressed D S B - i Intermediate 3^) B - B < D S 5 Codominant ^ 5<DsB + i Dominant 4ft D>5 + s SCALE 33 x3-3ft.  6 6 x 66ft.  9-9 x 9-9ft. 132 x 132ft. PIG. 44: I n i t i a l diameter matrix w i t h 10 per cent b i n o m i a l d i s t r i b u t i o n of m o r t a l i t y f o l l o w i n g p l a n t i n g . Run I I - 2 . The s c a l e s r e p r e s e n t 66 f t . at each sp a c i n g . 113 AGE 20 YEARS AGE 40 YEARS 5-4-OSin. S s ±0 '98 in »=±l-34in. AGE 60 YEARS > O 1 o m m ©<§>o <§> o > o o o m D= 9-94 in. 8 s" ± 183 in. AGE 80 YEARS o m o o o o o O . O ©j 5"- I4l3in. s=±2-57in. FIG. 45: The e f f e c t of amount and d i s t r i b u t i o n of m o r t a l i t y f o l l o w i n g p l a n t i n g on stand s t r u c t u r e . B i n o m i a l d i s t r i b u t i o n ' ( 1 0 per cent m o r t a l i t y ) . Spacing: 6.6 x 6 .6 f t . Run I I - 2 . AGE IOO YEARS J O © o mm CD. o ® o > • ® o o l _ 5-l8-59in. » - ±3 I Bin. FIG. 45: Continued. 1 1 5 INITIAL DIAMETER MATRIX AGE IO YEARS 5"— I-26 In. s-±0-42in. PIG. 46: I n i t i a l diameter matrix with 3 0 per cent binomial d i s t r i b u t i o n of morta l i ty fo l lowing p l a n t i n g . Run I I - 3 . 1 1 6 AGE 20 YEARS AGE 40 YEARS s - ±l-57ia AGE 60 YEARS o mi o # ( o e mm o 5= IO-63in. © ® @ Q s « ±211 in. o AGE SO YEARS • ® © O O © •© ® • o o ® o © # o © ® o © L i 5- I4 06in. s - ±2 91 in. PIG. 4 7 : The e f f e c t of amount and d i s t r i b u t i o n of m o r t a l i t y f o l l o w i n g . p l a n t i n g on stand s t r u c t u r e . B i n o m i a l d i s t r i b u t i o n ' ( 3 0 per cent m o r t a l i t y ) . Spacing: 6 . 6 x 6 . 6 f t . Run I I - 3 . AGE lOO YEARS 18 83 in. »-±3l3 in. FIG. 4 7 : Continued. 118 INITIAL DIAMETER MATRIX AGE IO YEARS o m o o • ® 8<§> # o O ® © CD o / d# © @ 5-1 26in. »-±0-42in. FIG 48: I n i t i a l - d i a m e t e r matrix w i t h 50 per cent binomial d i s t r i b u t i o n of m o r t a l i t y f o l l o w i n g p l a n t i n g . Run II-4 . 119 AGE 2 0 YEARS m© o © © ® c AGE 40 YEARS © o © r~© o© o ® o ' iv © © © © < • \ • • © ©©©OO* o © o © o C D O ( o ® @ O s ©8 © Q © _ © > / _ •© ©©a © < 0® • © © « J 5 - 4 -04^ s -±0-97in. @ g@QO © _0© 5-7 89 in. I f > J s -±l-60in. AGE 60 YEARS La* o © n © o o © o © © O i © © © o i © O [ _ • © f o © D=» IO-98in. i - ±2 04in. AGE SO YEARS o o o © © o o © o o 5= 14-46 in. • =±2-63in. FIG. 49: The e f f e c t of, amount and d i s t r i b u t i o n of m o r t a l i t y f o l l o w i n g , planting- on stand s t r u c t u r e . Binomial d i s t r i b u t i o n ' ( 5 0 per cent m o r t a l i t y ) . Spacing: 6.6 x 6.6 f t . Run I I - 4 . A G E l O O Y E A R S r o om or o Li 5=18 48in. 5 =±2 85in. PIG. 4 9 : Continued, 121 INITIAL DIAMETER MATRIX AGE IO YEARS 6= l-26in. » = ±0-42in. P I G . 5 0 : I n i t i a l d i a m e t e r m a t r i x w i t h 5 0 p e r c e n t u n i f o r m ( r e c t a n g u l a r ) d i s t r i b u t i o n o f m o r t a l i t y f o l l o w i n g p l a n t i n g . R u n I I - 5 -122 o @* . §® © ©o, , D" = 4-Q4in. s - ±0-98in. 5"= 7-9lin; t = ± i -S5 ia AGE 60 YEARS AGE S O YEARS o o L o o © o © • ® ® m © ® m ©o, , © ooo _J L 5= ll-45in. s - ±2-18in. o o © o © m o 5 » 14-92 in. s - ±2-7 8ln. PIG. 51 : The e f f e c t of amount a n d . d i s t r i b u t i o n of m o r t a l i t y f o l l o w i n g p l a n t i n g on stand s t r u c t u r e . Uniform ( r e c t a n g u l a r ) d i s t r i b u t i o n (50 per cent m o r t a l i t y ) , Spacing: 6 . 6 x 6 . 6 f t . Run I I - 5 . FIG. 51.: Continued. 124 INITIAL DIAMETER MATRIX AGE IO YEARS 5-1-26in; s - ±0-42in. FIG. 52: I n i t i a l diameter matrix w i t h two randomly l o c a t e d i n f e c t i o n c e n t r e s ( l 4 per cent m o r t a l i t y ) . Run II-6 . 125 AGE 20 YEARS AGE 40 YEARS #®©©@o©®©©©~^  D p© mmmom mm ©©11 O B o ® m ) 0 ® m < m / 1 ©00©© o@© © o f )©©©• 0 0 §~®®o © ©©< ©@ m m © s-g|©© o :3o©@@© <§>@©o©©<~ >o©, B = 4 05in; ©© • 0<§>i> 0©0© ©© • • O © © © © • i © © •©#©• © ©< ' ©© o o©@© o©©© s = ±098in. B = 7 I2in. • = ± l-42in. AGE 60 YEARS © © r ) o© o 1 © ©©© o< AGE SO YEARS o © o o © 1 o o o o o ©© o o B = IO-43in. s = ±209in. D= I484in. © J s = ±2-94 in. FIG. 53: The e f f e c t of amount and d i s t r i b u t i o n of m o r t a l i t y f o l l o w i n g p l a n t i n g on stand s t r u c t u r e . Two randomly l o c a t e d i n f e c t i o n c e n t r e s (l4 per cent m o r t a l i t y ) . Spacing: 6.6 x 6.6 f t . ' Run II-6. AGE lOO YEARS 5 = I924in. » = ±3-SOin. FIG. 53: Continued. 127 s p a c i n g . t e s t e d , 13.2 x 13.2 f t . , only the 30 per cent b i n o m i a l d i s t r i b u t i o n of m o r t a l i t y and the two random i n f e c t i o n c e n t r e s are approaching f u l l b a s a l area s t o c k i n g at age 100 y e a r s . G e n e r a l l y , i t can be seen t h a t the two clumped d i s t r i b u t i o n s take longer to r e c o v e r than the random (binomial) d i s t r i b u -t i o n s . There i s a l s o evidence to suggest t h a t , at the two widest spacings, the maximum b a s a l area f o r the two random i n f e c t i o n c e n t r e s w i l l not be as great as that reached by the stands w i t h b i n o m 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 . The reason f o r t h i s i s t h a t each i n f e c t i o n c e n t r e covers such a l a r g e area (36 and 64 m i l a c r e s at 9.9 x 9.9 and 13.2 x 13.2 f t . , r e s p e c t i v e l y ) t h a t the surrounding stand i s not capable of f u l l y occupying the area b e f o r e age 100 y e a r s . The l e n g t h of time b e f o r e the d i f f e r e n c e s i n the number of t r e e s per acre between the d i f f e r e n t types of m o r t a l i t y d i s a p p e a r s , i s ' l o n g e r than t h a t f o r b a s a l area ( P i g . 29, 32, 35 and. 38). T h i s i s because the s u r v i v i n g t r e e s have l e s s c o m p e t i t i o n and are t h e r e f o r e able to accumulate r a p i d , i n d i v i d u a l , b a s a l area growth. In the two clumped d i s t r i b u t i o n s t e s t e d , m o r t a l i t y i s more r a p i d than w i t h the random d i s t r i b u t i o n s i n the f i r s t two or three p e r i o d s a f t e r c o m p e t i t i o n s e t s i n . T h i s i s due to m o r t a l i t y o c c u r r i n g among the dense clumps of the s u r v i v i n g t r e e s . The diameter frequency d i s t r i b u t i o n s ( P i g . 41-43) are 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 of the m o r t a l i t y and r e t a i n the c h a r a c t e r i s t i c sigmoid shape of normal d i s t r i -b u t i o n s . A small secondary peak occurs towards the upper 128 l i m i t of the range a f t e r age 60 years i n the stand w i t h the two random i n f e c t i o n c e n t r e s ( P i g . 43). T h i s may be due t o the "edge e f f e c t " as a r e s u l t of which l a r g e t r e e s are pro-duced around each i n f e c t i o n c e n t r e . The development of the s t r u c t u r e of the stands under the e f f e c t s of the d i f f e r e n t types of p l a n t i n g m o r t a l i t y can be f o l l o w e d i n F i g . 44-53. The development of the stands e s t a b l i s h e d at other than the 6.6 x 6.6 f t . spacing, which d i f f e r s only i n the age a t which each stage Is reached, i s not reproduced here. I f i t i s assumed t h a t the f o u r diameter groupings shown are e q u i v a l e n t to the dominant, codominant, i n t e r m e d i a t e and suppressed crown c l a s s e s , i t can be seen t h a t there i s a ge n e r a l movement down, although between two and f i v e per cent of the t r e e s move up. M o r t a l i t y i s u s u a l l y c o n f i n e d t o the "suppressed" and " i n t e r m e d i a t e " c l a s s e s . Where the d i s t r i b u t i o n of the t r e e s i s clumped, m o r t a l i t y occurs f i r s t w i t h i n the clumps. These r e s u l t s do not d i f f e r from those t h a t might occur i n a c t u a l p l a n t a t i o n s ( G u i l l e b a u d and Hummel, 1949; Warrack, 1952). S i t e Q u a l i t y To t e s t d i f f e r e n c e s i n s i t e q u a l i t y , each t r e e i n the b a s i c diameter matrix used to develop the model was m u l t i -p l i e d by a constant t o reduce the mean d. b. h. t o 0.80 i n . (approximately S. I. 120) or to 1.92 i n . (approximately S. I. 160). The d i s t r i b u t i o n of the t r e e s i n the matrix i s there-f o r e the same as t h a t f o r the b a s i c model. 9 R e s u l t s S i t e i n d i c e s 120 and 160 and s i t e index 140 (the b a s i c model), are compared i n F i g . 54-58 f o r the 6.6 x 6.6 f t . s p a c i n g . Fig-. 54 shows t h a t the number of t r e e s per acre i s g r e a t e s t f o r the poorest s i t e and l e a s t f o r the best s i t e as expected. The p a t t e r n of mean d. b. h. growth i s s i m i l a r f o r a l l three s i t e s , the only d i f f e r e n c e being i n the value of the mean ( F i g . 55). The d i f f e r e n c e s i n b a s a l area per acre ( F i g . 56) are not as marked as those f o r d. b. h. but are i n p r o p o r t i o n t o the y i e l d t a b l e v a l u e s (Barnes (U. B. C. F o r e s t Club, 1959), McArdle et_ al_., 1949). The cumulative diameter frequency d i s t r i b u t i o n s are shown i n F i g . 57 and the stand s t r u c t u r e s at age 100 years i n F i g . 58. The l a t t e r i n d i c a t e t h a t the stand development of the two s i t e s has not been e x a c t l y p a r a l l e l . Some of the l o c a t i o n s occupied by t r e e s i n the s i t e index 160 stand are not occupied i n the s i t e index 120 stand where the m o r t a l i t y was l e s s . T h i n n i n g T h i n n i n g i s the most important s i l v i c u l t u r a l oper-a t i o n t h a t can improve the q u a l i t y and c o n d i t i o n of the stand once i t has been e s t a b l i s h e d . I t must not be confused w i t h e x p l o i t a t i o n f e l l i n g , which i s c a r r i e d out w i t h the s o l e purpose of o b t a i n i n g a monetary r e t u r n , r e g a r d l e s s of the c o n d i t i o n of the r e s i d u a l stand. T h i n n i n g p r a c t i c e on the Continent of Europe has, u n t i l r e c e n t l y at l e a s t , been c l a s s i c a l : "low" t h i n n i n g where the dead, dying and 130 4000r -3SOOU 3000r -UJ 2SOO OC CJ < oc UJ OL u> UJ 2000 oc t -oc UJ CO 2 1500 Z I OOD 500r -SITE INDEX 120 • U O 160 20 AO 60 AGE IN YEARS 80 IOO FIG. 54: The effect of s i t e on number of trees per acre1. Spacing: 6.6 x 6.6 f t . Runs 11-2,7,8. Ul / 1 3 1 SITE INDEX , 20p- / 120 / / j UO / / / / / / / •4~ / / / / / «> / / / * / / / / / ? I 2 " / / I / / / / • 1 0 / -a / / H 8 L / / / / / ct / \ I / / O / / / " ' / / '' / / / / / / / / / //•/ // / / _L 20 4 0 60 eo 100 AGE IN YEARS FIG. 55: The ef f e c t of s i t e on mean d. b. h. o. b. Spacing: 6 . 6 x 6 . 6 ft.. Runs 11-2,7,8. H in O O O T BASAL AREA IN SQUARE FEET PER ACRE in f to O O to s 8 o in O O o VJl OA CT\H3 • & X CD • CD 0\0 ct ct O • i-b CQ £ ct 3 CD CO O H 3 H I CT ro t» - CO -<] 03 OD 0) CD •a CD 03 o CD CO •a 03 o 3 cn V \ N O \ tn H m * Z _ O O z o m x O P i -< m > 3D \ 51 \ \ \ V \ \ \ \ N a> O \ 8' \ \ > \ ro DIAMETER AT BREAST HEIGHT IN INCHES FIG. 5 ? : Cumulative d. b. h. o. b. frequency d i s t r i b u t i o n s f or ( A ) s i t e index 1 2 0 and (B) s i t e index 1 6 0 . Spacing: 6.6 x 6.6 f t . Runs 11-7,8. (See also F i g . 21b.) M 134 SITE INDEX 120 SITE INDEX I60 o _ O O V 1 i © o © o CD. © ® © O O ~ l ® @ o 113 • ' • © J l _ •• © J B» 15211ns. s - ± 2 0 9 i n s . 5-23-581ns. • = ±358ins. FIG. 5 8 : The ef fect of s i t e on stand s tructure at age 1 0 0 years . Spacing: 6 . 6 x 6 . 6 f t . Runs 1 1 - 7 * 8 . (See a lso F i g . 2 5 . ) d i s e a s e d t r e e s and the "whips" are the f i r s t t o be removed, f o l l o w e d by the removal of suppressed and some in t e r m e d i a t e t r e e s to g i v e more growing space f o r the dominant and co-dominant t r e e s . In a d d i t i o n , "wolf" t r e e s are o f t e n removed at the time of the f i r s t low t h i n n i n g . Most of the t r e e s removed i n such t h i n n i n g s , p a r t i c u l a r l y those i n the e a r l y l i f e of the stand, were unusable and t h e r e f o r e there was l i t t l e f i n a n c i a l r e t u r n . To make t h i n n i n g more a t t r a c t i v e e c o n o m i c a l l y , "crown" t h i n n i n g was i n t r o d u c e d . A crown t h i n n i n g i s o l a t e s a l i m i t e d number of the best dominants i n the stand by removing the l a r g e r competitors surrounding them. Most of the unusable suppressed and i n t e r m e d i a t e t r e e s are l e f t s t a n d i n g . In both the low and crown t h i n n i n g s d e s c r i b e d above the t r e e s are s u b j e c t i v e l y chosen. The marking of t r e e s f o r t h i n n i n g Is t h e r e f o r e a slow process, r e q u i r i n g c o n s i d e r a b l e s k i l l . Because of t h i s , and the u n s a l e a b i l i t y of the s m a l l e r -s i z e d t r e e s , t h i n n i n g has not gained the same prominence i n the P a c i f i c Northwest as i t has i n Europe. To reduce the s u b j e c t i v i t y i n t h i n n i n g , t h i n n i n g p r e s c r i p t i o n s have been p r e s c r i b e d , d e s c r i b i n g the number of t r e e s or the amount of b a s a l area to be l e f t a f t e r each t h i n n i n g . The s i m p l e s t of these p r e s c r i p t i o n s i s row t h i n n i n g i n p l a n t a t i o n s (Spurr, 1948; L i t t l e and Mohr, 1963) where e n t i r e rows of t r e e s are cut, r e g a r d l e s s of the s i z e or q u a l i t y of the t r e e s , or the s i z e or q u a l i t y of the t r e e s i n the r e s i d u a l stand. Such a t h i n n i n g i s mechanical, 136 r e q u i r i n g l i t t l e s k i l l , and i s u s u a l l y p r a c t i c e d only i n young p l a n t a t i o n s . In South A f r i c a , numerical t h i n n i n g s have been car-r i e d out s i n c e about 1930 ( H i l e y , 1 9 5 9 ). Before marking a p l a n t a t i o n f o r t h i n n i n g , the marker measures out a one-tenth acre p l o t and marks the t r e e s on t h i s p l o t i n such a way t h a t the p r e s c r i b e d number of t r e e s are l e f t . He continues to mark a c c o r d i n g t o t h i s d e n s i t y but w i t h frequent checks to ensure he i s m a i n t a i n i n g h i s accuracy of marking. I t has been found t h a t t h i s can be done q u i c k l y and a c c u r a t e l y , by r e l a t i v e l y u n s k i l l e d l a b o u r . There i s s t i l l a c e r t a i n sub-j e c t i v e n e s s i n s e l e c t i n g the t r e e s t o be removed. Although the numerical schedules are s t r i c t l y adhered to, the t h i n n i n g p r a c t i c e s are u s u a l l y s i l v i c u l t u r a l l y , as w e l l as e c o n o m i c a l l y , a c c e p t a b l e . I t i s t h e r e f o r e d i f f i c u l t t o use the stand model t h a t has been developed f o r t h i s t h e s i s to t e s t the e f f e c t of v a r i o u s t h i n n i n g regimes. However, there are two pos-s i b l e methods t h a t can be used. F i r s t , knowing (from the matrix p r i n t e d f o r each f i v e - y e a r p e r i o d ) the d. b. h. of each t r e e and i t s l o c a t i o n i n the stand a t the end of any f i v e -year p e r i o d , i t i s p o s s i b l e t o s e l e c t those t r e e s which, i n the o p i n i o n of the reader, should be "thinned". The model i s then run f o r a f u r t h e r f i v e - o r ten-year p e r i o d and the pro-cess repeated. T h i s i s a s u b j e c t i v e method. An o b j e c t i v e method, and the one which i s used here, i s to p r e s c r i b e the removal of a l l t r e e s w i t h i n c e r t a i n diameter l i m i t s at the 137 end of each t h i n n i n g p e r i o d of ten or twenty y e a r s . The l i m i t s used depend on the mean d. b. h. and i t s standard d e v i a t i o n . In the present i n s t a n c e , they have been s e l e c t e d t o repr e s e n t a moderate and a severe low t h i n n i n g , and a crown t h i n n i n g or s e l e c t i o n f e l l i n g . The growth of an unthinned ( c o n t r o l ) stand i s summarized i n Table 3 . Moderate Low Thi n n i n g T h i s removes a l l t r e e s t h a t are l e s s than the mean d. b. h. minus one standard d e v i a t i o n (D - s) at the end of each t h i n n i n g p e r i o d ( a f t e r the f i v e - y e a r m o r t a l i t y has been removed). T h i n n i n g commences at the end of the p e r i o d i n which competition m o r t a l i t y f i r s t occurs. A c c o r d i n g t o the method which has been used i n the model to a l l o c a t e crown c l a s s e s t o each t r e e (see P i g . 2 3 ) , t h i s t h i n n i n g removes a l l the t r e e s i n the suppressed crown c l a s s and no t r e e s from any other crown c l a s s . I t was thought t h a t such a p r e s c r i p t i o n would be e q u i v a l e n t t o a li g h t - t o - m o d e r a t e low t h i n n i n g but, a c c o r d i n g to the d/D r a t i o (Warrack, 1959b)* the t h i n n i n g was more severe than expected. The r e s u l t s are shown i n P i g . 59 and 60 and i n Table 4 , f o r a ten-year t h i n n i n g p e r i o d . Although there i s c o n s i d e r a b l e r e d u c t i o n i n the number of t r e e s per acre com-pared w i t h the unthinned stand ( P i g . 59)* the r e d u c t i o n i n b a s a l area i s not so marked ( F i g . 6 o ). The i n c r e a s e i n net b a s a l area y i e l d over the unthinned stand i s at once apparent. Gross y i e l d (stand + t h i n n i n g s + m o r t a l i t y ) was s l i g h t l y TABLE 3 : The growth of an unthinned stand. S i te index: l 4 o . Spacing: 6 . 6 x 6 . 6 f t . Run I I - l . Age ( y r . ) Number of trees per acre Mean d . b . h . o . b . ( i n . ) Basal area per acre ( s q . f t . ) Stand 5 - y r . morta l i ty Stand 5 - y r . morta l i ty Gross cumulative 10 956 1.3 9 9 15 956 2 . 7 40 40 20 956 4 . 0 90 90 25 956 5 . 1 142 142 30 947 9 5 . 8 176 1 177 35 889 58 6 . 2 196 8 204 40 733 156 6 . 9 199 22 2g0 45 64o 93 7 . 7 213 14 258 50 551 89 8 . 4 220 18 283 55 458 93 9 . 2 219 25 307 60 396 62 1 0 . 1 228 19 334 65 342 53 11 .2 240 16 364 70 298 44 1 2 . 2 250 19 392 75 276 22 1 3 . 1 266 13 422 80 236 40 1 4 . 3 269 26 450 85 213 22 1 5 . 4 284 16 480 90 204 9 1 6 . 2 303 9 508 95 160 44 1 7 . 9 284 41 531 100 147 13 1 9 . 0 292 14 553 139 lOOO. 900 SOOU 700h IOOU O I I 1 u O 2 0 4 0 6 0 SO IOO AGE IN YEARS FIG. 59: The e f f e c t of thinning_ on number of t r e e s per a c r e . A l l t r e e s l e s s than (D - s) removed at 10-year i n t e r v a l s . Spacing: 6.6 x 6.6 f t . Run I I - 9 . 550i-500 Grot* Yield Net Thinned Stand U Unthinned T Thinned 450 40G+-UJ a o < 5 35°i a ui UJ u. SOO UJ ce 3 o in ± 250 < UJ ce < < 200 in < CO 150 I C O IOC-AGE IN YEARS PIG. 60: The e f f e c t of t h i n n i n g on gross, net (stand + t h i n n i n g s ) and stand b a s a l area y i e l d . A l l t r e e s l e s s than (D - s) removed at 10-year i n t e r v a l s . Spacing: 6.6 x 6.6 f t . Run II-9. TABLE 4 ; The e f f e c t of t h i n n i n g on growth and y i e l d . A l l t r e e s l e s s than (D - s) removed at i n t e r v a l s of 10 years. Spacing: 6 . 6 x 6 . 6 f t . Run I I - 9 . Age (yr . ) Number of t r e e s Mean rl " K V i r\ V i [ -i vt 1 d/D B a s a l area per acre (sq. f t . ) * 4 s. S. t w Stand Removed 5-yr. Stand Thin- L Li. U 1 U Stand Removed 5-yr. Net Cumul-(before as mort- (before nings (before as t h i n - mort- Y i e l d a t i v e t h i n - t h i n - a l i t y t h i n - t h i n - nings a l i t y (Stand gross ning) nings ning) ning) + t h i n - y i e l d D d nings) (net y i e l d + mor-t a l i t y ) 10 956 1.3 9 •9 9 • 15 956 2 . 7 - 4o . - 40 40 20 956 4 . 0 90 90 90 25 956 5 . 1 142 142 142 30 947 9 5 . 8 176 1 176 177 35 889 58 6 . 2 196 8 196 204 40 733 111 156 6 . 9 5 . 0 0 . 7 2 199 15 22 199 230 45 591 31 8 . 0 211 7 226 264 50 529 84 62 8 . 6 6 . 7 0 . 7 8 222 21 14 237 289 55 418 27 9 . 7 222 8 258 318 60 360 76 58 1 0 . 7 8 . 4 0 . 7 9 230 30 20 296 376'6 65 284 1 2 . 2 235 301 381 70 244 44 40 1 3 . 4 11 .2 0 .84 241 30 22 307 409 75 191 9 1 5 . 0 236 8 332 442 80 182 31 9 1 6 . 1 1 3 . 5 0 .84 260 31 8 356 473 85 151 1 7 . 6 257 384 501 90 151 ! 3 1 8 . 4 1 6 . 1 0 . 8 8 280 19 406 524 95 133 4 1 9 . 3 273 7 4 l 8 543 I'.lOO 120 13 13 2 0 . 0 1 7 . 7 0 . 8 8 263 23 26 408 559 T o t a l 372 476 168 150 i n c r e a s e d by t h i n n i n g . I n c r e a s i n g the t h i n n i n g p e r i o d t o twenty years ( F i g . 6 l and 62, Table 5) causes l i t t l e r e d u c t i o n i n the b a s a l area of the stand but, compared w i t h r e s u l t s from the ten-year t h i n n i n g p e r i o d , the net y i e l d i s reduced. Severe Low T h i n n i n g In t h i s t h i n n i n g none of the suppressed, codominant or dominant t r e e s were removed and only those t r e e s i n the i n t e r m e d i a t e crown c l a s s between (D - s) and (D - 0.5s) were thi n n e d . Such a t h i n n i n g could be j u s t i f i e d , s i l v i c u l t u r a l l y , on the grounds t h a t the suppressed t r e e s l e f t are u n l i k e l y t o s e r i o u s l y hamper the growth of the dominant t r e e s expected to form the f i n a l crop and, economically, on the grounds t h a t the i n t e r m e d i a t e t r e e s removed are more v a l u a b l e than the sup-pressed t r e e s removed i n the t h i n n i n g p r e v i o u s l y d e s c r i b e d . T h i s t h i n n i n g was t e s t e d w i t h a ten-year t h i n n i n g p e r i o d ( F i g . 63 and 64; Table 6) and a twenty-year p e r i o d ( F i g . 65 and 66, Table 7). On a ten-year c y c l e , net and gross b a s a l area y i e l d s were the h i g h e s t of the t h i n n i n g s t e s t e d . Crown T h i n n i n g or S e l e c t i o n F e l l i n g T h i s t h i n n i n g removed a small number of codominant t r e e s between (D + 0.75s) and (D + s) u s i n g a ten-year t h i n -ning p e r i o d . Although the number of t r e e s removed ( F i g . 61, Table 8) was the l e a s t of a l l the t h i n n i n g s t e s t e d (at ages 60 and 90 years there were no t r e e s w i t h i n the p r e s c r i b e d l i m i t s ) , the r e d u c t i o n In the b a s a l area of the stand was 143 PIG. 6 l : The ef f e c t of'thinning, on number of trees per acre. A l l trees less than (D - s) removed at 20-year i n t e r v a l s . Spacing: 6 . 6 x 6 . 6 f t . Run 1 1 - 1 0 . AGE IN YEARS PIG. 62: The e f f e c t of t h i n n i n g on gross, net (stand + t h i n n i n g s ) and stand b a s a l area y i e l d . A l l t r e e s l e s s than (D - s) removed at 20-year i n t e r v a l s . Spacing: 6.6 x 6.6 f t . Run 11-10. TABLE 5 : The e f f e c t of t h i n n i n g on growth and y i e l d . A l l t r e e s l e s s than (D - s) removed at i n t e r v a l s of 20 y e a r s . Spacing: 6 . 6 x 6 . 6 f t . Run 1 1 - 1 0 . Age (yr . ) 10 15 20 25 30 35 4o 45 50 55 60 65 70 75 8o 85 90 95 100 i T o t a l Number of t r e e s per acre ' Stand Removed 5-yr. (before as mort-t h i n - t h i n - a l i t y ning) nings 956 956 956 9^7 889 733 591 529 440 374 298 262 240 218 173 164 156 138 i l l 76 40 27 254 9 58 156 31 62 89 67 36 22 22 4 9 9 18 592 Mean d.b.h.o.b.(in.) Stand (before t h i n -ning) D Thin-nings d 1.3 2 . 7 4 . 0 5 . 1 5 . 8 6 . 2 6 . 9 8 . 0 8 . 6 9 . 5 1 0 . 4 1 2 . 0 1 3 . 0 1 4 . 0 1 5 . 1 1 6 . 8 1 7 . 7 1 8 . 5 1 9 . 4 5 . 0 7 . 9 1 2 . 0 1 7 . 0 d/D r a t i o 0 . 7 2 0 . 7 6 0 . 8 0 0 . 8 8 B a s a l area per acre (sq. f t . ) Stand (before t h i n -ning) Removed 5-yr. as t h i n - mort-nings a l i t y 9 40 90 142 176 196 199 211 222 222 228 237 247 262 275. 269 284 293 285 15 26 332 42 114 l 8 22 7 14 24 22 18 13 17 4 10 11 26 198 Net Y i e l d (Stand + t h i n -nings) 9 40 90 142 176 196 199 226 237 237 243 278 288 304 316 3^1 357 366 357 Cumul-a t i v e gross y i e l d (net y i e l d + mor-t a l i t y ) ! 9 40 90 142 177 204 230 264 289 313 341 376 404 433 463 492 517 538 555 146 lOOOr-900r-BOO ui I 700| ce ui a a ce UJ CO 3 Z 600 SOO 400 300 200r-lOOl Unthinned Thinned "20 40 •it R5o AGE IN YEARS PIG. 63: The e f f e c t , of t h i n n i n g on number of t r e e s per a c r e . A l l t r e e s between (D - s) and (D - 0.5s) removed at 10-year i n t e r v a l s . Spacing: 6 .6 x 6.6 f t . Run 11-11. J. - J . , _> A 6 E , N YEARS , / . , , PIG. 64: The e f f e c t of t h i n n i n g on gross, net (stand + thinnings}_ and stand b a s a l area y i e l d . A l l t r e e s between (D - s) and (D - 0 . 5 s ) removed at 1 0-year i n t e r v a l s . Spacing: 6 . 6 x 6 . 6 f t . Run 11-11 . TABLE 6 : The e f f e c t of t h i n n i n g on growth and y i e l d . A l l t r e e s between (D - s) and (D - 0.5s) removed at i n t e r v a l s of 10 years. Spacing: 6 . 6 x 6 . 6 f t . Run 11-11 . Age Number of t r e e s Mean d/D B a s a l area per acre (sq. f t .) ( y r . ) per acre d.b.h. o.b.(in.) r a t i o Stand Removed 5-yr. Stand Thin- Stand Removed 5-yr. Net Cumul-(before as mort- (before nlngs (before as t h i n - mort- Y i e l d a t i v e t h i n - t h i n - a l i t y t h i n - t h i n - nings a l i t y (Stand gross ning) nlngs ning) ning) + t h i n - y i e l d -. D d nings) (net y i e l d + mor-t a l i t y ) 10 956 1 .3 9 9 9 15 956 2 . 7 40 40 40 20 956 4 . 0 90 90 90 25 956 5 . 1 142 142 142 30 947 9 5 . 8 176 1 176 177 35 889 58 6 . 2 196 8 196 204 40 733 120 156 6 . 9 6 . 0 0 . 8 7 199 24 22 199 230 45 573 • 4o 7 . 9 203 6 227 263 50 524 111 49 8 . 6 7 . 5 0 . 8 6 221 34 9 245 290 55 396 18 9 . 8 213 6 271 322 60 351 44 44 1 0 . 8 9 , 4 0 . 8 8 229 22 14 287 352 65 284 22 1 2 . 2 236 : 7 315 387 70 266 22 18 13 .1 1 1 . 4 0 . 8 7 255 16 10 335 417 75 240 4 1 4 . 1 268 3 363 448 80 209 31 31 1 5 . 4 1 3 . 6 0 . 8 9 274 32 21 369 475 85 169 9 1 6 . 9 267 4 394 504 90 156 18 13 1 8 . 0 1 6 . 4 0 . 9 1 280 26 12 406 528 95 138 1 9 . 0 275 428 550 100 125 18 13 2 0 . 0 18-. 5 0 . 9 2 274 33 19 427 568 T o t a l 364 484 186 141 149 l O O O r -900h 8 0 0 -700h Ul O 600J-ce UJ a ui 500h UJ ce cc 400r-UJ CD 2 3 Z Thinned 300 r 200r ICOh ± 20 40 60 80 P I G . 65: 100 AGE IN YEARS The e f f e c t of thinning, on number^ of trees per acre. A l l trees between (D - s) and (D - 0.5s) removed at 20-year i n t e r v a l s . Spacing: 6.6 x 6.6 f t . Run 11-12.. 550i-500 450 400p-tt O < tt UJ OL UJ UJ u. UJ ce < a in 350 300 z 250r-< UJ ce < ^ 200f-M < CD I50f-lOOt-SO Gross Yield Net Thinned Stand U Unthinned T Thinned TC" PIG. 6 6 : The e f f e c t of t h i n n i n g ^ r f f r o s s , net (stand + T o thinnings_}_ and stand basal area y i e l d . A l l t r e e s between (D - s) and ( D - 0 . 5 S ) removed at 2 0 - y e a r i n t e r v a l s . Spacing: 6 . 6 x 6 . 6 f t . Run. 1 1 - 1 2 . TABLE 7: The e f f e c t of t h i n n i n g on growth and y i e l d . A l l t r e e s between (D - s) and (D - 0.5s) removed at I n t e r v a l s of 20 years. Spacing: 6.6 x 6.6 f t . Run 11-12. Age (yr . ) Number of t r e e s d.b.h. Mean o.b.(in. ' d/D V l Q T 1 (~) B a s a l area per acre (sq. f t .) per acre Stand Removed 5-yr. Stand Thin- ' • l a U X U Stand Removed 5-yr. Net Cumul-(before as mort- (before nlngs (before as t h i n - mort- Y i e l d a t i v e t h i n - t h i n - a l i t y t h i n - t h i n - nings a l i t y (Stand gross ning) nings ning) D d ning) + t h i n -nings) y i e l d (net y i e l d + mor-t a l i t y ) 10 956 1.3 9 9 9 15 956 2.7 40 4o 40 20 956 4.0 90 90 90 25 956 5.1 142 142 142 30 947 9 5.8 176 1 176 177 35 889 58 6.2 196 24 8 196 204 40 733 120 156 6.9 6.0 0.87 199 22 199 230 45 573 - 4o 7.9 203 6 227 263 50 524 49 8.6 221 9 245 290 55 467 58 9.4 231 15 255 315 60 378 49 89 10.4 8.9 0.86 230 21 27 253 34l 65 311 18 11.7 237 6 282 376 70 289 22 12.5 255 13 300 406 75 267 22 13.4 270 15 315 436 8o 231 44 36 14.6 12.7 0.87 275 39 23 320 464 85 178 9 16.1 259 6 343 493 90 164 13 17.3 274 10 358 519 95 147 18 18.5 276 20 360 540 100 138 18 9 19.2 17.4 0.90 283 29 . 14 367 560 T o t a l 231 606 113 194 152 lOOOr-AGE IN YEARS PIG. 6 7 : The e f f e c t of t h i n n i n g on number o f _ t r e e s per ac r e . A l l t r e e s between (D + 0 . 7 5 s ) and (D + s) removed at 1 0-year i n t e r v a l s . Spacing: 6 . 6 x 6 . 6 f t . Run 1 1 - 1 3 . TABLE 8 : The e f f e c t of t h i n n i n g on growth and y i e l d . A l l t r e e s between (D + 0 . 7 5 s ) and (D + s) removed at i n t e r v a l s of 10 years. Spacing: 6 . 6 x 6 . 6 f t . Run 1 1 -13 . Age (y r . ) Number of t r e e s per acre Stand Removed 5-yr. (before as . mort-t h i n - t h i n - . a l i t y ning) nings Mean d.b.h.o.b.(in.) Stand Thin-(before nings t h i n -ning) D d d/D r a t i o B a sal area per acre (sq. f t . ) Stand (before t h i n -ning) Removed as t h i n -nings 5-yr. mort-a l i t y Net Y i e l d (Stand + t h i n -nings) Cumul-a t i v e gross y i e l d (net y i e l d + mor-t a l i t y ) 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 T o t a l 956 956 956 956 947 889 733 618 524 427 391 324 285 244 196 173 164 151 146 44 53 9 18 4_ 128 9 58 156 71 93 44 36 40 31 49 4 9 13 4_ 684 1.3 2 . 7 4 . 0 5 . 1 5 . 8 6 . 2 6 . 9 7 . 6 8 . 4 9 . 0 1 0 . 8 1 1 . 9 1 3 . 0 1 4 . 3 1 5 . 0 1 6 . 0 1 7 . 3 1 8 . 2 8.1 9 . 9 14.5 17.4 21.8 1.17 1.18 1.22 1.22 1 .20 40 90 142 176 196 199 202 209 198 215 219 231 236 230 225 242 258 277 16 28 10 30 12 "96" 9 9 40 40 90 90 142 142 1 176 177 8 196 204 22 199 230 11 218 260 20 225 285 11 243 314 10 260 341 21 264 367 16 275 395 12 291 423 32 284 448 3 309 476 9 326 502 9 342 527 3 361 549 188 154 the most marked ( P i g . 68) due to the h i g h m o r t a l i t y among the sma l l e r t r e e s i n the r e s i d u a l stand, as w e l l as the l o s s due t o . t h i n n i n g . Net y i e l d was a l s o reduced and, above age 75 years, the gross y i e l d of the thinned stand was a l i t t l e , l e s s than t h a t of the unthinned stand. The r e s u l t s of these t e s t s , although only c o v e r i n g a small p r o p o r t i o n of many d i f f e r e n t types of t h i n n i n g , i n d i -cate t h a t the stand model can be used f o r t e s t i n g the e f f e c t s of t h i n n i n g on the growth a n d . y i e l d of the stand. Any o b j e c t i v e method of t h i n n i n g , such as those used above, can be t e s t e d , p r o v i d e d the FORTRAN program f o r ' t h e model i s modified to s u i t the i n d i v i d u a l ' s requirements. Height Growth In a l l the t e s t s so f a r c a r r i e d out u s i n g the model, no attempt has been made to p r e d i c t h e i g h t growth. The reasons f o r t h i s omission have been s t a t e d e a r l i e r i n t h i s t h e s i s . I t i s g e n e r a l l y accepted t h a t , w i t h i n . a stand, t r e e h e i g h t i s c l o s e l y c o r r e l a t e d with d. b. h., the r e l a t i o n s h i p u s u a l l y being c u r v i l i n e a r . The r e l a t i o n s h i p w i l l vary among stands due to d i f f e r e n c e s in' s t o c k i n g , d e n s i t y of t r e e s , age and s i t e . To study t h i s r e l a t i o n s h i p , data f o r 869 Douglas f i r sample t r e e s were obtained from permanent sample p l o t s l o c a t e d on the U n i v e r s i t y Research F o r e s t , Haney, the U n i v e r s i t y Campus F o r e s t , Vancouver, and from the experimental p l o t s of the Research D i v i s i o n of the B. C. F o r e s t S e r v i c e at Cowichan Lake, Vancouver I s l a n d . The l a t t e r s e r i e s of p l o t s was l o c a t e d i n p l a n t a t i o n s of Douglas f i r which had 550r-500I— 450 400 ui a o < 350 3 ui ui 300f-ui ce < o Z 250I ce < 20C4-150-lOOf-5C+-Gross Yield Net Thinned Stand U Unthinned T Thinned AGE IN YEARS PIG. 68: The e f f e c t ' o f ' t h i n n i n g on gross, net (stand + thinnings_)_ and stand. basal_ area y i e l d . A l l t r e e s between (D + 0.75s) and (D + s) removed at 10-year i n t e r v a l s . Spacing: 6.6 x 6.6 f t . ' Run 11-13. 156 been thinned, l e a v i n g the r e s i d u a l stands at d e n s i t i e s which v a r i e d between "normal" and "open-grown". Using these data, a r e g r e s s i o n c a l c u l a t i o n was car-r i e d out of t r e e h e i g h t on d. b. h., number of t r e e s per acre, b a s a l area per acre, s i t e index and age. The r e s u l t s of t h i s r e g r e s s i o n c a l c u l a t i o n are summarised i n Table 9. A. s e r i e s of r e g r e s s i o n equations i s given, each equation having one l e s s independent v a r i a b l e than the p r e v i o u s equation. The v a r i a b l e e l i m i n a t e d each time was the one making the l e a s t a b s o l u t e 2 c o n t r i b u t i o n t o R . I t appears t h a t the r e g r e s s i o n equation H = -11.083 + 8.27095D + 0.160482B - 0.154019D2, where H i s the t o t a l h e i g h t of the t r e e i n f e e t , D i s the d. b. h. o. b. i n inches and B i s the b a s a l area of the stand i n square f e e t per acre, i s the most s u i t a b l e f o r p r a c t i c a l purposes. I n c l u s i o n of f u r t h e r v a r i a b l e s does not g r e a t l y decrease the 2 2 r e s i d u a l v a r i a n c e , s , whereas e l i m i n a t i n g D almost doubles. the r e s i d u a l v a r i a n c e . Examples of h e i g h t / d . b. h. curves, c a l c u l a t e d from the above equation, are shown i n P i g . 69 f o r stand b a s a l areas between 60 and 240 sq. f t . per a c r e . C o n c l u s i o n s The model has proved to be s a t i s f a c t o r y f o r the v a r i o u s t e s t s performed i n t h i s part, of the t h e s i s . I t i s not p o s s i b l e at present, due to the u n a v a i l a b i l i t y of f i e l d data, to c o n f i r m t h a t the a b s o l u t e v a l u e s f o r number of t r e e s , mean d. b. h. and basal' area are a c c u r a t e . However'', there are no s i g n i f i c a n t departures, except when c a r r y i n g out TABLE 9= Regression of t r e e height on d. b. h., number of t r e e s per a c r e , b a s a l area per acre, s i t e index and age. The v a r i a b l e making the l e a s t a b s o l u t e c o n t r i b u t i o n to R2 i s e l i m i n a t e d each time. Number of t r e e s = 869. Source of data: B. C. F o r e s t S e r v i c e , U n i v e r s i t y Research F o r e s t and Campus F o r e s t . Mean he i g h t = 53.7 f t . Standard d e v i a t i o n = + 34.74 f t . Regression of h e i ght on: Constant D.b.h. ( i n . ) No. of B.A./ac. t r e e s / a c . ( s q . f t . ) S i t e Index Age (yr . ) (D.b.h.) 2 (D.b.h. ) 3 L ° S e (d.b.h.) R 2 2 s Regres-s i o n . coef-f i c i e n t s -15.415 8.3478 -0.005859 0.141596 0.060795 0.210877 -0.195037 0.001067 -2.1966 0 .946 65.88 - 4.633 8.5384 -0.007905 0.179185 - 0.055343 -O.203969 0.001293 -2.6728 0 .945 66.85 - 3.073 8.3799 -0.008939 0.189941 - - -0.191386 0.001041 -2.5069 0 .945 66.92 -11.025 10.9741 0.157393 - - -0.273890 0.001818 -6.8723 0.939 74.26 -10.132 9.7477 0.158176 - - -0.192917 -4.8099 0 .939 74.27 -11.084 8.2710 0.160482 - - -0.154019 - 0 .938 74.94 - 6.099 11.0728 - - - -0.222466 - 0 .890 133.00 9.005 6.5691 - - - - - 0 .844 187.48 Mean S.D. r f o r Ht. 6.81 +4.86 0.919 727 120 +396 +65 -0.407 0.806 139 +26 0.107 35 17 0.608 69.96 954.0 +103.84 +2240.6 0.803 0.661 1.6651 +0.7475 0.893 158 159 t e s t s on some of the stands w i t h an i n i t i a l s p acing of 3-3 x 3-3 f t . , from t h a t which might be expected t o occur i n nature. Reasons f o r the l i m i t a t i o n s of the model at c l o s e spacings have been given i n Part I I of t h i s t h e s i s . Of p a r t i c u l a r i n t e r e s t , as i t has seldom been d e s c r i b e d i n n a t u r a l stands or p l a n t a t i o n s , i s the development of the s t r u c t u r e of the stand a f t e r v a r i o u s amounts and d i s -t r i b u t i o n s of m o r t a l i t y have been a p p l i e d t o the stand a f t e r p l a n t i n g . T h i s w i l l help the f o r e s t manager t o decide whether to r e p l a n t a young p l a n t a t i o n which has s u f f e r e d e a r l y mort-a l i t y . The s i t e i n d i c e s t e s t e d cover the middle of the range of the s i t e q u a l i t i e s encountered i n the c o a s t a l r e g i o n of B r i t i s h Columbia. To t e s t s i t e i n d i c e s o u t s i d e the range given, i t may be necessary t o modify the model to s t a r t at age f i v e years f o r b e t t e r s i t e s , so that the t r e e s have not a l r e a d y begun t o compete wi t h each other, or at age f i f t e e n years f o r poorer s i t e s , by which time the m a j o r i t y of the t r e e s should have reached b r e a s t h e i g h t . The t h i n n i n g t e s t s that have been d e s c r i b e d are examples of how the model might be used r a t h e r than an exhaus-t i v e survey of such t e s t s . The model can be mod i f i e d by r e w r i t i n g the FORTRAN program t o meet the requirements of each i n d i v i d u a l case. SUMMARY AND SUGGESTIONS FOR FUTURE DEVELOPMENT The object of th i s thes is was to develop a mathematical model to describe the growth of Douglas f i r stands under the conditions commonly encountered.in the coastal region of B r i t i s h Columbia. This has been done by descr ib ing the growth in d i a -meter of each tree in a matrix of 225 trees , assuming the trees were located at the in tersect ions of a square l a t t i c e . It was assumed that , at age ten years , these trees were open-grown. The f ive -year per iod ic growth i n d. b. h. was ca lculated for each tree using a regress ion of d. b. h. on d. b. h. at the beginning of the per iod , d . b. h. at age ten years and t o t a l age, obtained from a sample of eighteen trees on the Saanich Peninsula, Vancouver Is land. These growth data were i n general agreement with those from younger trees co l l ec ted at several other loca-t i o n s . The crown width of each tree was also ca lculated from the d. b. h . , using regressions of crown width on d. b. h. obtained for 426 trees sampled In the i n t e r i o r and i n the coastal regions of B r i t i s h Columbia and Washington. Trees were assumed to continue to grow at the open-growth rate u n t i l the crowns overlapped. The f ive -year per iod ic diameter growth of each tree was then reduced by an amount depending on the proport ion of the circumference of the crown occupied by the crowns of surrounding trees . This was repeated at f ive -year in t erva l s to age 100 years using an I . B. M. 7090 e l e c t r o n i c computer. Morta l i t y was obtained by assuming that 160 161 a t r e e was dead i f the f i v e - y e a r p e r i o d i c d. To. h. was reduced below a c e r t a i n percentage of the d. b. h. of the t r e e . T h i s minimum percentage growth expected f o r s u r v i v a l v a r i e d from f i v e per cent at age ten years t o 0 . 1 per cent at age 45 years or above. The model worked s a t i s f a c t o r i l y over the range of c o n d i t i o n s t e s t e d u n t i l the number of t r e e s was reduced to. below 2 5 . T h i s occurred only above age 40 years at the 3 . 3 x 3 . 3 f t . spacing so t h a t the model i s t h e r e f o r e of l i m i t e d use when c a r r y i n g out t e s t s at o l d e r ages at t h i s s p acing. The b a s i c model was developed u s i n g a matrix of diameters at breast h e i g h t obtained from a p l a n t a t i o n of Douglas f i r on the U n i v e r s i t y Research F o r e s t at Haney, B. C. These diameters were modified t o s u i t the requirements of spacing and age i n the model. These data have been used as a " c o n t r o l " i n t e s t -i n g the model under other c o n d i t i o n s of growth. The model does not attempt to d e s c r i b e h e i g h t growth. Instead, the r e l a t i o n s h i p between he i g h t and d. b. h. has been s t u d i e d over a wide range of s i t e q u a l i t y , d e n s i t i e s of stock-i n g and age, and a m u l t i p l e r e g r e s s i o n d e r i v e d . T h i s may be used t o determine the height of any t r e e , given i t s d. b. h. and the b a s a l area of the stand. Volume growth has not been estimated e i t h e r , although t h i s can be assumed t o be c o r r e l a t e d w i t h b a s a l area. A l t e r n a t i v e l y , knowing the hei g h t and d. b. h. of each t r e e , an estimate of t r e e volume can be obtained u s i n g the method of Newnham (1958) or of Smith and Breadon ( 1 9 6 4 ) . 162 The assumptions t h a t have been made i n the model must be b i o l o g i c a l l y j u s t i f i e d . The f i r s t of these i s t h a t competi-t i o n can be measured by the p r o p o r t i o n of the crown of each t r e e occupied by the crowns of surrounding t r e e s . In the model, t h i s has been m o d i f i e d t o allow a 40 per cent crown o v e r l a p before the amount of competition i s s u f f i c i e n t l y great to reduce diameter growth. Measurements taken i n seven-year-o l d Douglas f i r p l a n t a t i o n s on the U n i v e r s i t y Research F o r e s t support t h i s assumption. I t has been noted i n Part II. t h a t r o o t spread i s c l o s e l y r e l a t e d to crown spread. For growth, the t r e e r e q u i r e s l i g h t f o r p h o t o s y n t h e s i s and n u t r i e n t s and moisture from the s o i l . I f the crown of a t r e e i s over-lapped by the crowns of surrounding t r e e s , shading of the lower branches occurs, and p h o t o s y n t h e s i s and the p r o d u c t i o n of carbohydrates i s reduced w i t h the r e s u l t a n t r e d u c t i o n i n t r e e growth. I f the r o o t systems of t r e e s o v e r l a p , the com-p e t i t i o n f o r the a v a i l a b l e s o i l n u t r i e n t s and moisture i s in c r e a s e d u n t i l a p o i n t i s reached where the r o o t system i s not able t o o b t a i n the maximum amount of n u t r i e n t s and mois-tu r e t h a t i t i s capable of t r a n s p o r t i n g t o the a e r i a l p a r t of the t r e e . L o g i c a l l y , c o m p e t i t i o n must t h e r e f o r e be r e l a t e d to the p r o p o r t i o n of the circumference of the t r e e occupied by the crowns of surrounding t r e e s . The assumption t h a t the optimum r a t e of diameter growth i s reduced by the p r o p o r t i o n of the circumference of the crown ( a f t e r r e d u c t i o n t o allow f o r overlap) occupied by the crowns of competitors, i s d i f f i c u l t t o j u s t i f y 163 q u a n t i t a t i v e l y , due to l a c k of Information. The diameter growth of open-grown t r e e s i s known and i t i s known t h a t , as competi-t i o n s e t s In, diameter growth i s reduced to the p o i n t where the t r e e d i e s . The r e l a t i o n s h i p between the amount of competition and r e d u c t i o n i n diameter growth us e d • i n t h i s t h e s i s was chosen because i t was simple to use. I t has been assumed i n the model t h a t , i f a t r e e was r e l e a s e d from a l l competition, i t would resume the r a t e of growth of an open-grown t r e e . T h i s ignores any "shock" e f f e c t . Reasons f o r t h i s "shock" e f f e c t have been d i s c u s s e d i n Part I. Shock i s most l i k e l y to occur when a stand, which has been allowed to become very dense w i t h p o o r l y developed crowns, i s suddenly opened up. Nowhere are these c o n d i t i o n s found i n the model so that i t i s safe to ignore "shock". I t i s probable, however, t h a t there Is a t i m e - l a g before the t r e e resumes f u l l y the optimum r a t e of growth. T h i s t i m e - l a g i s not taken i n t o account by the model. The e r r o r s i n v o l v e d are not thought to be s e r i o u s , although Reukema (1964) has shown t h a t s h o r t -term growth of crown was not r e l a t e d to r e l e a s e . The model used here should apply g e n e r a l l y . , Once a s a t i s f a c t o r y model had been obtained i t was next p o s s i b l e to t e s t the e f f e c t s of d i f f e r e n t d i s t r i b u t i o n s and amounts of m o r t a l i t y f o l l o w i n g p l a n t i n g (Table 10), d i f f e r e n t s i t e q u a l i t i e s (Table 11), and d i f f e r e n t t h i n n i n g regimes (Table 12) on the growth of the stand. The e f f e c t s of d i f f e r e n t i n i t i a l spacings (Table 13) had a l r e a d y been t e s t e d w h i l e d e v e l o p i n g the model. In the summary t a b l e s the 6.6 x TABLE 10: The e f f e c t o f amount and d i s t r i b u t i o n o f m o r t a l i t y f o l l o w i n g p l a n t i n g on s t a n d g r o w t h . S i t e I n d e x : 140. Mean d.b.h.o.b. a t age 10 y e a r s : 1 . 2 6 i n . S p a c i n g : 6.6x6.6 f t . Run D i s t r i - Amount Number Age a t Number B a s a l Age 100 y e a r s b u t i o n o f o f o f t r e e s w h i c h o f t r e e s a r e a p e r a c r e mean p e r a c r e p e r m o r t a l i t y m o r t a l -f o l l o w i n g i t y a t age d.b.h. when a c r e p l a n t i n g •(*) 10 y e a r s o.b. i s t h e mean when 12 i n . d.b.h." mean Number B a s a l o.b. i s d.b.h. o f Mean a r e a 12 i n . o.b. i s t r e e s d.b.h. p e r 12 i n . p e r o.b. a c r e ( s q . f t . ) a c r e ( i n . ) ( s q . f t . ) I I - 1 N e g l i g i b l e - 956 69 302 248 147 1 9 . 0 292 I I - 2 B i n o m i a l 10 907 70 319 257 156 18.6 302 I I - 3 B i n o m i a l 30 711 67 325 2 6 l 151 1 8 . 8 300 I I - 4 B i n o m i a l 50 516 66 317 250 138 1 8 . 5 263 I I - 5 U n i f o r m ( r e c t - 50 498 63 298 228 138 1 9 . 3 286 a n g u l a r ) I I . 6 2 R a n d . I n f . 14 858 68 305 247 .142 1 9 . 2 296 C e n t r e s TABLE 11: The e f f e c t of s i t e q u a l i t y on stand growth. Amount of m o r t a l i t y f o l l o w i n g p l a n t i n g : n e g l i g i b l e . Spacing: 6.6 x 6.6 f t . Run S i t e Index Age 10 years Age at Number of B a s a l which t r e e s per area per mean acre when acre when d.b.h. mean mean d.b.h. o.b. i s d.b.h.o.b. o.b. i s 12 12 i n . i s 12 i n . i n . ( s q . f t . ) Age 100 years Number Mean of t r e e s d.b.h.o.b. per acre ( i n . ) Number Mean B a s a l of t r e e s d.b.h. area per acre o.b. per ( i n . ) a c r e ( s q . f t . ) II-7 120 I I - l 140 II-8 160 956 0.80 956 1.26 956 1.92 84 280 245 69 302 2 4 8 55 338 276 213 15.2 274 1 4 7 19.0 292 107 23.6 331 TABLE 12: The e f f e c t of thinning on stand growth. Site index: l4o. Mean d. b. h. o. b. at age 10 years: 1.26 i n . Amount of mortality following planting: n e g l i g i b l e . Spacing 6.6 x 6.6 f t . Trees removed between: lower upper l i m i t l i m i t 0 0 D - s T>s Thin- Num-ning ber Per- of i o ' d thin-(yr.) nings be-fore age 100 years Age at which mean d.b.h. Num-ber of trees per 11-131 D+ 10 . 7 5 s D+s o.b.is acre 12 i n . when mean d.b.h. o.b.is 12 i n . Basal area per acre when mean d.b.h. o.b.is 12 i n . (sq. f t . ) Thinnings Num-ber of trees per acre Basal area per acre (sq. f t . ) T otal j basal — -area I Num-Age 100 years y i e l d (sq. f t . ) ber of trees per acre Mean Basal Thinnings d.b.h. area o.b. per (in.) acre (sq. f t . ) Num-ber of trees per acre Basal area per acre (sq. f t . ) T otal basal area yield! (sq. i f t . ) l - - 69 302 248 i | - 1 248 147 19.0 292 - - 292 10 6 64 284 228 i 271 66 ; 294 : 120 20.0 263 359 146 . 409 20 3 65 298 237 | 187 4l : 278 ; 138 19.4 285 227 73 357 10 6 64 289 230 i 275 79 | 309 125 20.0 274 346 153 427 20 3 67 303 244 169 45 •299 138 19.2 283 213 84 366 10 4 71 270 224 \ 106 54 : 278 . 146 18.2 277 124 84 361 TABLE 13: The effect of i n i t i a l spacing (planting distance) on stand growth. S i te index: 140. Mean d . b . h . o . b . at age 10 years: 1.26 i n . D i s t r i b u t i o n of morta l i ty fol lowing p lant ing: n e g l i g i b l e . Run II-1 I n i t i a l Number of spacing trees per ( f t . ) acre at age 10 y r . 3.3 6.6 9.9 13.2 16.5 19.8 Age at which mean d . h . b . o . b . i s 12 i n . 3822 956 425 239 153 106 63 69 64 52 51 51 Number of trees per acre when mean d . b . h . o . b . i s 12 i n . 510 302 319 239 153 106 Basal area per acre when mean d . h . b . o . b . i s . 12 i n . (sq. f t . ) 406 248 258 198 125 87 Age 100 years Number of trees per acre 213 147 134 146 149 106 d Mean b . h . o . b . ( i n . ) 1999 19.0 17.9 18.4 19.6 21.6 Basal area per acre (sq. f t . ) 466 292 240 272 318 273 1 6 8 6 , 6 f t . spacing of the b a s i c matrix has been taken as a " c o n t r o l " , w i t h which the r e s u l t s of the v a r i o u s t e s t s may be compared. The age at which the stand reaches a.mean d. b. h. of twelve inches i s an i n d i c a t i o n of the minimum r o t a t i o n (Smith et a l , , 1 9 6 2 ) . In terms of a b s o l u t e v a l u e s i t i s not p o s s i b l e t o c o n f i r m that these r e s u l t s are a c c u r a t e . However, the b a s i c model g i v e s s a t i s f a c t o r y f i t s t o the y i e l d t a b l e data of Barnes (U. B. C. F o r e s t Club, 1 9 5 9 ) and McArdle et a l . . ( 1 9 4 9 ) . I t should be r e a l i s e d a l s o t h a t the r e s u l t s of these t e s t s are based on one "run" i n each case-. A l t h o u g h . i t i s p o s s i b l e to c o n t r o l s i t e d i f f e r e n c e s more r e a d i l y i n the stand model by u s i n g i d e n t i c a l diameter d i s t r i b u t i o n s , t r e e d i s t r i b u t i o n s or s i t e q u a l i t i e s as may be r e q u i r e d , t h i s i s e q u i v a l e n t to a f i e l d experiment i n which there i s only one r e p l i c a t e of each treatment. The "runs" have been r e p l i c a t e d i n "degree" but not i n " k i n d " due t o the amount of computer time r e q u i r e d . I t i s probable t h a t , i f not completely a c c u r a t e i n a b s o l u t e v a l u e s , the r e s u l t s g i v e a c c u r a t e , r e l a t i v e e stimates of growth under the range of c o n d i t i o n s t e s t e d . To c o n f i r m the accuracy of the r e s u l t s , f u r t h e r f i e l d data, l a r g e l y not a v a i l a b l e at present, would have to be secured. S e v e r a l r e c e n t l y e s t a b l i s h e d spacing and t h i n n i n g . t r i a l s may p r o v i d e more s u i t a b l e data i n the long run. T h i s t h e s i s has shown how a stand model can be developed f o r Douglas f i r and has shown how the model may be used.to study the growth of stands under v a r i o u s c o n d i t i o n s . 169 So f a r the model does not give estimates of volume growth although these may be obtained - from i n d i v i d u a l diameter and h e i g h t data. T h i s could be b u i l t i n t o the model by modifying the FORTRAN program. The model a l s o assumes t h a t , a f t e r the i n i t i a l m o r t a l i t y f o l l o w i n g p l a n t i n g , a l l m o r t a l i t y t h a t occurs i n the stand i s due to s u p p r e s s i o n . In n a t u r a l stands, besides suppression m o r t a l i t y , , there i s u s u a l l y a small amount of random m o r t a l i t y due to i n s e c t s , d i s e a s e , or extremes of c l i m a t e . In the stand model, such m o r t a l i t y could be obtained by a.method s i m i l a r to t h a t d e s c r i b e d f o r stand model IIA (Newnham, 1 9 6 4 ) . At the end of each f i v e - y e a r p e r i o d there i s a t h r e e - d i g i t random number a s s o c i a t e d with.each t r e e . I f t h i s random number i s l e s s than the random m o r t a l i t y t h a t has been p r e s c r i b e d , the t r e e " d i e s " . I f i t i s d e s i r e d to p r e s c r i b e a "clumped" m o r t a l i t y at a p a r t i c u l a r age, t h i s can be accom-p l i s h e d by s e t t i n g the random numbers of the t r e e s to be " k i l l e d " equal to zero at t h a t age. The model can be used t o t e s t any p r e s c r i b e d t h i n n i n g by modifying the FORTRAN program t o meet the requirements of each case. In c o n j u n c t i o n w i t h the computer program of Kozak and Munro (1963)> i t can be used to f i n d the t h e o r e t i c a l d i s t r i b u t i o n g i v i n g . t h e best f i t to the a c t u a l d i s t r i b u t i o n of the t r e e s i n the stand a t the end. of any five-year, p e r i o d . From t h i s i t would be p o s s i b l e to f i n d the best s i z e of quadrat to use as a b a s i s of sampling at the v a r i o u s i n i t i a l spacings and stand ages. For these t e s t s , a . l a r g e r b a s i c 170 diameter matrix (at l e a s t 30 x 30 t r e e s ) could be r e q u i r e d t o g i v e s u f f i c i e n t quadrats f o r s a t i s f a c t o r y d i s t r i b u t i o n f i t t i n g . A b a s i c square spacing has been assumed In a l l the t e s t s performed w i t h the model. The p a t t e r n of the spacing can be v a r i e d t o a c e r t a i n extent, as has been done i n t e s t i n g the d i f f e r e n t d i s t r i b u t i o n s of m o r t a l i t y f o l l o w i n g . p l a n t i n g , by o m i t t i n g t r e e s from some of the l o c a t i o n s on the square l a t t i c e . I t i s a l s o most' convenient to use a square p a t t e r n when programming the computer. Although the model can be modi f i e d t o use a b a s i c t r i a n g u l a r , or p o l y g o n a l , p a t t e r n the r e s u l t i n g program would be more complicated than that f o r square spacing. I t i s not known whether i t w i l l be p o s s i b l e t o adapt the model r e a d i l y f o r use w i t h other t r e e s p e c i e s . In.theory i t should be p o s s i b l e , p r o v i d i n g . t h e necessary crown w i d t h / d. b. h. and diameter growth r e g r e s s i o n s are known f o r the s p e c i e s . Some d i f f i c u l t y may be encountered i n o b t a i n i n g s a t i s f a c t o r y crown width/d. b. h. r e g r e s s i o n s f o r s p e c i e s , such as some of the spruces, which have a columnar crown h a b i t . In i t s f i n a l form, i t should be p o s s i b l e t o use the model t o estimate, g i v e n . s u f f i c i e n t b a s i c Information, the volume (even, perhaps, by grade) and value s of sawn products, pulpwood, plywood, and waste f o r any Douglas f i r stand, grown under v a r i o u s c o n d i t i o n s . A l l t h a t i s r e q u i r e d i s a wealth of i n f o r m a t i o n on u t i l i z a t i o n , a l a r g e computer, the p a t i e n c e to w r i t e the necessary program, and time t o t e s t and r e f i n e the b i o l o g i c a l assumptions i n v o l v e d . REFERENCES ADAMS, W. R. 1936. Changes r e s u l t i n g from t h i n n i n g i n young pine p l a n t a t i o n s . J . For., _3_4, 1 5 4 - 9 . ANDERSON, R. T. 1937 . The a p p l i c a t i o n of F o u r i e r ' s s e r i e s i n f o r e s t mensuration. J . For., 3_5> 2 9 3 - 9 -ANON. 1955 . P o r i a w e i r i i r o o t r o t ' o f Douglas f i r . Rep. P a c i f i c Nthwest. For. Range Exp. Sta., 1954, 28 pp. ANSCOMBE, F. J.. 1949. The s t a t i s t i c a l a n a l y s i s of i n s e c t counts based on the negative b i n o m i a l d i s t r i b u t i o n . B i o m e t r i c s , 5., 1 6 5 - 7 5 . ANUCIN, N. P. i 9 6 0 . Determining the c u r r e n t increment of stands by the l a t e r a l (cambial) s u r f a c e area of the t r e e s . Voprosy L e s o v e d e n i j a i Lesovodstva, Moscow, 3 4 6 - 6 1 . (F. A. 2 2 , 2 2 5 8 ) . BAKER, F. S. 1 9 2 3 . Notes on the composition of even aged stands. J . For., 21, 7 1 2 - 7 . BAKER, F. S. 1 9 5 0 . P r i n c i p l e s of s i l v i c u l t u r e . McGraw-Hill Book Co., Inc., N. York, 4 l 4 pp. BARNES, G. H. 1956. Intermediate y i e l d s of Douglas f i r as i n t e r p r e t e d from B r i t i s h y i e l d t a b l e s . J . For., 5 4 ( 3 ) , 1 7 7 - 9 . BRIEGLEB, P. A. 1 9 5 2 . An approach to d e n s i t y measurement i n Douglas f i r . J . For., 5 0 ( 7 ) , 5 2 9 - 3 6 . BRIGHT, G. A. 1 9 l 4 . The r e l a t i o n of crown space to the volume of present and f u t u r e stands of western y e l l o w p i n e . F o r e s t r y Q u a r t e r l y , 12 , 3 3 0 - 4 0 . BUCKMAN, R. E. 1962. Growth and y i e l d of red pine i n Minnesota. Tech. B u l l . U. S. 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S i t e q u a l i t y deter-mination i n young Douglas f i r . For, Chron., _35(l), 22-9. THOM, H. C. S. 1954. Frequency of maximum wind speeds. Amer. Soc. C i v i l . E ngin. P r o c , .80(539), 11 PP. THOMAS, G. P., and CRAIG, H. M. 1958. Winter i n j u r y and sub-sequent m o r t a l i t y t o Douglas f i r . Bi-m. Progr. Rep. Div. For. B i o l . Dep. A g r i c , Can.,.14(4), 3. TINNEY, W. A., and MALMBERG, D. B, 1948. Tree management and marking r u l e s . Second growth D o u g l a s - f i r . Univ. Wash. C o l l . For., 33 PP. TUCKER, H. G. 1962. An i n t r o d u c t i o n t o p r o b a b i l i t y and mathematical s t a t i s t i s s . Academic Press, N. York, 228 pp. U. B. C. FACULTY OP FORESTRY. 1959- The f i r s t decade of management and r e s e a r c h - U. B. C. F o r e s t . Univ. B. C. P a c For., 82 pp. U. B. C. FOREST. CLUB. 1959. Forestry, handbook of B r i t i s h Columbia. Univ. B. C , Pac. For., F o r e s t Club, 800 pp. Van SLYKE, A. 1964a. Study s p a c i n g and t h i n n i n g w i t h N e l d e r 1 s . new systematic d e s i g n s . Univ. B. C. Fac. For., 11 pp. Mimeo. 1 Se-van SLYKE, A. 1964b. A n a l y s i s of Nelder systematic spacing designs. Univ. B.. C. Pac. For., 12 pp. Mlmeo. VEZINA, P. E. 1962. O b j e c t i v e measures of t h i n n i n g grades and.methods. For. Res. Br. Can. Dep. For., P r o j e c t Q-85, .17 pp. Mimeo. VEZINA, P. E. 1963. Some aspects of the development of n a t u r a l l y o c c u r r i n g , densely grown even-aged balsam f i r stands In Quebec. For. Res. Br. Can. Dep. For., P r o j e c t Que. 63-3, 15 PP. Mimeo. WALTERS, J . 1954. A system of i n d i r e c t c o n t r o l of Douglas f i r beetle,. Dendroctonus pseudotsugae Hopk. Univ. B. C. Fac. For., unpublished.M. P. t h e s i s , l l 6 pp. WARE, L. M., and STAHELIN, R. 1948. Growth of southern pine p l a n t a t i o n s of v a r i o u s s p a c i n g s . J . For., _46(4), 267-74. .WARRACK, G. 1952. 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 . F or. Chron., 28(2), 46-56. WARRACK, G. C. 1959a. Crown dimension, i n i t i a l diameter and diameter growth i n a young stand of Douglas f i r . F o r . Chron., 35(2), 150-3. WARRACK, G. C. 1959b. F o r e c a s t of y i e l d i n r e l a t i o n t o t h i n n i n g regimes i n Douglas f i r . Tech. Publ. B. C. For. Serv., T. 51, 56 pp. WEIR, J . R., and HUBERT, E. H. 1919. The i n f l u e n c e of t h i n -n i n g on western hemlock and grand f i r i n f e c t e d w i t h Echinodontium t i n c t o r i u m . J . For.,._17, 21-35. WILEY, J . J . J r . , 1959. C o n t r o l techniques f o r -managed even-aged stands. J . For., 57(5), 343-7. WORTHINGTON, N. P. 1961. Some ob s e r v a t i o n s on y i e l d and e a r l y t h i n n i n g i n a Douglas f i r p l a n t a t i o n . J . For., 59(5), 331-4. WRIGHT, K. H., and LAUTHERBACH, P. G. 1958. A 10-year study of m o r t a l i t y i n a D o u g l a s - f i r sawtimber stand i n Coos and Douglas Counties, Oregon. Res. Pap. P a c i f . Nthwest. For. Range Exp. Sta., 2J_, 29 pp. YERKES, V. P. i960. E f f e c t of t h i n n i n g on form of young growth D o u g l a s - f i r t r e e s . Res. Note P a c i f . Nthwest. For. Range Exp. Sta., 194, 5 PP. 181 APPENDIX I THE DISTRIBUTIONS ENCOUNTERED IN FOREST RESEARCH I n t r o d u c t i o n Most r e s e a r c h e r s i n f o r e s t r y have at some time encountered the more common d i s t r i b u t i o n s d e s c r i b e d i n f o r e s t r y l i t e r a t u r e (Smith and Ker, 1957). I t i s the purpose of t h i s s e c t i o n t o present some of these d i s t r i b u t i o n s i n a u n i f o r m manner based on.the p r i n c i p l e s of mathematical s t a t i s t i c s . I t i s f i r s t necessary t o give some d e f i n i t i o n s . D e f i n i t i o n 1: I f X i s a random v a r i a b l e , then the d i s t r i -b u t i o n f u n c t i o n of X, P v, i s a f u n c t i o n d e f i n e d by f o r every r e a l number x (Tucker, 1962). The term on the r i g h t -hand s i d e of the above equation means "the p r o b a b i l i t y of the event t h a t the random v a r i a b l e , X, w i l l take on a value l e s s than, or equal t o , x, a r e a l number". Because i t i s a prob-a b i l i t y , E^(x) must take on a value between zero and one. D e f i n i t i o n 2: The d i s c r e t e d e n s i t y f u n c t i o n , f^.(x), i s d e f i n e d by r P X ^ x i f x = x n n i f x ^ x f o r a l l n. 182 It should be noted that in d i scre te d i s t r i b u t i o n s the values of x can only be integers and also that the density f u n c t i o n f x ( x ) , takes the value zero outside the range of the values of x on which the d i s t r i b u t i o n i s def ined. D e f i n i t i o n 3- A random v a r i a b l e , X, i s said to have.an absolute ly continuous d i s t r i b u t i o n i f there ex i s t s a funct ion, f^(x) , such that r X F x ( x ) = f Y ( t ) d t for every r e a l number, x. The funct ion f x ( x ) i s the density funct ion of the random var iab le X (Tucker, 1 9 6 2 ) . D e f i n i t i o n 4: I f X ^ , . . . , X are n independent observations on a random v a r i a b l e , ' X , then the mean, X n = (X.^  + . . . + X )/n 2 1 ^ 2 ^ 2 and the var iance , s = ]T X. - ( £ X.) / n ) . n n " 1 i= l 1 i= l 1 For information about the propert ies of d i s t r i b u t i o n funct ions , the various methods of obtaining estimates of the parameters descr ib ing a d i s t r i b u t i o n from samples, expecta-t ions and consistent and unbiased est imators, reference should be made to any standard text on mathematical s t a t i s t i c s (e. g. Kendall and Stuart , 1 9 5 8 , 1 9 6 1 ; Tucker, 1 9 6 2 ) . D i s t r i b u t i o n s encountered i n f o r e s t r y research A. D i s c r e t e d i s t r i b u t i o n s 1. The binomial d i s t r i b u t i o n This d i s t r i b u t i o n occurs where the experiment con-s i s t s of a sequence of n ( B e r n o u l l i ) t r i a l s which s a t i s f y the f o l l o w i n g requirements: ( i ) the outcome of each t r i a l can only be one of two p o s s i b l e incompatible events, "success" (S) or " f a i l u r e " ( p ) , ( i i ) the outcome of e a c h . t r i a l i s independent of the other t r i a l s , and ( i i i ) the p r o b a b i l i t y , p, of S o c c u r r i n g does not vary from t r i a l t o t r i a l . I f these c o n d i t i o n s are met then the binomial d e n s i t y f u n c t i o n i s f x ( x ) = P[X = x] = Q p X ( l - p ) n - X = (n - x j i x l P X d " P)" " X , x = 0,1,2,...,n where 0 < p < l and x = number of "successes" i n n t r i a l s . P r o p e r t i e s : = np; s n = n p ( l - p ) . Estimator: p = X /n. rr Tables of the binomial d i s t r i b u t i o n f u n c t i o n are given i n Burington and May (1953). 2. The Poisson d i s t r i b u t i o n The binomial d i s t r i b u t i o n approaches the Poisson d i s t r i b u t i o n as a l i m i t when n — > o o and p—=?> 0 i n such a way tha t np = m Is a constant (Burington and May, 1953)• The d e n s i t y f u n c t i o n of the Poisson d i s t r i b u t i o n i s f x ( x ) = P [ X = xj = m ~, , x = 0,1,2, .. m>0 Because of the above l i m i t i n g p r o p e r t y , the Poisson d i s t r i -b u t i o n may be used t o approximate the bi n o m i a l d i s t r i b u t i o n and v i c e v e r s a . _ 2 P r o p e r t i e s : X = m; s = m * n n E s t i m a t o r : m = X n Tables of the Poisson d i s t r i b u t i o n are given i n Burington and May (1953). 3. The negative b i n o m i a l d i s t r i b u t i o n The d e n s i t y f u n c t i o n of the negative b i n o m i a l d i s t r i -b u t i o n i s f x ( x ) = P[x = x] = (1 + ^ ) _ k i f x = 0 w k + x - 1\ m -k m \ X / . . . , where m>0, k;>0. The Poisson d i s t r i b u t i o n i s obtained as a l i m i t as k — ^ C x a . ~ . . TT 2 m^ P r o p e r t i e s : . X^ = m; = m + . E s t i m a t o r s : A = X ; k = p n _ (Anscombe, 19^9) 185 Tables of the negative b i n o m i a l d i s t r i b u t i o n have been pub-l i s h e d by Grimm ( 1 9 6 2 ) . The w r i t e r has developed a FORTRAN program f o r the I, B. M. 1 6 2 0 e l e c t r o n i c computer which c a l c u l a t e s t a b l e s of the d i s t r i b u t i o n over the range of value s of m and k normally encountered i n p r a c t i c e . In d e c i d i n g which of the above three d i s t r i b u t i o n s should be used t o f i t e m p i r i c a l data, the r e l a t i o n s h i p between — 2 — 2 X and s should be s t u d i e d . I f X > s , the b i n o m i a l d i s t r i -n n n n' 2 b u t i o n i s i n d i c a t e d ; i f Xfi = s^, the Poisson d i s t r i b u t i o n i s _ 2 i n d i c a t e d ; and i f X n < s n , the negative b i n o m i a l d i s t r i b u t i o n i s i n d i c a t e d . The Poisson d i s t r i b u t i o n occurs when the i n d i v i d u a l s i n the p o p u l a t i o n are randomly d i s t r i b u t e d (see P i e l o u , 1959, f o r a d i s c u s s i o n on randomness). I f the v a r i a n c e i s l e s s than the mean, the i n d i v i d u a l s are more u n i f o r m l y d i s , t r i b u t e d . The negative b i n o m i a l d i s t r i b u t i o n i n d i c a t e s that the i n d i v i d u a l s occur i n clumps or aggregates. The uniform ( r e c t a n g u l a r ) d i s t r i b u t i o n i n d i c a t e s extreme dumpiness. 4. The un i f o r m ( r e c t a n g u l a r ) d i s t r i b u t i o n The d e n s i t y f u n c t i o n of t h i s d i s t r i b u t i o n i s 1/N i f x = 0 , 1 , 2 N 0 otherwise where N > 0 . The u n i f o r m d i s t r i b u t i o n g i v e s an equal p r o b a b i l i t y of occurrence t o each value of x w i t h i n the range. P r o p e r t i e s : = N/2 E s t i m a t o r : N = max j x j 1<1£N B. A b s o l u t e l y continuous d i s t r i b u t i o n s 1. The gamma d i s t r i b u t i o n The gamma d i s t r i b u t i o n has a d e n s i t y f u n c t i o n f x ( x ) : „< + i * e "'r i f x >0 r ( - < + , l ) ° < - + 1 0 otherwise .oo where <*>>.-1,^ >0 and r ( ° < + l ) = y°Vydy =«< P(<*) . I f n i s an i n t e g e r (n>0) then P(n) = (n - 1) I In p a r t i c u l a r , P ( l ) = 1 and Vii) = Jn . P r o p e r t i e s : X n = + l ) ; s 2 = p 2 ( o < + l ) E s t i m a t o r s : By the method of moments — 2 2 * = 1; ^ s n A n These are not n e c e s s a r i l y the most e f f i c i e n t e s t i m a t o r s ( K e n d a l l and S t u a r t , . 1 9 6 l ) . Thorn (19^9) gave approximate s o l u t i o n s t o the maximum l i k e l i h o o d e s t i m a t o r s of o( and |S>. These are A 1 V1 + I ( l o s X n - 7J i l 0 « x i > « - _ ' n — • - 1 4(106 X - i Z log X,) ' o<+ 1 1 8 7 Tables of the gamma f u n c t i o n are given In Burington and May (1-953) • 2. The normal d i s t r i b u t i o n The normal d i s t r i b u t i o n has d e n s i t y (x -/M 2 T5 f{x)=-rL~e 2 c r , - o ° < x « * o where -O O < / J L < O O and cf2>0. T h i s d i s t r i b u t i o n i s o f t e n w r i t t e n 2 2 as N(^, CT ), and cT being the parameters d e s c r i b i n g the d i s t r i b u t i o n . Tables of the N(0,.l) d i s t r i b u t i o n f u n c t i o n are given i n most s t a t i s t i c a l t e x t s or i n F i s h e r and Yates (1957). P r o p e r t i e s : I f X i s a random v a r i a b l e having a N(0, 1) d i s t r i b u t i o n , then the d i s t r i b u t i o n of X i s gamma with &(= and = 2 (Tucker, 1962) . I f X^,...,Xn i s a sequence of n independent v a r i a b l e s each w i t h a N(0, l ) d i s t r i b u t i o n , then the d i s t r i -2 2 b u t i o n of X^ + ... + X^ i s chi-square w i t h n degrees of freedom. A _ ^ 2 E s t i m a t o r s : }^ = X ; cr = s / n n The f o l l o w i n g theorem i s sometimes found u s e f u l when working w i t h d i s t r i b u t i o n problems. The c e n t r a l l i m i t theorem: Let X-^ , . . .•,X be a sequence of independent, i d e n t i c a l l y d i s t r i b u t e d , random v a r i a b l e s 2 w i t h f i n i t e common e x p e c t a t i o n , Jl, and v a r i a n c e , d . Let Y n = ("X1 + ... + ;X n)/cT/n" 188 Then cm e n as n—><*> u n i f o r m l y i n x (TUcker, 1962) . Because of the l i m i t i n g nature of t h i s theorem, when n i s s u f f i c i e n t l y l a r g e , the normal d i s t r i b u t i o n may be used to approximate any other d i s t r i b u t i o n . Pearl-Reed growth curves Pearl-Reed growth.curves, o r i g i n a l l y developed to d e s c r i b e the growth of the p o p u l a t i o n of the United States ( P e a r l and Reed,. 1920), have been used i n f o r e s t r y t o . d e s c r i b e diameter d i s t r i b u t i o n s ('Osborne and Schumacher,. 1935) . The Pearl-Reed equation i s of the form where a, b, and c are constants having p o s i t i v e v a l u e s . In the o r i g i n a l paper ( P e a r l and Reed, 1920), y was the popu-l a t i o n s i z e and x the time. When used to d e s c r i b e diameter d i s t r i b u t i o n s , y i s the cumulative number of t r e e s and x i s the diameter. T h i s equation may be w r i t t e n i n the form T h i s equation has the p o s s i b l e disadvantage t h a t i t produces a symmetrical frequency d i s t r i b u t i o n . Osborne and Schumacher (1935) used a : m o d i f i e d Pearl-Reed equation of the be ax y = ax 1 + ce 188 form y = c + fTx) where,c i s the lower asymptote, . k + c i s the upper asymptote and. m i s an a r b i t r a r y constant. The term, f ( x ) , i s of the form 2 n f ( x ) = b^x + b 2 x + . . . + b f i x T h i s gave asymmetrical d i s t r i b u t i o n s which d e s c r i b e d the d i s t r i b u t i o n of diameters of even-aged stands of red gum (Liquidambar s t y r a c i f l u a L.) w i t h remarkable accuracy. 19P APPENDIX I I A DETAILED DESCRIPTION OP THE FORTRAN PROGRAM FOR STAND MODEL I I 1. A l i s t of the more important v a r i a b l e s i n the program wi t h t h e i r meanings. A ( a l s o MA) ACOMP. ASTART A ST OP A l A1C, A2C BAST B1,B2,B3,B4 B1C, B2C D DAP5 DINC DIO FLA ST PN Age i n y e a r s . Age at the beginning of the p e r i o d i n which m o r t a l i t y f i r s t occurs. Age at the s t a r t of the program. Age at which the program stops ( u s u a l l y 100 y e a r s ; ASTOP ^ 1 0 0 ) . Constant term i n the r a d i a l growth r e g r e s s i o n . Constant terms i n the two crown width r e g r e s s i o n s . B a s a l area per a c r e . R e gression c o e f f i c i e n t s i n the r a d i a l growth r e g r e s s i o n . R e gression c o e f f i c i e n t s In•the two crown* width r e g r e s s i o n s . D. B. h. o. b. at the beginning of each f i v e - y e a r p e r i o d . D. B. h. o. b. at the end of each f i v e -year p e r i o d . Minimum percentage f i v e - y e a r diameter growth f o r s u r v i v a l . D. b. h, o. b. at age 10 y e a r s . Number of t r e e s per acre at beginning of each f i v e - y e a r p e r i o d . Number of t r e e s at end of each f i v e - y e a r p e r i o d . 19G FPL o r K L M MATIO NDIST NMAT NT NOCT PD P I PS REDFAC REDINC S O C ( I , J ) S(M) THETA Number of l i v e t r e e s i n the matrix at age 10 y e a r s . Row number i n matrix ( 0 < 1 ^ 2 0 , u s u a l l y 1 ^ 1 5 ) . Column number i n m a t r i x . ( 0 < J £ 2 0 , u s u a l l y J ^ 1 5 ) . Row number of "competitors" i n matrix. Column number of "competitors" i n matrix. The number of l o c a t i o n s t h a t the competing t r e e In each octant i s away from the t r e e being s t u d i e d . Number of t r e e s per row and per column i n input matrix ( 6 ^ M A T I O ^ 2 0 ) . Number of p l a n t i n g d i s t a n c e s ( i n i t i a l s p a c i n g s ) . Number of t r e e s per row and per column i n working matrix. (6 £ N M A T ^ M A T I O ) . Number of t r e e s per acre ( i n i n p u t ) . Number of oct a n t . P l a n t i n g d i s t a n c e ( i n i t i a l s p a c i n g ) . Tf = 3 . 1 4 1 5 9 6 . P l o t s i z e i n acres (of working m a t r i x ) . Reduction f a c t o r t o reduce c a l c u l a t e d crown width t o " c o m p e t i t i v e " crown width. Increment f o r REDFAC. P r o p o r t i o n of the circumference of the crown i n the I , J t h . p o s i t i o n occupied.by competitors. D i s t a n c e of the Mth. competing t r e e from the t r e e being s t u d i e d ( P D £ S ( M ) ^ 8 x P D ) • Angle subtended at the centre of the crown by the two p o i n t s of i n t e r s e c t i o n of the "c o m p e t i t i v e " crowns, d i v i d e d by two ( 0-^THETA£ TT ) . 1 9 2 2 . D e s c r i p t i o n of the FORTRAN program. Line No. MAINLINE PROGRAM 4 - 2 3 Input s e c t i o n . The parameters, r e g r e s s i o n c o e f f i c i e n t s and the b a s i c data are read i n t o the computer. 2 4 - 3 0 P r i n t s out b a s i c . m a t r i x of diameters. 3 1 - 4 9 3 Loop f o r each p l a n t i n g d i s t a n c e . 3 7 - 3 9 C a l c u l a t e s the d i s t a n c e of each p o s s i b l e competitor from the study t r e e . 49-37 .4 Loop t o c a l c u l a t e f i v e - y e a r p e r i o d i c diameter growth. 5 3 - 3^9 Loop t o c a l c u l a t e the competitive s t a t u s , SOC, of each t r e e by o c t a n t s . 5 4 - 9 7 Values are g i v e n . t o the v a r i a b l e s r e q u i r e d t o c a l c u l a t e the p o s i t i o n s i n the matrix of "competitors", D ( K , L ) , with r e s p e c t to each t r e e , D ( l , J ) . The values of the v a r i a b l e s are determined a c c o r d i n g t o the octant being con-s i d e r e d (see P i g . 1 2 ) . 9 8 - 1 0 4 C a l c u l a t e s the l o c a t i o n of the f i r s t t r e e p o s i -. t i o n i n . t h e o c t a n t . (The f u n c t i o n subprogram K P I N D ( K M ) i s d e s c r i b e d below.) 1 0 5 T e s t s t o see i f there i s a t r e e i n t h i s p o s i -t i o n , ( l i v e t r e e s > 0 ) . 1 0 6 I f a t r e e i s present i n the f i r s t p o s i t i o n the subroutine CROWN,(described below) i s c a l l e d and THETA i s c a l c u l a t e d . The next statement ( 1 0 7 ) c a l c u l a t e s the competitive s t a t u s of the t r e e and the program then branches to the next oct-ant. (To save o p e r a t i n g time, THETA i n the program i s h a l f the angle subtended at the centre of the crown by the i n t e r s e c t i o n of the "co m p e t i t i v e " crown p e r i m e t e r s . The p r o p o r t i o n of the circumference of the crown occupied by each competitor Is t h e r e f o r e d i v i d e d by Tt , and not.2Tf , t o ob t a i n , the p r o p o r t i o n of the crown occupied, SOC). 193 L i n e No. 109 I f there i s no t r e e i n the f i r s t p o s i t i o n the program branches to the second p o s i t i o n i n the oct-ant and the process i s repeated u n t i l a t r e e i s found. I f no t r e e i s found i n any of the f o u r t e e n p o s i t i o n s i n the octant the program branches t o the next oc t a n t . 370-1 C a l c u l a t e s the d. b. h. of each t r e e at the end of each f i v e - y e a r p e r i o d (DAP5). 373 I f D ( l , J ) ^ 0 , DAP5(I, J) i s set equal t o zero. 380 T e s t s f o r m o r t a l i t y . 381 Sets D(I,J) equal t o DAP5(l,j) f o r a l l l i v e t r e e s . 383 Sets D ( l , J ) equal t o -DAP5(L,J) f o r a l l t r e e s t h a t have "died d u r i n g the c u r r e n t f i v e - y e a r p e r i o d . 385-95 The new diameter matrix i s p r i n t e d out. 396-404 I n i t i a l i s e s frequency d i s t r i b u t i o n t a b l e loop. 405-18 C a l c u l a t e s the number of t r e e s t h a t have " d i e d " i n each one-inch diameter c l a s s i n the past f i v e y e a r s . 420-9 C a l c u l a t e s the number of l i v e t r e e s i n each one-i n c h diameter c l a s s . 433-51 C a l c u l a t e s means, standard d e v i a t i o n s , b a s a l area, e t c . 452-82 P r i n t s out diameter frequency d i s t r i b u t i o n . t a b l e . 483 T e s t s t o see i f m o r t a l i t y has occurred before the c u r r e n t p e r i o d . 4 8 4 Tests t o see i f m o r t a l i t y has s t a r t e d i n the cu r r e n t p e r i o d . 485 I f m o r t a l i t y has occurred d u r i n g the c u r r e n t p e r i o d , ACOMP i s set equal t o the age at the s t a r t of the p e r i o d . 486-8 I f m o r t a l i t y has occurred, e i t h e r before or d u r i n g the c u r r e n t p e r i o d , REDFAC i s m o d i f i e d . 492 T e s t s f o r end of run. 194 FUNCTION KFIND(KM) When determining the p o s i t i o n of competitors i n the matrix t h i s subprogram ensures t h a t the mai n l i n e program does not branch out of the matrix. SUBROUTINE CROWN C a l c u l a t e s the value of THETA, the angle subtended at the centre of the crown by the i n t e r s e c t i o n of the crown perimeters d i v i d e d by two. 4 R = .5* REDFAC so that RI = .5*(A1C + B 1 C ( D ( I , J ) ) * REDFAC 5-12 C a l c u l a t e s the "c o m p e t i t i v e " crown r a d i u s (Rl) of the t r e e being s t u d i e d and of the p o t e n t i a l competitor (R2). 13 Tests t o see i f the " c o m p e t i t i v e " crowns o v e r l a p . 14 Te s t s t o see i f the crown of the t r e e being s t u d i e d (the I,Jth) completely overlaps that of the competitor (the K,Lth.). 15 I f the answer t o 13 i s "no" or to 14 i s "yes", THETA Is set equal t o zero and c o n t r o l i s ret u r n e d to the ma i n l i n e program. 17 T e s t s t o see i f the crown of the competitor over-l a p s t h a t of the t r e e being s t u d i e d . 18 I f the answer t o 17 i s "yes", THETA i s set equal t o TT . 20,21 . C a l c u l a t e s the o r d i n a t e s of the p o i n t of I n t e r -s e c t i o n ( i n the f i r s t quadrant) of the "c o m p e t i t i v e " crowns. 22 Tests t o see i f THETA.is g r e a t e r than, equal t o or l e s s than 71/2. THETA i s c a l c u l a t e d a c c o r d i n g l y (see F i g . 70) and c o n t r o l r e t u r n e d t o the main-l i n e program. 3. Program output An example of the output given at the end of each f i v e -year p e r i o d by the FORTRAN program f o r the I. B. M. 7090 i s shown i n F i g . 71. 195 i. n/2-««-*n 2. O -ce -<JV2 e«arctan(Y/X) FIG. 7 0 : Ca lcu la t ion of 0 in the FORTRAN program. C . r l . h . G . B . F R E Q U E N C Y DISTRIBUTION TABLE 0 • fl • H • NO. OF NO. OF TREES 5YR. MORTALITY 5YR. MORTALITY CLASS TREES PER AC. PER AC. 2 0 0. 2 8.9 3 1 / 75.6 2 8. 9 4 28 124.4 4 I 7.8 5 3C 133.3 4 1 t.tt 6 36 168.9 6 2 6 . 7 7 29 128.9 1 4 .4 8 15 66. 7 0 0. 9 13 57.8 0 0 . 10 5 22.2 0 0. 11 1 4.4 0 0 . 12 1 4.4 c 0. TCTAL = 177 786.7 19 84.4 Kt AN = 5.99 INS. 4.72 INS. VARIANCE 3.6G05 STANDARD DEVIATION = 1.90 TREE CF MEAN 8.A. = 6.28 BASAL AREA PER AC. = 169.4 S'j. F t . 11.1 SO. F T . P .A. I . (BASAL AREA) = 5.1 C .A. I . (BASAL AREA) = 5.6 REDFAC = 0.6059  FIG. 71: Example of program output. STAND MODEL II - NEWNHAM FOr* THAN SCUHCfc LIST MOOfIR ISK SOURCfc STATEMfcNT 1 DI P E N S 1 ON P D 1 5 1 . P S I 5 ) , N T I 5 > , D I N C 1 2 0 1 , D U O , 2 0 ) , 0 1 0 1 2 0 , 2 0 1 , N D 6 I 3 0 > I I 1 2 DIMtNSION D A P i l 20 , 201 , S O C I 2C • 201 , S I 5 0 1 , N N O B I 3 0 1 ,DI SI 151) I 1 I 2 .-. C O M M U N R E O F AC , A 1 C t A2C, rl I C i ( 3 2 C , 1 I J , K , L « M , NMA T »S» THt T A , 1) 1 I 3 4 RE AO 1 5 , 1 ) NMA T, M A T 10, N I ) 1 S T , A S T A R T , A S T G P . F A C R E D , R E D I n t C 1 1 4 10 1 F O R M A T 1 3 I 3 , 4 F 6 . C ) 1 1 5 1 1 R L A 0 I 5 . 2 ) t P D { [ ) , I = T , N D I S T ) , < P S ( ! ) , I M , N O I S T ) 1 1 6 22 2 F O R M A T1 1 0 F 6 . 0 I 1 1 7 23 R t A D I 5 , 5 ) A l C , A 2 C , l t l C , 8 2 C I I 8 24 5 F U R M A T 1 4 F 1 0 . 0 I 1 1 9 2b R L A O I 5 , o ) I D I N C I L A ) , L A - 2 , 1 9 1 1 1 10 32 6 F U R M A T I 1 8 F 4 . 3 ) I 1 1 1 3 3 R E A D I 5 , 1 2 0 0 1 I I 1 1 S T I 1 ) , 1 ' 1 , 4 H ) 1 1 12 40 1 2 0 0 FORMAT 1 1 2 1 6 . 4 ) 11 13 41 R t A O l 5 , 3 1 F P L O T , I N T I 1 1 , 1 » I , N 0 l S 1 ) 1 1 14 46 3 FIJRMAf ( F 6 . 6 . 5 1 6 ) 1 I 15 47 R E A C I 5 . 4 ) A l , b l . B 2 , B 3 , B 4 . b A M A T 1 I 16 50 4 F O R M A 1 1 6 F 1 0 . G I 11 17 51 200 R t A O l 5 , 1 2 0 1 ) 1 1 1 8 52 1 2 0 1 O F U R M A T I 3 0 X . 4 2 H / 3 0 X , 7 5 H 11 1 9 1 ) 1 1 2 0 5 3* D U 7 1 - 1 . M A T 1 0 II "21" 54 7 R E A D 1 5 » H ) 1 D 1 0 I I , J 1, J * 1 , M A T 1 0 1 1 1 22 6 2 8 F C R M A t 1 2 0 F 3 . 1 1 1 1 23 63 t . R | T E I 6 , 9 > 1 1 24 64 9 C F U R M A T I 1 H 1 , 5 1 X , 1 5 H S T A N D M O D E L I I G / 1 MO, 3 2 H I J R I Gl N A L O . B . H . O . B . OF t A I I 25 ICh T R E t / l 1 1 76 ~ 6 5 W R I 1 E I 6 , 1 2 0 1 1 " " I I i t -6 6 01) 10 [ M . N M A t II 78 67 10 H K I T E I 6 . 1 1 I 1 0 1 0 1 I . J I . J - l . N M A T ) I 1 29 /5 u F O R M A T ! I H 0 , F 4 . 1 , 1 9 F S . I I 1 1 30 76 DC 1 3 0 l l > l , N D I S T 1 I 11 77 W R I T E C 6, 1 2 I N T I 1 1 1 , P S I 1 1 1 , P D I I I 1 , M A T 1U ,F A C R L D , R F O I .YC 1 1 32 100 12 O F O R M A I I 1 H W 5 1 X , 1 5 H S T A N 0 M O D E L 1 I G / / / 4 7 X . 2 2 H N U . U F T R E E S P E R ur. -irr '33 ' "" '" " I 5 / / 4 9 X , 1 I H P L O I S 1 ZE » F H . 5 , 4 h AC . / / 4 6 X , 1 9 H P L A N T 1NG D I S T A N C L » F 5 . 1,411 14 2 H F I . / / / / 1 0 X . 8 H M A T R I X = 1 3 , 8 H S Q U A R E , , 9 H R E D I AC * F 6 . 3 , 9 ) 1 R L D 1 N C " F ( l 1 i t 3.4) 11 lb 101 X » P D I 1 1 1 11 37 1 0 2 D C 1 2 1 0 1-1 ,48 11 38 1 0 3 1 2 1 0 S1 I ) • DT S T ( I ) • X 11 »9 " " 1 0 5 B A S T . H A M A T / P S I I 1) 11 4 0 1 0 6 R E D F A C A F A C R E D 11 41 1 0 7 F L A S T » F P L U T 11 4 2 110 A « A S T A R T 11 43 111 A C O M P - 0 . I 1 44 117 OC 14 I =T . NM.A T" . . . j J- " . . 5 " 113 CC 14 J ' l . N M A I 11 46 114 14 DI I , J ) = 0 1 C ( I . J I 11 4 7 117 P 1 = 3 . 1 4 1 5 9 11 4rt 1 2 0 15 DC 9 3 1 M . N M A T 11 4 9 121 OC 9 3 J M . N K A T 11 DC 122 S O C I I , J ) » 0 . 11 .1 1 2 3 I F I D I I . J I ) 9 2 , 9 2 , 1 6 11 'J2 1 2 4 16 DC 91 N U C T = l , d 11 53 1 2 5 G O TO 1 1 7 , 1 7 , 1 8 , 1 8 , 1 7 , 1 7 , 1 8 , 1 8 ) , N U C T 11 54 1 2 6 17 1 1' 1 11 55 1 2 7 J 1 " J 11 56 1 3 0 G O TO 19 11 5 7 1 3 1 l a l l = J I I o „ 1 3 2 J 1 • I 11 59 133 1"! G O TO 1 2 0 , 2 0 , 2 1 , 7 1 , 2 1 , 2 1 , 2 0 , 2 0 ) , N U C I 11 60 1 3 4 20 1 M I = - 1 11 61 1 3 5 I M 2 —2 11 <-2 1 3 6 I M 3 " - 3 11 6 3 1 3 7 I M 4 = - 4 II 64 1 4 0 I M 5 —5 11 65 141 IM6=-fc 1 1 66 1 4 2 I M 7 . - 7 11 67 1 4 3 I M 8 — 8 11 68 144 I M 9 « - 9 11 6 9 1 4 5 I M I O ' - I O I T "70 1 4 6 GO T O 22 11 11 1 4 7 21 1M1 -1 11 72 1 5 0 I M 2 > 2 11 11 151 I M 3 - 3 11 l< 1 5 2 I M 4 - 4 I I / 5 1 5 3 I M 5 - 5 " " " " ~ " I T ~76 1 5 4 I M 6 - 6 11 77 155 I M 7 - 7 11 l i 1 5 6 I M 8 « 8 11 7 9 1 5 7 I M 9 « 9 11 80 1 6 0 IM10-10 11 M 161 22 G u " t o T 2 3 , 2 4 , 2 3 T ? 4 , 2 4 7 2 3 , 2 4 , 2 3 ) " ; N U C I " rr ' 82 1 6 2 2 1 J M l . - t 11 83 1 6 3 J M 2 - - 2 11 >i4 1 6 4 J M 3 — 3 11 K5 1 6 5 J M 4 - - 4 11 •J6 1 6 6 J M 5 — 5 11 87 1 6 7 JM . 6 —6 11 " " ' 8 8 _ 1 7 0 J M 7"— 1 11 8 9 171 GU 10 75 11 90 1 7 2 2 4 J K l ' l 11 9 1 1 7 3 J M 2 - 2 11 12 1 7 4 J M 3 ' 3 11 9 3 1 7 5 " J M 4 - 4 ~ ri" T4 1 76 J M 5 - 5 11 95 1 7 7 J M 6 - 6 11 9 6 200 J M 7 « 7 11 1 7 2 0 1 25 M " l 11 98 202 I U l l - l ) 2 7 , 2 6 , 2 7 11 99 2 0 3 2 6 K - x F i N i r m t m r r " ' 11 100 2 0 4 L ' J 1 I I 1 0 1 2 0 5 GO TO 20 11 l u 2 2 0 6 27 K « J 1 11 l o i 2 0 7 L ' K F I N D I 1 1 . 1 M 1 I 11 1 0 4 2 1 0 28 I F I O I K . L I I 3 0 , 3 0 , 2 9 11 10 5 " 2 1 1 2 9 " C A L L CROHN IT" | 0 6 2 1 2 S O C I 1 , J I - S O C I l , J I « . 5 « T H E T A / P I 11 11)7 213 GO FO 91 11 1 0 8 2 1 4 30 M-2 11 1 0 9 2 1 5 I H I l - 1 ) 3 2 . 3 1 , 3 2 II 110 ISN SOURCE STATEMENT 216 217 31 L-KFINDI J U J M l l GO TO 33 II 111 II 112 220 32 K»KF1ND<J1»JMI> II 113 221 I F I O I K . L I I 34,34,33 II 114 222 33 CALL CROWN 11 115 223 SOCI I , J I - S O C I l , J I * . 5 o T H E T A / P l II 116 224 GO TO 91 II 117 225 34 M-3 II 118 226 I F I U - I I 36,35,36 II 119 227 35 K-KFINDI I 1UM2I I I 120 230 L - J l I I 121 231 GC TO 37 1 1 122 232 36 K - J l II 123 233 L-KFINOI11*IM2I 11 124 234 37 I F I O I K . L I I 39,39,38 I t 125 235 3a CALL CKUWN II 126 236 SOCII.Jl-SOCI I , J)».5»THEIA/PI II 127 237 GO TO 91 II 128 240 39 M-4 1 1 129 241 I F I I l - I I 41,40,41 II 130 242 40 L-KFINDIJ14JMII II 131 243 GO 10 42 II 132 244 41 K - K F I N D I J K J M l ) 1 1 133 245 42 I F I O I K . L I I 44.44,43 II 134 246 43 CALL CROHN 11 115 247 SOCII,Jl-SOCII,J>»THETA/PI II 116 250 GO TO 91 II 117 251 44 M-5 II 138 252 I F I I l - I I 46,45,46 II 139 253 45 L - K F I N D I J U J M 2 I II 140 254 GO TO 47 II 141 255 46 K -KFINOIJ1»JM2I I 1 142 256 47 I F I O I K . L I I 49,49,48 II 143 257 43 CALL CROHN II 144 260 SOCII,JI-SOCIl,J)«.S«THfcfA/PI 1 1 145 261 GO TO 91 1 1 146 262 49 M-6 II 14/ 26 3 I H I l - l l 51,50,51 1 I 148 264 50 K-KFINOI I U I M 3 I 1 1 149 265 L - J l I I 150 266 GU TO 52 11 151 267 51 K - J 1 1 1 152 2 70 L-KF1NDII1.1M3I II 153 211 52 I F I O I K . L I I 54,54,53 II 154 2 7 2 T T CALL—CRUWN I I 155 273 SOCII,J I -SUCIl,JI».5»IHtTA/PI 1 1 156 274 GO TO 91 II 15/ 275 54 M-7 II 158 276 I F I I l - I I 56,55,56 II 159 277 55 L - K F I N O l J l + J M l ) 1 1 160 300 GO TO 57 II 161 301 56 K-KFINDIJIFJM1I I I 162 302 57 I F I O I K . L I I 59,59,58 11 163 303 58 CALL CROHN 1 I 164 304 305 SUCI 1,JI-SOCII,JI.THETA/PI GO TO 91 I 1 165 II 166 306 59 K-8 II 16/ 30 7 IFI 1 1-1 ) 6*1,60,61 II l o 8 310 60 L-KFINDIJ1.JH2I II 169 311 GU TO 62 II I/O 312 61 K-KFINDIJ1*JM2 1 II 1/1 313 62 I F I O I K . L I I 64,64,63 II 172 314 63 CALL CROHN II 1/3 315 s u c i i , J I - S O C I i , j f* IHE ra/p i II 1/4 316 GO TO 91 II 1/5 317 64 M-9 I I 1/6 320 I F I U - I I 66,65,66 1 1 1 / 7 321 65 K-KFINDI I UIM4I 1 1 178 322 L. J 1 II 179 323 GO TO 67 II lliu 324 66 K - J l I I 161 125 L-KFINOI I 1*IM4I 11 1U2 326 327 67 I F I O I K . L I I 68.68,67 CALL CROHN 1 I 183 1 1 184 3J0 STT SOCII.JI-SOCII,JI*.5«THETA/P | 1 1 1H5 GO TO 91 II 186 332 68 H- 10 II 18/ 333 I F I I l - I I 70,69,70 11 168 334 69 L-KFINDIJ1»JM1) 1 I 189 335 GU 10 71 11 K O 336 70 K - K F I N D I J l t J M l l I I 191 337 71 I F I O I K . L I I 73,73,72 II 192 340 72 CALL CROHN 11 113 341 SOCI 1,JI-SOCIl,JI«THEIA/PI 1 1 194 342 GU TO 91 11 195 343 73 M - l l 1 1 196 344 345- •ff— I F I I l - I I 75,74,75 II 19 7 K-KFlNOI | 1HM3I 1 1 198 346 l-KFINDIJ1»JM3> 11 199 347 GU TO 76 1 1 200 350 75 K-KFINDIJl»JMi) II 201 351 L-KFINUII1.IM3) 1 1 202 352 76 I F I O I K . L I I 78,78,77 1 1 203 553 77 CALL CROHN I 1 204 354 SOCII,Jl-SOCII.J)».5«IHEIA/PI 1 1 205 355 GU TO 91 1 1 206 356 78 M-12 1 1 207 157 I F I I l - I I 80,79,80 11 206 360 79 K-KFINOII1*IM4I 1 1 209 361 L-KFINOIJl»JM2l II 210 362 GO TO 180 1 1 2 1 1 363 80 K-KFINOI J U J M 2 I 1 1 212 364 L-KFINDI IWIM4I II 213 365 180 I F I O I K . L I I 81,81,181 I 1 214 366 181 CALL CROHN II 215 361 suci i , jy-soci r. J ) • T H E T A / P I I I 216 170 GO 10 91 1 1 217 371 81 M-13 I 1 218 172 MM-0 1 1 219 373 I F I U - I ) 83,82,83 II 220 ISN SUURCE STATENtNI 374 82 375 L-KF1NIH J l « J M 3 l cn ra i* 221 222 376 83 K - K F I N U I J t « J N 3 ) 377 84 IFIOIK.LII 86,86,85 400 85 C a l l CRUUN 401 SOCII.JI -SOCII.JI . IMEI4/CI 402 MMM _403 86 IFII1-II 88.87.68 223 224 225 226 227 22B 404 87 K-KFINOI I 1HM5I 405 L - J l 406 GU TO 89 407 88 K ' J l 410 L'KF 1N0I IHIM5I 411 89 IFIOIK.LII 2lH.11a.ta 229 230 2)1 232 231 234 CALL CROHN SUCI I, J I-SOC I I . J I » . » « T H £ T A / P I GO IC 91 1FIMMI 299,299.91 M»15 JFIIl-II 10l,J00,301_ K"KFINO( IIMM5I L-KFINDI J l . J H H GO TO 302 K'KFINOIJltJMll L-KFINDI I1HM5I IH0JK.LJJ303,30J,72 K"16 IFI I 1-1 I 305,304,305 1-KFINDIJKJH2I GO TO 306 K ' K F I N O I J 1 » J K 2 > IFIOIK.LII 307,307,72 «»!' IFI 11 — 11 309,308,309 K-KFINOI I IHH4) L - K F I N D I J 1 » J M 4 I GO TO 310 K « K F 1 N D ( J I * J M 4 ) 1-KF1ND"( |T»IM»T IFIOIK.LII 311.311,90 IF I 11- I > 313,312,313 K-KFINDI IHIrl'J) L = K F I n O ( J I » J M 3 ) _ ~ GlT"TO' 3 1 4 ~ K-KFINDl J10H3I L-KF1N01I1.IM5I F F4 LI I K • L) I 315,315,72 M» 19 IFI I 1- 1 I 311,316,317 K.KFl',01 I KIH6I L"J l GO TO 318 KOI L -KF INDI l l . lHb) IFIOIK.LII 319,319,90 M = 20 IFI 11-1 I 321,320,321 L » K F ! M U ( J l » J M l l CU 10 322 K«KFr< O I J l * J M l ) IFIOIK.LII 323,323,72 K»21 IFI 11-1 1 325,324,325 L°KFI,'<0< J l » J M 2 > GU TO 326 K.KFINOI JUJM2) IFIOIK.LII 327. 327, 72 H-22 _ IFI If- II 329,328, 329 K..K.FINOI M»IM5I L »KF L'nOI J 1 *JM41 GO 10 3291 K.KFIN0IJIOM4I L-KF1N0II1MH5I IFIOIK.LII 330, 330, 72 M-2 1 IFI | 1 - I) 332, 331,332 K-KF |Mil | KIH6I L ' K F H I ) ( J 1 » J M 3 I GO TO 133 K-KFIMCI J H J H l l " -L'KFINOI I 1HH6I IFIOIK.LII 334,334,72 H«24 IFII1-I) 316,335,336 K.KF I.MOI I 1.IM7I L - J l GO TO 337 K-JI L'KF I«0{Il»IM/l IFIOIK.LII 338,338,90 M»25 MM*II : — • • IF I II-I I 340,339,340 L > K F H 0 < J I « J M 1 ) GU TO 341 K'KfI lUIJl<JH11 IFIOIK.LII 343,343,342 CALL CROW*, SUCII.J I'SUCI I.JI.THETA/PI H H « l IFI11-II 345,344,345 K-KI'MUI I1MM5I L 'KFl |D(J1.JH5I GO TO 346 K.-KFINOI J10M5I l - K F I N U I l l « I M 5 ) IFIOIK.LII 347,347,90 IKMMt 348,348,91 235 236 211 218 219 240 ~24T~ 242 241 244 245 246 741" 248 249 250 251 252 251 254 255 256 257 258 "i'59— 260 26 1 262 263 2h4_ ~26"5 266 267 2o8 2o9 2 I0_ _ "2/1 lit 2/3 2 74 2/5 . 2 76 2/7 276— -279 280 28 1 282 2 8 l _ 2»4 285 286 287 2tfH 289 V-TlH 2)1 292 291 294 295 ?•>» 2)7 293 299 300 301 T 0 2 " 3::i 304 irj5 106 30 7 ii.8 109 110 311 312 313 I T T 315 316 31/ 318 311 T20~ 1 2 1 322 323 324 125 "37 6"" 32 7 128 129 110 ISN SOURCE STATEMENT 552 346 M-27 1 1 331 553 IFI l l - l 1 350,349.350 1 I 332 554 349 K-KFINOI11»IM6> 1 1 333 555 L-KFINOIJ1»JM4I I I 334 556 GO TO 351 1 I 335 55/ 350 K-KFINDIJ1«JM4I 1 I 336 560 L-KFINDI I1HM6I II 337 561 351 IFIOIK.LII 352,352,72 II 338 562 352 M»28 1 1 339 563 I F I I l - I I 354,353,354 1 1 340 564 353 K " K F I N D I 1 1 . IH7I II 341 565 L-KFINDIJ1.JM2I 11 342 566 GO TO 355 1 1 343 567 354 K-KFINOI JMJM2) 1 1 344 570 L-KFlNDl11.1H7! 1 I 345 571 355 IFIOIK.LII 356.356,72 1 I 346 572 356 M-29 II 34 7 573 I F I I l - I I 358,357,358 II 348 574 357 L-KFINDIJ1»JM3I 11 3 4 9 575 GO 70 359 11 350 576 358 K-KFINDI J K J M 3 I II 351 577 359 IFIOIK.LII 360,360,72 1 I 352 600 360 M-30 1 1 353 601 I F I I l - I I 362,361,362 II 3 5 4 602 361 K-KFINDI 1U1M6I 1 1 3 5 5 603 l-KFINDI J U J M 5 ) I I 356 604 GO TO 363 I I 357 605 362 K-KFINOIJ1«JM5I 1 1 358 606 L-KFINDII1 .1H6I 1 1 359 607 363 IFIOIK.LII 364.364.72 1 1 360 610 364 M-31 1 1 361 611 I F I I l - I I 366.365.366 I I 362 612 365 K - K F I N U I rmnai 1 1 363 613 L - J l 1 1 364 614 GU TO 367 11 365 615 366 K - J l 1 1 366 616 L-KFINDII1+IN8) II 367 617 367 IFIDIK.LII 91,91,90 11 3 0 8 620 91 CUNTINUE 1 1 369 622 0DAP5II .JI-OIl ,J)»2.-IAl»Bl-OII J)tA«IU2*04-AI»B3-DlO<1.Jl1• I I 370 111.-SOCII.J>) I I 371 623 GO TO 93 I I 372 624 92 DAP5I1.J)-0. 1 1 373 625 93 CONTINUE II 174 630 U - . 2 - A * . 5 I I 175 631 A=A»5. I I 176 632 DU 97 I-l.NMAT I I 377 6 3 3 DU 97 J-l,NMAT II 378 634 IFIDAP5Il ,JI) 95,95,94 I 1 379 635 94 IFIIDAP5II,Jl-DI1,Jl)/DI1,JI-D1NCILA)1 96,95,95 11 380 6)6 95 D I l , J I - D A P 5 l I , J 1 1 1 3H1 637 GO TO 97 1 1 382 640 96 01 1,J) — DAP5Il,JI I I 383 641 97 CONTINUE 1 1 384 644 MA-A I I 385 645 M M A - A - . 0 5 * . ! 1 1 386 646 HMA-MHA-20 II 387 64 7 IFIMA-MMA) 102,98,102 1 1 388 650 98 HR1TEI6.99) MA I 1 389 651 99 FORMATI 1 HI, 57X, 5HAGE - I3/1HO,17HD.8.H.O.B. MATRIX) 1 1 390 652 DO 100 I-1,NMAT I I 1 9 1 653 100 HRITEI6.101I 1 U I I , J l , J - l . N H A T I 1 I 392 661 101 FORMAT 1 1H0.F6.2,15F7.2) 1 1 193 662 NRITEI6,10151 1 I 394 663 1015 FORMAT 11H0,19HI0EAD T R E E S ARE - 1 1 I 1 3)5 664 102 00 103 13-1,30 I I 19 6 665 NN08I131-0 I I 397 666 103 NOB!I3>=0 1 1 398 670 SD-O. 11 3 9 9 671 SDO-0. 1 I 400 672 NNO--0 I I 401 673 SND-O. I I 402 6 74 SNOO-0. 1 I 403 675 NDEAO-0 I 1 404 676 OU 112 l-l .NMAT I I 405 ~sn 00 112 J-l.NMAT 1 I 406 700 I F I D I I . J I ) 104,112,108 I I 40/ 701 104 ON —01 I . J 1 1 1 408 702 NCEAD-NDEAOtl 11 409 703 SND-SNO+DN I I 410 704 SNCO-SNDODN-ON I I 411 705 14-1 I I 412 706 105 TI3-I3 I I 413 707 TI3-T13».5 I I 414 710 1FIDN-T13) 107,106,106 1 1 415 711 106 13-13*1 I 1 416 712 IFI13-30) 105.107,107 1 1 417 7 1 3 107 N N D B I I J I - N N O B I I 3 ) » l I I 419 714 GO TO 112 I I 419 715 108 NNOW-NNOH*! 1 1 420 716 SU-SO.DI1,Jl I 1 421 717 S D D - S D D . O I 1 , J l - D I1 , J l I I 422 720 13-1 I I 423 ' 2 1 109 T I J - I J II 4 2 4 722 T I 3 - T I 3 . . 5 1 1 425 723 IFIDI1.JI-TI3) 1 1 1 , 110 1 1 0 I I 426 724 U O 1 3 - 13 . 1 1 1 427 7 2 5 IFII3-30) 109,111 ,111 I I 42B 726 111 N08II3I-N0BI I3)«l I I 429 727 1 1 2 CONTINUE 1 1 43U 732 IFINNOW«NOEADI 130,130 1 1 3 I I 431 733 113 IFINOEAD) 1 1 5 , 1 1 5 , 1 1 4 I I 432 734 114 FOEAD-NDEAD 1 1 433 735 08ARN-SND/F0EA0 I I 4 3 4 736 F C A - F D E A O / P S I 1 1 1 1 1 435 737 8AN-P 1 -5NDC /I 5 7 6 . - P S I 11 11 II 436 7 4 0 GO TO 1 1 5 1 I I 437 741 1 1 5 DBARN-O. I I 43S 742 FCA-O. 1 1 439 743 BAN-O. 1 1 4 4 0 201 ISN S O U R C E S T A T E M E N T 7 4 4 1 151 I F I N N O W I 1 1 6 , 1 1 6 , 1 1 5 2 II 441 7 4 5 E N - N N C W I 1 442 " 7 4b DBAR«SD/FN 11 443 747 V A R - I S O D - S D ' S D / F N I / I F N - l . 1 1 I 444 7 5 0 S I G M A » S U R T ( V A R I 1 1 445 7 5 1 D 2 B A R = S U R T < SOO/FNI 1 1 4 4 6 7 5 2 BA'IM'SDD/I 5 7 6 . ' P S I l . l 1 1 1 1 44 7 7 5 3 A V A I B R A / A I I 448 7 5 4 C A I = I B A - B A S T I - . 2 1 1 449 755 B A S T - B A I 1 4 5 0 746 FNTA=FN/PSI11 1 I I 4 5 1 757 116 MMA=A*.l+.l 1 1 4 5 2 7 6 0 M M A = M M A « 1 0 I 1 4 5 3 7 6 1 1F1 MA—MHA) 1 2 5 1 , 1 1 6 1 , 1 2 5 1 1 1 454 7 6 2 1 1 6 1 W R I T E ( 6 , 1 1 7 I MA 1 1 455 7 6 3 1 1 7 O F O R M A T I I H l , 5 7 X , 5 H A G E • 1 3/IHO, 3 9 H 0 . B .H.O. B. F R E U U E N C V DISTRIBUTION I 1 4 5 6 1 I A B L E / I H O , 1 2 X . 6 0 H D . B . H . NO. UF NU. O F TREES 5VR. MORTALITY 5 Y R I 1 4 5 7 2. MOR1ALI T V / I H ,12X,25HCLASS IREtS PER AC.,25X,7HPER AC.//) 1 1 4 5 8 7 6 4 DU 1 2 5 1 3 = 1 , 3 0 II 4 5 9 7 6 5 I F I ' I D B I I 3 I » N N 0 8 1 1 3 ) 1 1 2 5 , 1 2 5 , 1 1 8 I 1 4 6 0 "766 T|8 I F I N D B I I 3 I I 1 2 0 , 1 2 0 , 1 1 9 I I 41.1 767 1 1 9 F N 0 8 = NI)H( 13) 1 1 462 7 70 F N C A C = F N 0 8 / P S I 1 I ) 1 I 463 7 7 1 F N N O A C = 0 . I I 464 772 GU FU 1 2 1 I 1 465 773 1 2 0 F N C A C = 0 . II 4r,6 7 7 4 1 2 1 " 1 F I N N D H I I 3 I I 1 2 3 , 1 2 3 , 1 2 2 I 1 4 6 7 775 1 2 2 FNNDB'NNDBI13) 1 I 468 7 7 6 FNN[)AC = F N N D B / P S I 1 I 1 11 41.9 77 7 1 2 3 W R I T E ( 6 , 1 2 4 ) 1 3, NOB 1 I 3 11F NO AC t NNOB ( 13) .FNNOAC 1 I 4 7 0 1 0 0 0 1 2 4 F O R M A T I I H , 1 2 X , 1 4 , 1 9 , F 1 2 . 1 , 1 1 4 , F 1 8 . 1 1 I 1 4 7 1 1 0 0 1 1 2 6 C O N T I N U E 1 1 4 72 " 1 0 0 3 GU TU 1 2 5 3 " I T 473 1 0 0 4 1 2 5 1 k'RIIFIn, 1 2 5 2 1 MA 1 1 474 1 0 0 5 1 2 5 2 F U K M A T I 1 H 1 , 5 7 X , 5 H A G E =13/) II 4/5 1 0 0 6 1 2 5 3 0 U R I T E I 6 , 1 2 6 ) NNUW,FNTA,NDEAD,F0A,UBAR,nuARN,VAR,SIGMA,D28AR ,6A,BA4I 1 4/0 l.CAI.AVAl 1 1 4 / 7 10.9I 1 2 6 O F O R M A T I 1)10, 1 3 X , 7HTI) TAL • 1 5, F I 2. 1,114 ,F I 8. 1/IHO, 14X , 6 H X L AN = F 6 . 2 . 5 H I I 4/8 1 INS. ,"15X,F6.2, 5 H INS./IHO, 1 0 X , 1 0 H V A R T ANC1T B F 9 . 4 / I H O , 2 G N S I A N D A R O TIT 4/9 2EV1 AT I ON = F 6 . 2 / 1 H O , 2 0 H IRtE CF M E A N B . A . » F 6 . 2 / I H O . 2 0 H C A S A L AREA P l l 4au 3ER AC. = F 7 . l , B H Su. FI.,11X,F/.1,8H Sc. F 1 . / I H O , 2 0 H P . A . I . I HASAl ARI I 481 4EA) = F 7 . 1 / 1 H 0 . 2 0 H M . A . I . I B A S A L AREA) *F7.1/7/1 11 482 1 0 1 0 I F I A C O M P ] 1 2 7 , 1 2 7 , 1 2 7 2 11 483 1 0 1 1 1 2 7 I F I F L A S T - F M I 1 2 8 , 1 2 8 , 1 2 7 1 II 484 1 0 1 2 ACTjMP = A - " 5 . " " " " 1 1 4.15 " 1 0 1 3 1 2 7 2 X«A-ACUMP I 1 486 1 0 1 4 Y - X - 2 0 . II 48/ 1 0 1 5 R E U F A C = R E D F A C * R E D I N C M X » . 0 1 . Y » A B S < Y I ) 1 1 4un 1 0 1 6 1 2 8 kR1TEI 6 , 1 2 9 ) R C O F A C II 4.'19 1 0 1 7 1 2 9 FORMAT I I H . 8 H R E 0 F AC - F 7 . 4 I 1 1 490 1 0 2 0 FLAST-FN 1 1' 4") 1 1 0 2 1 I F I A - A S I O P I 1 5 , 1 3 0 , 1 3 0 II 49,/ 1 0 2 2 1 3 0 C O N I I N U E 11 493 1 0 2 4 GU TU 200 1 1 494 1 0 2 5 E N D 11 4 9 3 ".CUU 1 1 - NEkNHAM FUR I R A N SUURCt LIST K F I N U I S N S O U R C E S T A T E M E N T 1 F U N C T I O N K F I N U I K M ) KFINCCOl 2 D I M E N S I O N S I 5 0 ) , D I 7 0 , 2 0 I KFlNUGOl 3 COMMON R E D F A C , A l C , A 2 C , B l C , 0 2 C , l , J , k , L , M . N M A T , S . T M t f A . O " K F I N 0 G 0 3 4 I F I K M I 1,1,2 KFIN0GO4 5 I K F I N O - K M . N M A T KF1NU00j 6 R E T U R N KFI1I0C06 7 2 IFIKM-NMAT) 4,4,3 K F I N U G O / 1 0 3 KFINO-KM-NMAT K F INDGOtJ 1 1 R E T U R N " " KlT"NUGU9 1 2 4 X F I N D » K M Kf I'.UGIO 1 3 R E T U R N K F 1 N O G 11 14 END- K F I N C C 1 2 STAND MODEL II - NEWNHAM FORTRAN SOURCE LIST CROWN ISN SOURCE STATEMENT 1 SUBROUTINE CRUWN CRUkNGOl 2 0IMENS1UN 51501,0170,201 CRUWNG02 3 COMMON REDFAC, A1C. 42C.81C ,"B2C','l ,'J ,K,'L .MVNMA'I, S, THE TA",D' " " ' CRO-nGOT" 4 R«.5»RE0FAC CROW )G04 5 IFIOI I, J 1 - 3 . 1 1,2,2 CRO.UGO'. 6 1 RIMA1C*RIC-DII , JI>-R CRDWNG06 7 GO TO 3 CkOkNGO/ 10 2 _ R l " l A2C»B2C»DI I , J ) l»R _ _ _ CROfcNGOB 11 3 I F I D I K , L l ' - 3 . l 4,5,5 -. . . ........ . . _ .... _ t H'tl^.-j&o-* 12 4 R2"lALC*BlC>DtK,L)MR CRU-NGIO 13 GO TO 6 CROkNull 14 5 R2"<A2CtB2C«0IK,L))«A CR0WNG12 15 6 IFI R U R 2 - S I M I I 8,8,7 CKUa'NGl 3 16 7 IFIR1-R2-SI M) I 9,8 J8_ CH0WNGI4 1/8 TFETA.C. ' - - - CRI!«~,SI > 20 RETURN CRIIWNG16 71 9 IFI R 2 - R I - S I M I I 11,10,10 CR0W..C1 / 22 10 THHTA"3.14159 CR0WNC18 23 RETURN CROwNGl I 2 4 _ U X"UI-R1-R?«R2»S<MI»S<") 1/12.-SIM) I CR0.NG2U 25 " V - S U R I i S T i R l - X i X I " " ' . . . . CWS5CTI— 2 6 1FIX) 12,13,14 CKO-..C22 27 12 X»-x CR0WNG23 30 ThCTA'll.570BO*ATANIX/V>l>R2/RI CR0WN024 31 RETURN CRQkNG25 32 13 TH6TA-1.5 7080'R2/m CR0WNG26 J'J R E T U R N " " " ZWGZtilifl 34 1* ThF.TA'ATANIY/XI>R2/Rl CR0WN028 35 RETURN CR0WJG29 ;6 END CRUWNU30 

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