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A least squares analysis of inventory data to compare yields of pure and mixed stands in British Columbia… Yang, Richard C. 1978

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A LEAST SQUARES ANALYSIS OF INVENTORY DATA TO COMPARE YIELDS OF PURE AND MIXED STANDS IN BRITISH COLUMBIA FOREST ZONES by Richard C. Yang B.S.A., National Taiwan Un i v e r s i t y , 1963 M.S., Louisiana State U n i v e r s i t y , 1972 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES (Forestry) We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1978 (c) Richard C. Yang In presenting th i s thes is in p a r t i a l fu l f i lment of the requirements for an advanced degree at the Univers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make it f ree l y ava i lab le for reference and study. I fur ther agree that permission for extensive copying of th is thes is for scho la r l y purposes may be granted by the Head of my Department or by h is representat ives . It is understood that copying or pub l i ca t ion of th is thes is for f i n a n c i a l gain sha l l not be allowed without my wri t ten permission. Forestry Department of _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ The Univers i ty of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 D a t e October 19, 1978 - i i -Supervisor: Dr. A. Kozak ABSTRACT The author.developed a s t a t i s t i c a l procedure to analyze i r r e g u l a r , unbalanced inventory data by the l e a s t squares p r i n c i p l e . The method i s found useful i n f o r e s t r y where data c o l l e c t e d are often unbalanced i n nature. It provides a unique means to incorporate q u a l i t a t i v e as w e l l as quantitative variables i n forest y i e l d analyses. Inventory data f o r three major species — Douglas-fir, spruce, and lodgepole pine were analyzed i n connection with the study of growth and y i e l d of pure (81% or more of the overstory i s of a si n g l e species) and mixed stands i n up to 12 B.C. forest inventory zones i n which they occurred. More than 50% of Douglas-fir, spruce, and lodgepole pine stands occur n a t u r a l l y i n pure stands. If there are any adverse e f f e c t s on the establishment of pure stands, these should have been well r e f l e c t e d q u a n t i t a t i v e l y i n the data provided by the B.C. Forest Service. Estimates of s i t e index from the inventory data might support that pure stands deteriorate s o i l conditions; however, the higher s i t e indices i n mixed stands may be a t t r i b u t e d to the better s i t e conditions when the stands were o r i g i n a l l y established. Mixed con i f e r stands tend to grow more trees per acre than pure or hardwood mixed type stands. Among the three species investigated, Douglas-fir required more growing space than the others. The r e l a t i v e stand density based on basal area per acre also indicates that stand density i s higher i n co n i f e r mixed stands than i n pure or hardwood mixed type stands. - i i i -The mean annual increment i s higher i n hardwood mixed stands than i n pure or conifer mixed ones. But stand age i n hardwood mixed types i s much l e s s . The mean annual basal area increment of conifer mixed stands i s cons i s t e n t l y higher than that of the other two types. Zonal v a r i a t i o n s i n the mean annual basal area growth are apparent. The mean annual volume increment follows a trend s i m i l a r to that of the mean annual basal area increment. Douglas-fir stands growing on the Coast and i n the I n t e r i o r were compared. Mean annual volume growth i s 84.00 cubic feet per acre for the Coast stands and 25.53 cubic feet for the I n t e r i o r stands. The e f f e c t of species composition on net volume y i e l d i s s i g n i f i c a n t i n I n t e r i o r Douglas-fir stands, but non-significant i n Coast Douglas-fir, I n t e r i o r spruce, and I n t e r i o r lodgepole pine stands. That the e f f e c t of forest inventory zones i s highly s i g n i f i c a n t i n the In t e r i o r Douglas-fir, spruce, and lodgepole pine stands j u s t i f i e s the zonation unless adjustments are made for stand density. Interactions for types and zones are s i g n i f i c a n t i n the Coast Douglas-fir, the In t e r i o r spruce, and the I n t e r i o r lodgepole pine stands but are non-s i g n i f i c a n t i n the I n t e r i o r Douglas-fir stands. The difference i n y i e l d i n the I n t e r i o r Douglas-fir stands i s a t t r i b u t a b l e to species composition types and forest inventory zones alone. The establishment of Douglas-fir conifer mixed type stands i n the I n t e r i o r e f f e c t i v e l y increases forest p r o d u c t i v i t y by 21%. - i v -Interpretations of the i n t e r a c t i o n s lead to the conclusion that the advantages of monocultural or m u l t i c u l t u r a l practices cannot be over-generalized. Pure type stands are more productive i n some zones but l e s s i n the others. The same i s true f o r m u l t i c u l t u r a l p r a c t i c e s . Growth of forest trees i s e s s e n t i a l l y site-dependent. Before a d e c i s i o n i s reached on what species composition type to establish,, f o r e s t e r s should c a r e f u l l y investigate the l o c a l s i t e q u a l i t y and past y i e l d h i s t o r y of various f o r e s t types to ensure that the maximum p o t e n t i a l p r o d u c t i v i t y of a p a r t i c u l a r s i t e can be r e a l i z e d . Further analyses to t e s t the hypothesis that no differences i n volume y i e l d e x i s t among three species composition types for stands growing on same s i t e conditions reveal that the e f f e c t s for species types and inventory zones as well as i n t e r a c t i o n s thereof are not s i g n i f i c a n t for Coast Douglas-fir, however, for I n t e r i o r Douglas-fir stands, the e f f e c t of species composition i s s i g n i f i c a n t . I t i s shown that on s i m i l a r s i t e conditions, Douglas-fir co n i f e r mixed stands y i e l d s u b s t a n t i a l l y more than pure or hardwood mixed stands i n the I n t e r i o r . The species composition e f f e c t i s not s i g n i f i c a n t i n I n t e r i o r spruce stands while zonal e f f e c t s and i n t e r a c t i o n s f o r types and zones are s i g n i f i c a n t . In I n t e r i o r lodgepole pine stands, e f f e c t s of composition types, zones, and i n t e r a c t i o n s thereof d i f f e r s i g n i f i c a n t l y . In a l l three species groups investigated, that the e f f e c t s of hardwood mixed type c o n s i s t e n t l y shows negative 'values implies that hardwood mixed type stands are the least d esirable stand composition structure - v -for these species i n the I n t e r i o r . Differences i n volume between pure and mixed type stands r e s u l t p r i m a r i l y from the i n e q u a l i t y i n basal area per acre. The v a r i a b l e s , height x basal area and basal area are most important i n y i e l d table analyses. In addition, stand age, r e l a t i v e basal area, and forest inventory zone are a l l highly s i g n i f i c a n t i n contributing to the v a r i a t i o n s i n volume y i e l d of the Coast Douglas-fir stands. For I n t e r i o r Douglas-fir, the most s i g n i f i c a n t v a r i a b l e s are, i n addition to the above two v a r i a b l e s , stand age, and r e l a t i v e stand density. E f f e c t s of species composition type and forest inventory zones are non-significant. For I n t e r i o r spruce, the prominent variables i n y i e l d table analysis are height x basal area, basal area, species composition, stand age, height, and r e l a t i v e stand density. A l l v a r i a b l e s being equal, pure spruce stands outyield stands of mixed spruce-hardwood and mixed spruce-conifer. The r e s u l t s provide good evidence that establishment of pure spruce stands i s more desirable than of spruce and conifers or hardwood mixed stands. For lodgepole pine, the most s i g n i f i c a n t v a r i a b l e s i n y i e l d tables analysis are height x basal area, species composition types, and forest inventory zones. The high s i g n i f i c a n c e of zonal e f f e c t s suggests that a separate y i e l d table f or lodgepole pine i n each zone i s warranted, unless appropriate adjustments are made for s i t e index and stand density. Y i e l d of pure lodgepole pine stands exceeds those of lodgepole pine co n i f e r mixed type and lodgepole pine hardwood mixed types. Therefore, for high y i e l d s the establishment of pure lodgepole pine type stands i s preferred. - v i -Appl i c a t i o n of these methods to the temporary sample pl o t data has c l e a r l y demonstrated the widespread d i s t r i b u t i o n of pure stands and lack of s u b s t a n t i a l e f f e c t s of monocultures on y i e l d . Nevertheless, the fa c t that higher y i e l d s may r e s u l t from some multicultures should encourage establishment of long term studies of spacing and mixtures of species. - v i i -TABLE OF CONTENTS Page ABSTRACT i i TABLE OF CONTENTS v i i LIST OF TABLES x i LIST OF FIGURES xv LIST OF MAPS x v i i ACKNOWLEDGEMENTS x v i i i 1.0 INTRODUCTION 1 2.0 REVIEW AND BACKGROUND INFORMATION 6 2.1 Analysis of Unbalanced Data 6 2.2 Pure Versus Mixed Stands 19 2.2.1 D e f i n i t i o n 19 2.2.2 S i l v i c u l t u r a l Considerations of Pure and Mixed Stands 20 2.2.3 Comparison of Growth and Y i e l d of Pure and Mixed Stands 25 2.3 Forest Inventory Zones 29 2.3.1 D e f i n i t i o n 29 2.3.2 Regional P r o d u c t i v i t y of Forest Stands. . 31 2.3.3 Regionality and Bi o c l i m a t i c Zones Studies i n Other Regions 36 2.4 Chapter Summary 39 3.0 METHODOLOGY 41 3.1 Linear Model 41 3.2 Normal Equations 43 - v i i i -3.3 Imposing R e s t r i c t i o n s 47 3.4 Absorption Process 48 3.5 Estimation of Constants 50 3.6 P a r t i t i o n i n g the Tot a l Sum of Squares 56 3.7 Least Squares Means, Variances, and Hypothesis Tests 58 3.8 Computer Program 62 3.9 Chapter Summary 65 4.0 DATA BASE 66 4.1 Chapter Summary 69 5.0 RESULTS AND DISCUSSIONS 72 5.1 Occurrence of Douglas-fir-, Spruce-, and Lodgepole Pine-dominated Forest Types i n B r i t i s h Columbia. . 72 5.2 Si t e D e t e r i o r a t i o n and Pure Stands . 76 5.3 Number of Trees Per Acre 79 5.4 Relative Stand Density 81 5.5 Average Stand Age, Mean Annual Height, Basal Area, and Volume Growth 83 5.5.1 Average Stand Age 83 5.5.2 Mean Annual Height Growth 88 5.5.3 Mean Annual Basal Area Growth 92 5.5.4 Mean Annual Volume Growth 97 5.6 Difference i n Growth and Y i e l d Between Coast and In t e r i o r Douglas-fir Stands 97 - i x -5.7 Comparison of Volume Y i e l d by Species Composition Types and Forest Inventory Zones 103 5.7.1 Douglas-fir 104 5.7.2 Spruce 119 5.7.3 Lodgepole Pine 122 5.7.4 Summary of Volume Y i e l d by Species Composition and Inventory Zones 125 5.8 Influence of Species Composition Types on Volume Y i e l d 127 5.8.1 Douglas-fir 136 5.8.2 Spruce 136 5.8.3 Lodgepole Pine 137 5.9 Y i e l d of Douglas-fir and Conifer Mixed Type Stands i n the I n t e r i o r . . , 138 5.10 Y i e l d Table Construction 152 5.10.1 Douglas-fir 162 5.10.2 Spruce 164 5.10.3 Lodgepole Pine 166 5.11 Test of Homogeneity of Variance 167 5.12 Least Squares Analysis 171 5.13 Chapter Summary 174 6.0 SUMMARY AND CONCLUSIONS 177 7.0 LITERATURE CITED 188 APPENDICES 196 1. B.C. Forest Inventory Zones 197 2. Computer Program 201 - X -3. Least Squares Equations 212 3.1 Coast Douglas-fir 212 3.2 I n t e r i o r Douglas-fir 213 3.3 I n t e r i o r Spruce 215 3.4 I n t e r i o r Lodgepole Pine 217 - x i -LIST OF TABLES Table Page 2-1 Y i e l d s of pure hemlock and hemlock-spruce admixed stands (Eidman, 1952) . 27 2- 2 C l a s s i f i c a t i o n of forest regions of B r i t i s h Columbia (Stanek, 1966.) 3.0 3- 1 Western hemlock inventory data c l a s s i f i e d by species types and inventory zones ( F . I . Z . ) 45 3-2 The l e a s t squares equations formulated from the western hemlock data 46 3-3 The reduced set of l e a s t squares equations for the western hemlock data 51 3-4 Matrix inverse to the complete variance-covariance matrix f o r western hemlock data .53 3-5 Estimated constants for western hemlock mean annual growth data 54 3-6 Complete set of estimated constants for western hemlock mean annual growth data 35 3-7 Analysis of variance table f or western hemlock mean annual growth data 59 3- 8 Comparison of the incidence matrix between the program and BMDP general l i n e a r hypothesis program 63 4- 1 The number of sample plo t s by species for data provided by B.C. Forest Service Inventory D i v i s i o n . . .68 4-2 Comparison of species types and forest type groups . . .70 5-1 Numbers of sample p l o t s by species and forest species type 73 5-2 The frequency of sample p l o t s by forest types and forest inventory zones 75 5-3 Averages of s i t e index by species types and forest inventory zones 78 5-4 Averaged numbers of trees per acre i n various species types and forest inventory zones 80 - x i i -0 5-5 Relative stand density f o r various species type and forest inventory zone combinations 82 5-6 Age, mean annual height, basal area, and volume growth by species types and forest inventory zones . . .84 5-7 Mean height of dominant and codominant trees by types and inventory zones 85 5-8 Mean basal area per acre of Douglas-fir, spruce, and lodgepole pine by species types and inventory zones 86 5-9 Mean net volume of Douglas-fir, spruce, and lodgepole pine by species types and inventory zones. . .87 5-10 Comparison of Douglas-fir stands grown on the Coast and i n the I n t e r i o r 102 5-11 Estimated constants for Coast Douglas-fir volume y i e l d . 105 5-12 Estimated constants f o r I n t e r i o r Douglas-fir volume y i e l d 106 5-13 Analysis of variance for the Coast Douglas-fir net volume y i e l d by species types and inventory zones . . . 107 5-14 Analysis of variance f o r the I n t e r i o r Douglas-fir net volume y i e l d by species types and inventory zones . . . 108 5-15 P o t e n t i a l y i e l d of Douglas-fir at age 100 i n B.C. Forest Inventory Zones 109 5-16 Estimated constants f o r I n t e r i o r spruce net volume y i e l d 110 5-17 Analysis of variance f o r net volume y i e l d s of I n t e r i o r spruce stands by species types and inventory zones . . I l l 5-18 P o t e n t i a l y i e l d of I n t e r i o r spruce stands at age 100 by forest inventory zones and types 112 5-19 Estimated constants for lodgepole pine net volume y i e l d by species types and inventory zones 113 5-20 Analysis of variance for lodgepole pine net volume y i e l d by species types and inventory zones 114 - x i i i -5-21 P o t e n t i a l y i e l d s of I n t e r i o r lodgepole pine at age 100 by species types and forest inventory zones . . 115 5-22 Analysis of variance for net volume y i e l d of the Coast Douglas-fir stands adjusted f o r s i t e index and stand age 128 5-23 Estimated constants from volume y i e l d data of Coast Douglas-fir stands 129 5-24 Analysis of variance for net volume y i e l d of the I n t e r i o r Douglas-fir stands adjusted f o r s i t e index and stand age 130 5-25 Estimated constants f o r net volume y i e l d s of I n t e r i o r Douglas-fir stands 131 5-26 Analysis of variance f o r net volume y i e l d s of the I n t e r i o r spruce stands adjusted for s i t e index and stand age 132 5-27 Estimated constants from volume data of the I n t e r i o r spruce stands . 133 5-28 Analysis of variance f o r net volume y i e l d s of I n t e r i o r lodgepole pine stands adjusted f o r s i t e index and stand age 134 5-29 Estimated constants f o r net volume y i e l d s of I n t e r i o r lodgepole pine stands 135 5-30 Comparison of height, number of trees, basal area, and volume per acre f o r pure Douglas-fir and Douglas-fir c o n i f e r mixture stands grown I n t e r i o r on site, class I 139 5-31 Comparison of height, number of trees, basal area, and volume per acre f o r pure Douglas-fir and Douglas-fir conifer mixture stands grown on I n t e r i o r s i t e class II 140 5-32 Comparison of height, number of trees, basal area, and volume per acre f o r pure Douglas-fir and Douglas-fir conifer mixture stands grown on I n t e r i o r s i t e c l a s s I I I 141 5-33 Analysis of variance f o r the net volume y i e l d of Coast Douglas-fir stands 154 5-34 Analysis of variance f o r net volume y i e l d s of I n t e r i o r Douglas-fir stands 155 5-35 Analysis of variance for net volume y i e l d data of I n t e r i o r spruce stands 156 - x i v -5-36 Analysis of variance for net volume y i e l d of I n t e r i o r lodgepole pine stands 157 5-37 Comparison of F-values i n Coast Douglas-fir y i e l d analysis 158 5-38 Comparison of F-values i n I n t e r i o r Douglas-fir y i e l d analysis 159 5-39 Comparison of F-values i n I n t e r i o r spruce y i e l d analysis 160 5-40 Comparison of F-values i n I n t e r i o r lodgepole pine y i e l d a n alysis 161 5-41 Y i e l d functions for Douglas-fir, spruce, and lodgepole pine 165 5-42 Standard deviations of Douglas-fir volume y i e l d data by species types and f o r e s t inventory zones . . . . 168 5-43 Standard deviations of spruce volume y i e l d data by species composition types and inventory zones . . . . 169 5-44 Standard deviations of lodgepole pine volume y i e l d data by species composition types and inventory zones . .170 - XV -LIST OF FIGURES Figure Page 5-1 Comparison of the mean height increments of dominant and codominant trees of Douglas^fir stands by types and zones . 89 5-2 Comparison of the mean annual height increments of dominant and codominant trees by types and zones of I n t e r i o r spruce 90 5-3 Comparison of the mean annual height increments of dominant and codominant trees of lodgepole pine i n I n t e r i o r zones 91 5-4 Comparison of the mean basal area increments of Douglas-fir stands by types and zones 93 5-5 Comparison of the mean basal area increment of spruce stands by types and forest inventory zones 94 5-6 Comparison of the mean annual basal area increments of lodgepole pine stands by types and inventory zones. . .95 5-7 Comparison of the mean volume increments of Douglas-fir by types and zones 98 5-8 Comparison of the mean annual volume increments of spruce by types and zones 99 5-9 Comparison of the mean annual volume increments of lodgepole pine by types and zones • 100 5-10 P o t e n t i a l y i e l d of Douglas-fir at age 100 by types and inventory zones 117 5-11 P o t e n t i a l y i e l d s of I n t e r i o r spruce at age 100 by species types and inventory zones 120 5-12 P o t e n t i a l y i e l d s of I n t e r i o r lodgepole pine stands at age 100 123 5-13 Height growth of I n t e r i o r Douglas-fir on s i t e c l a s s I . . 142 5-14 Height growth of I n t e r i o r Douglas-fir on s i t e c l a s s I I . . 143 5-15 Height growth of I n t e r i o r Douglas-fir on s i t e c l a s s I I I . .144 - x v i -5-16 Basal area growth of I n t e r i o r Douglas-fir on s i t e c l a s s I . .'.146 5-17 Basal area growth of I n t e r i o r Douglas-fir on s i t e c l a s s II.".;. 147 5-18 Basal area growth of I n t e r i o r Douglas-fir on s i t e c l a s s I I I . .148 5-19 Volume growth of I n t e r i o r Douglas-fir on s i t e c l a s s I .149 5-20 Volume growth of I n t e r i o r Douglas-fir on s i t e c l a s s II . . . . .150 5-21 Volume growth of I n t e r i o r Douglas-fir on s i t e Class I II . . . . 151 - x v i i -LIST OF MAPS Map Page 1 Forest Inventory Zones in British Columbia 32 - x v i i i -ACKNOWLEDGEMENT S Without the e f f o r t , encouragement, and guidance of several people, t h i s thesis would not have been possible. I wish to extend my sincere thanks to my supervisor, Dr. A. Kozak (Forestry), for sharing h i s time, wisdom, and experience i n h i s r o l e as an academic advisor. The moral support he gave me throughout the study, h i s guidance, understanding, patience and the a d d i t i o n a l e f f o r t he made to make my studies a warm and rewarding experience are h e a r t i l y acknowledged. I am indebted to Dr. J.H.G. Smith (Forestry) who suggested to undertake the study, provided valuable advice i n the course of data analysis and reviewed co n s t r u c t i v e l y and c r i t i c a l l y the manuscript. Special thanks are extended to my Committee Members: Dr. S. W. Nash (Mathematics), Dr. D.D. Munro (Forestry), and Dr. J.P. Demaerschalk (Forestry) for t h e i r invaluable suggestions and advice. E d i t o r i a l assistance provided by, and some equations derived from Dr. Nash are c o r d i a l l y acknowledged. Grateful thanks go to Dr. Sagary Nokoe, a former fellow graduate student, for h i s help i n t r a n s c r i p t of the o r i g i n a l data. I wish also to thank the B.C. Forest Service Inventory D i v i s i o n and the B.C. Forest Service P r o d u c t i v i t y Committee for providing the data for t h i s study. - xix -F i n a n c i a l assistance was given i n the form of Faculty of Forestry Teaching/Research Grants, the McPhee Forest Fellowship, and the Youth Summer Employment Program of the Province of B.C. I am g r a t e f u l to a l l those who made these possible. Thanks are due to Dr. J.W. Wilson (Forestry) who supported my transfer from Wood Science to Biometrics. I owe my deepest gratitude to my family, my wife Judy, my ch i l d r e n John and Marjorie, who have undergone many s a c r i f i c e s to make i t a l l possible. - 1 -1.0 INTRODUCTION Forest inventories, one of the basic tools of the forest manager to describe a f o r e s t and c o n t r o l managed stands, are c a r r i e d out to assess the quantity and q u a l i t y of forest trees and many c h a r a c t e r i s t i c s of the land area upon which trees grow. It goes without saying that well-designed inventories always contain an accumulation of material providing a s u i t a b l e basis for various s p e c i a l i z e d studies. By appropriate analyses and correct i n t e r p r e t a t i o n s of f o r e s t inventory data one can gain i n s i d e knowledge of stand c h a r a c t e r i s t i c s . H i s t o r i c a l l y , data c o l l e c t e d for growth and y i e l d studies from permanent or temporary p l o t s were analyzed by graphical methods before early t h i s century. With the advances of s t a t i s t i c a l methods, regression analysis has gradually replaced the t r a d i t i o n a l graphical method. Forest mensurationists have found regression analysis an indispensable technique i n growth and y i e l d studies. Indeed, i t i s a powerful t o o l i n e x p l o i t i n g data i f factors of i n t e r e s t are quantitative i n nature e.g. volume, basal area, height, and age. If factors involved are q u a l i t a t i v e i n nature, such as f o r e s t types, inventory zones, the data are conventionally analyzed by the analysis of variance or covariance. However, comparing data from forest inventory surveys to those from well-designed experiments, the former are frequently notorious for t h e i r i r r e g u l a r i t y : the numbers of sample p l o t s surveyed often d i f f e r - 2 -i n main classes as w e l l as i n subclasses. T r a d i t i o n a l analysis of variance methods, i n terms of well-designed experiments, are generally applicable only to balanced data (exceptions are the s p e c i f i e d patterns of L a t i n square designs, balanced incomplete block designs, and d e r i v a t i v e s thereof). Hence for i r r e g u l a r unbalanced f o r e s t inventory data, analysis of variance i n i t s t r a d i t i o n a l framework i s inap p l i c a b l e . Therefore, the f i r s t objective of t h i s study was to search f o r and develop, i f necessary, a methodology of analyzing and t e s t i n g data c o l l e c t e d from forest inventory surveys. The second objective of t h i s thesis was to test the hypothesis that no d i f f e r e n c e i n net volume y i e l d e x i s t s among pure, co n i f e r mixed, and conifer-hardwood mixed type stands. Advantages and d i s -advantages of pure and mixed stands have been the subject of many con t r o v e r s i a l arguments since the l a s t century. The problem of mono- and multi-culture i s not only of academic i n t e r e s t but also of p r a c t i c a l importance i n B r i t i s h Columbia. Y i e l d s of pure and mixed fo r e s t types are often compared on a very l i m i t e d number of p l o t s . In t h i s study, the aforementioned hypotheses w i l l be tested by a large set of data c o l l e c t e d across B.C., which i s e s s e n t i a l to draw a generalized conclusion on the c o n t r o v e r s i a l problem. The t h i r d objective was to determine the s t a t i s t i c a l s i g n i f i c a n c e of any influence of inventory zones on stand y i e l d s other than through difference i n average s i t e and stand density. In B r i t i s h Columbia, forested area (Public Sustained Y i e l d Units and Tree Farm Licences) has been grouped i n t o twelve inventory zones as a basis for - 3 -planning growth and y i e l d and l o s s factor studies. Although zonal volume and,age curves (VAC) have been constructed i n B.C. for various forest types and employed for p r e d i c t i n g volume y i e l d s and determining annual allowable cut, how the y i e l d of a forest type d i f f e r s from one zone to the other has not yet been tested s t a t i s t i c a l l y . Knowledge of the zonal growth changes i s necessary for various reasons. For example, i t may be h e l p f u l f o r managing foresters interested i n assessing the p r o f i t a b i l i t y of f o r e s t c u l t i v a t i o n on d i f f e r e n t s i t e s and i n various c l i m a t i c regions. In the following chapter the background information on the analysis methods f o r unbalanced data, the s i l v i c u l t u r a l and mensurational aspects of pure and mixed type stands, and inventory zones are concisely reviewed. The basic problem tackled i s to develop a s t a t i s t i c a l l y v a l i d method f o r analyzing inventory p l o t s . The strength of inventory plots i s t h e i r large number. Their weakness, however, i s the lack of balance i n classes and lack of representation i n some subclasses. The strength has been gradually recognized by research foresters as i l l u s t r a t e d by an increasing number of studies based on data from inventory p l o t s . The s t a t i s t i c a l methods used i n those studies are mainly regression. A f t e r a long process of t r i a l and error, i t was found that the l e a s t squares analysis i s a procedure which s u i t s to the present objectives best. Therefore, Chapter 3 describes the procedures of the l e a s t squares analysis and the development of a computing program which handles problems with empty c e l l s i n subclasses. - 4 -Sources of the inventory data used i n the study are b r i e f e d i n Chapter 4. In the following chapters, three major B.C. commercial species, Douglas-fir (Pseudotsuga menziesii (Mirb.) Franco), spruce ^ (Picea spp.), and lodgepole pine (Pinus contorta Dougl.) were chosen and analyzed to i l l u s t r a t e the a p p l i c a t i o n of the method developed i n Chapter 3 i n connection with the s i l v i c u l t u r a l and mensurational objectives set f o r t h above. Species other than the three could have been included; however, time and space do not allow f o r extensive study on a l l species presently growing i n B.C. Results and discussions are presented i n Chapter 5 where means of stand c h a r a c t e r i s t i c s i n species composition type and inventory zone combinations are f i r s t l y examined and then the le a s t squares method developed i n Chapter 3 i s applied to the inventory data to asce r t a i n the e f f e c t s of species composition types and forest inventory zones. F i n a l l y a l l stand parameters, q u a l i t a t i v e as w e l l as quantitative, are included i n y i e l d table analyses to assess t h e i r r e l a t i v e importance i n y i e l d table construction. Chapter 6 summarizes the r e s u l t s and conclusions. Admittedly, the only way i n which one can obtain d e f i n i t i v e r e s u l t s to s a t i s f y the l a s t two objectives set f o r t h above i s to e s t a b l i s h plantations i n which species are appropriately tested over a range of spacing and planting configurations. We are forced into studies such as t h i s thesis because data are not a v a i l a b l e but many speculations have been made and there has been much pub l i c concern about the problem of monocultures. I t i s believed that the analyses of inventory data w i l l - 5 -shed some light on the issue in the absence of direct experimental data. - 6 -2.0 REVIEW AND BACKGROUND INFORMATION 2.1 Analysis of Unbalanced Data Forest inventory data contain much information f o r various s p e c i a l i z e d studies. But, when the data are c l a s s i f i e d into classes and subclasses designated f o r a s p e c i f i e d study, the numbers of sample pl o t s (observations) i n main classes and subclasses are often found to d i f f e r . In s t a t i s t i c a l a n a l y s i s , two data structures are generally recognized: balanced (complete) and unbalanced (incomplete). Data where these numbers are the same are known as equal-numbers data, or more frequently, as balanced data. The s t a t i s t i c a l procedure appropriate to the case where the numbers i n various subclasses are equal i s simple, andihas been very f u l l y developed. In contrast, data with unequal numbers of observations i n subclasses, i n c l u d i n g perhaps some that contain no observations at a l l (empty subclasses, or empty c e l l s ) are c a l l e d unequal-numbers data, or usually, unbalanced data, or sometimes "messy data" (Searle, 1971). The e v i l of lack of balance i s most frequently encountered with data r o u t i n e l y c o l l e c t e d as part of currently operative information systems such as, for example, with those providing " o f f i c i a l s t a t i s t i c s " based on censuses and surveys. However, lack of balance also can a r i s e i n data c o l l e c t i n g a c t i v i t i e s which were designed to give complete records as i n balanced experiments ( J e f f e r s , 1965). So frequent i s the occurrence of incomplete data that i t has become one of the more - 7 -important problems i n s t a t i s t i c a l a n a l y sis. While the widespread occurrence of the incomplete data problem may be regarded as a "necessary e v i l " r e a l i s t i c a l l y associated with data c o l l e c t i o n , the l i t e r a t u r e provides ample evidence that s t a t i s t i c i a n s have expended considerable e f f o r t s to "make a v i r t u e of the v i c e " (Hartley and Hocking, 1971). Brandt (1933) was the f i r s t s t a t i s t i c i a n to treat the analysis of variance i n a 2 x S table with disproportionate frequencies. Yates (1934) considered the more general case of a p x q table, and suggested c e r t a i n corrections which appear to be necessary i n Brandt's methods. He proposed two procedures f o r analyzing multiple c l a s s i f i e d unbalanced data: (1) a method of f i t t i n g constants for the cases i n which i n t e r a c t i o n s do not e x i s t ; and (2) a method of weighted squares of means for the cases with i n t e r a c t i o n s present. (1) Method of f i t t i n g constants In the general p x q table the ordinary procedure of the method of l e a s t squares gives the following equations for m, a^, b^ the estimates of y , a., (3 . . - 8 -Leading Term m Nm +N a +N a + + N b +N b + . .. =SS x * x z • z . x x a Z ' Z y • a l N l . m + N l . a l + n l l b l + n12 b2 +-'' = S S y l . a 2 N^m + N ^ + n 2 1 b l + n22 b2 +'• ' = S S y 2 b l N . . I m + n l l a l + n21 a2 + •'• + N . l b l = S S y . l b2 N . 2 m + n 1 2 a l + n 2 2 a 2 + + N.2 b2 = S S y . 2 The r u l e f o r formation i s as follows. Write N equations for the N observed values of the form Y. = y + a. + 8. + e. (2-1) 13k I 3 l j k where e^^y_ i - s a n error term. To form the equation i n which m i s the leading term, sum the squares of the c o e f f i c i e n t s of y i n the N equations and the products of these c o e f f i c i e n t s with the corresponding c o e f f i c i e n t s of a^, (X^J . . . 8^ 5 ••• i - n turn. In t h i s case the c o e f f i c i e n t f o r y i s always unity so the f i r s t sum i s N. The c o e f f i c i e n t of i s unity i n the N^ equations f o r the observations i n a l l a^ classes and zero elsewhere, so that the sum of the products i s N^ , and so on. These - 9 -sums give the c o e f f i c i e n t s of m, a^, a^, ..., b^, b^, ... i n the f i r s t equation. The SS i s the sum of products of each observed y.. value with the c o e f f i c i e n t of y i n the corresponding equation (2-1). The other equations are formed i n the same manner. The whole set of equations gives the values of y, the a's and the B's which make the sum of the squares of the error terms e^yi a minimum. Only p + q - 1 of the above p + q + 1 equations are independent, the equation with leading term m being the same as the sum of a l l the equations with an 'a' as leading terms, and of a l l the equations with a, 'b' as leading term. Methodologies to solve the l i n e a r equations are many. A f t e r the values of the a's, b's and m have been obtained, the reduction i n the sum of squares due to f i t t i n g the constants can be calculated as the sum of the products of each constant with the numerical term of the equation of which i t i s the leading term. Yates (1934) pointed out that the general r u l e applicable to any groups of f i t t e d constants i s to f i n d the part of the sum of squares accounted for by f i t t i n g the constants and deduct from i t the part of the sum of squares accounted f o r by f i t t i n g a l l the constants except those to be tested. (2) Method of weighted squares of mean For a two-way analysis of variance model, when i n t e r a c t i o n i s - 10 -present, Yates (1934) described the method of weighted squares of mean which furnished unbiased estimates of the main e f f e c t s . Assume the model y . ., = y + a. + B. + (aB) . . + e. (2-2) 3 x j k i J xj x j k 2 i=l , 2 , . . . r ; j=l,2,...s; k=l,2,.. .n. .; and e. ., »>_/NID(0,O' ). Define xj i j k y'. and y 1 . as the unweighted row and column means, res p e c t i v e l y , x. . . j • y.. y.. y' . = £ —^- and y' . = E - 1 1 " x.. s . j . . r 3 i On the basis of the assumption of variance homogeneity of the c e l l s , the variance of y! i s x.. Y i i 1 Var(y!^ ) = Var(Z - ^ ) = \ S Var(y ) j s s j 2 2 1 _ - - - 2 _ s 1 •2 . n. . 2 . n. . s J xj s j xj a 2 N. x v, 1 _ 1 y 1 where — = —z h N. 2 . n.. x s x xj (2-3) These weights 1SL w i l l be used i n c a l c u l a t i n g the A main e f f e c t sum of squares. - 11 -where The sums of squares f o r A i s defined as SS A = E (y^ - C ) 2 (2-4) j Z. N. y' . where C = Z- N. x 1 1 2 I t can be shown that — - r EN.(y! -C) has expectation r-1 . x •'x.. x 2 , 1 _ M ,2 0 r ^ l E i a ' i E N. a . 1 x a l = a i " ^w. . X X The sums of squares f o r the B e f f e c t i s defined analogously; SS = E N (y' -D) 2, " 3 • 3 • 3 E N y 1 1 1 j J ' 3' where — = — Z — and D = J r j xj j j 1 2 Also, under the assumed model, — r - E N . (y' . -D) S J- . I • "1 • J - 12 -has expectation E N. 6. ^ 3 3 a 2 + 2 N. B ' 2 , where 8'. = 6. - J v . T s-1 • 3 3 3 3 S N 3 Both main e f f e c t mean squares are tested by comparing them against the error (within) mean square. For c a l c u l a t i o n purposes the following i d e n t i t i e s are u s e f u l : .2 and a N.y- y E N.(y! _ m - C ) 2 = E N ( y j / ^ " ( 2 - 5 ) 1 x . 1 (E N.y'. ) .2 . „ x2 j J -J' 2 E N . (y'. - D) = EN.(y' ) - J . ( 2 - 6 ) i . j . . 1 -3- E N . 3 3 , 3 Following Yates' p u b l i c a t i o n on the analysis of variance for unequal frequencies i n subclasses, Snedecor and Cox ( 1 9 3 5 ) , Nair ( 1 9 4 1 ) , Cochran ( 1 9 4 3 ) , Rao ( 1 9 4 6 ) , Snedecor ( 1 9 5 6 ) , Stevens ( 1 9 4 8 ) , and Henderson ( 1 9 5 3 ) presented analysis of variance procedures f o r unbalanced c l a s s i f i c a t i o n s . Day and Fisher ( 1 9 3 7 ) , Wilks ( 1 9 3 8 ) , Hazel ( 1 9 4 6 ) , Rao ( 1 9 4 6 ) , Henderson ( 1 9 5 3 ) , Das ( 1 9 5 3 ) , Federer ( 1 9 5 5 , 1 9 5 7 ) , and Hartley ( 1 9 6 7 ) , discussed covariance analysis f o r unbalanced c l a s s i f i c a t i o n s . B a r t l e t t ( 1 9 3 6 , 1 9 3 7 ) , Quenouille ( 1 9 4 8 ) , Federer ( 1 9 5 5 ) and Outhwarte and Ruther ( 1 9 5 5 ) , among others, have discussed - 13 -the use of a dummy covariable to remove the e f f e c t of disproportion i n p a r t i c u l a r unbalanced c l a s s i f i c a t i o n s . Gosslee and Lucas (1965) investigated the analysis of variance of disproportionate data when i n t e r a c t i o n i s present. According to the authors, the analysis of variance of disproportionate data can be computed by several methods which f a l l i nto two major groupings, namely, addit i v e sums of squares method and l e a s t squares methods. Additive sums of squares methods produce sums of squares for e f f e c t s and int e r a c t i o n s that add to the sum of squares representing v a r i a t i o n among the c e l l means. But each sum of squares does not, i n general, follow a chi-square d i s t r i b u t i o n under the n u l l hypothesis nor a noncentral chi-square d i s t r i b u t i o n under the n u l l hypothesis. Thus, the r a t i o s of mean squares used to test hypotheses do not, i n general, follow the F d i s t r i b u t i o n . Least squares methods define tests of hypotheses which follow an F d i s t r i b u t i o n under the n u l l hypothesis, and thus have actual l e v e l s of s i g n i f i c a n c e equal to t h e i r normal l e v e l s . Also each test s t a t i s t i c follows a noncentral F d i s t r i b u t i o n under the a l t e r n a t i v e hypothesis so that the power of the test can be obtained r e a d i l y . However, the computed sums of squares are generally not addit i v e . Examples of l e a s t squares methods are the method of f i t t i n g constants (Yates, 1934), the method of weighted squares of means (Yates, 1934; Snedecor and Cox, 1935), and the modified method of weighted squares of means (Kramer, 1955). Examples of the addit i v e - 14 -sums of squares methods are the method of unweighted means (Yates, 1934; Snedecor and Cox, 1935) and the method of expected subclasses numbers (Snedecor, 1934). When i n t e r a c t i o n e x i s t s i n disproportionate data the sums of squares calculated to test main e f f e c t depend on the parameters i n the model. Thus, when i n t e r a c t i o n i s assumed to exi s t a method should be chosen which uses appropriate r e s t r i c t i o n s even though i t may not be the simplest or the most e f f i c i e n t method (Gosslee and Lucas, 1965). In the absence of in t e r a c t i o n s a method of analysis can be chosen on the basis of i t s e f f i c i e n c y alone (Finney, 1948). Bancroft (1968) treated the analysis of variance and covariance f o r unequal subclass frequencies i n great d e t a i l . He examined exact methods and approximate methods for two-way and three-way c l a s s i f i c a t i o n s , and extended the exact and approximate methods to n-way c l a s s i f i c a t i o n with unequal and disproportionate subclass numbers. Computational procedures for unbalanced disproportionate data are more complicated than f o r the balanced data, so that p r i o r to the present era of computers, approximate methods such as unweighted means were generally preferred to exact l e a s t squares methods. With the appearance of high-speed e l e c t r o n i c computers, the analyses of data with unequal numbers of observation have been f a c i l i t a t e d . The le a s t squares methods have been generally favored because of the advantages previously mentioned as w e l l as s i m p l i c i t y i n matrix expression. - 15 -The a p p l i c a t i o n of l e a s t squares theory to the analysis of variance has long been known to s t a t i s t i c i a n s under the name " l i n e a r s t a t i s t i c a l models" or "general l i n e a r hypothesis". Most researchers i n f o r e s t r y are f a m i l i a r with regression analysis which applies the l e a s t squares p r i n c i p l e to obtain unbiased estimates of parameters i n question. In regression analysis the v a r i a b l e s concerned are p r i m a r i l y continuous or quantitative v a r i a b l e s . The l e a s t squares theory applies equally well to "counter" or q u a l i t a t i v e v a r i a b l e s such as the analysis of variance. In f a c t , under the normal d i s t r i b u t i o n assumption, the r e s u l t from maximum l i k e l i h o o d method i s i d e n t i c a l to that from the l e a s t squares methods. In d e f i n i n g the analysis of variance, regression analysis and the analysis of covariance, Scheffe (1959) said the analysis of variance i s a body of s t a t i s t i c a l methods of analysis measurements assumed to be of the structure, Y i = X l i 3 l + X 2 i 3 2 + ••• + X p l e p + e i (i=l>2,...,n) (2-7) where the c o e f f i c i e n t s a r e integers usually 0 or 1. In the analysis of variance the ^ j - j ^ a r e t n e values of "counter" v a r i a b l e s or " i n d i c a t o r " v a r i a b l e s which r e f e r to the presence or absence of the e f f e c t 3 i n the conditions under which the observations are taken; j X.. i s the number of times 8. occurs i n the i - t h observation, and t h i s i s u s u a l l y 0 or 1. If the {X..} are values taken on i n the observations - 16 -not by counter v a r i a b l e but by a continuous v a r i a b l e , we say we have a case of regression analysis. If there are some {X } of both kinds, we have an analysis of covariance. The analysis of l i n e a r models by l e a s t squares theory, including point and i n t e r v a l estimation and tests of hypotheses, i s well-supported by rigorous mathematical theory (Scheff£, 1959; Kendall and Stuart, 1966; G r a y b i l l , 1961; Rao, 1965; Searle, 1971). Searle (1971) began with a simple explanation of regression and proceeded to multiple regression, to give a u n i f i e d treatment for t e s t i n g a general l i n e a r hypothesis. He introduced models not of f u l l rank by discussing regression on dummy (0,1) variables and showing i t s equivalence to l i n e a r models. He then treated the non-full-rank model i n great d e t a i l , u t i l i z i n g generalized inverse matrices and giving a u n i f i e d procedure for evaluating any testable l i n e a r hypothesis. Three methods to compute the analysis of variance from view-points of multiple regression have been examined by Freeman and J e f f e r s (1962): (1) the d i r e c t s o l u t i o n of normal equations or use of formulae based on them, (2) the i t e r a t i v e s o l u t i o n using the language of vector space, and (3) the approach through the variance-covariance matrix. A l l of the methods require, at l e a s t i m p l i c i t l y , the s o l u t i o n of a set of normal equations. The sums of squares i n the general unequal numbers analysis of variance f o r an n-way or n-factor c l a s s i f i c a t i o n can be obtained i n general terms from standard regression theory. However, the computing formulas using the usual normal equation and s o l u t i o n procedures become - 17 -unmanageable even with the la r g e s t computers whenever the number of factors and/or the number of l e v e l s become moderately large. With the use of the calculus of f a c t o r i a l s developed by Kurkjian and Zelen (1962) , Federer and Zelen (1966) were able to set f o r t h r e l a t i v e l y simple computational procedures f o r obtaining sums of squares i n the analysis of variance f o r any e f f e c t while eliminating a l l other e f f e c t s . For sums of squares of i n t e r a c t i o n to the order two, the computations may be performed on an ordinary c a l c u l a t o r . As l i n e a r models have become more widely u t i l i z e d , they have simultaneously become les s w e l l s p e c i f i e d . Urquhard et a l . (1973) reconsidered the d e f i n i t i o n of a l i n e a r model with s p e c i a l reference to i t s as s o c i a t i o n with the experimental content. They proposed an alternate kind of l i n e a r model which i s c l o s e l y i d e n t i f i e d with the experimental context. The means of the populations sampled i n the experimental context serve as i t s parameters. The proposal was echoed by Hocking and Speed (1975) who found i t eliminated much of the confusion, p a r t i c u l a r l y i n the area of hypothesis t e s t i n g with unbalanced data. There are many procedures to obtain estimates f o r parameters of l i n e a r models. Rao (1965) and Searle (1971) used a generalized inverse method to solve the simultaneous equation system. A generalized inverse i s characterized by the following properties (Searle, 1971): - 18 -(1) when G i s a generalized inverse of X'X. then G' i s also a generalized inverse of X'X; (2) y&X'X = X; i . e . GX' i s a general inverse of X; (3) XGX' i s inv a r i a n t to G; and (A) XGX' i s symmetric, whether G_ i s or not. The generalized inverse technique provides a convenient procedure i n the analysis of variance for l i n e a r models. But, i f the in v e s t i g a t o r i s interested i n the estimate of a p a r t i c u l a r e f f e c t , the question of e s t i m a b i l i t y a r i s e s (Searle, 1971). A unique s o l u t i o n to the l e a s t squares equation cannot be obtained u n t i l they are reduced i n number to equal the number of degrees of freedom. Although numerous r e s t r i c t i o n s may be imposed i n order to accomplish t h i s , the r e s t r i c t i o n that the constants for the main e f f e c t s sum to zero over each row and column i s probably the most s a t i s f a c t o r y (Harvey, 1960). The l i n e a r mathematical model i t s e l f suggests t h i s set of r e s t r i c t i o n s since the main e f f e c t s , i n t e r a c t i o n e f f e c t s , and the error term e.., are expressed as deviations from the i l k mean y. Using the r e s t r i c t i o n which requires the s e t t i n g of one of the constants i n each set of main e f f e c t s equal to zero (say, a and b ) P q suggests that the (ab). , the (ab) . and the (ab) i n t e r a c t i o n constants i q P3 pq could be also set equal to zero, thereby, allowing the d e l e t i o n of a l l these equations. Although a unique s o l u t i o n to the equation can be obtained when t h i s i s done, the estimates for the constants are not s a t i s f a c t o r y (Harvey, 1960). - 19 -Freeman and J e f f e r s (1962) described an algorithm to estimate means and standard errors i n the analysis of non-orthogonal experiments. Mulet and Walsh (1974) reported a generalized inverse algorithm. Recently, Hemmerle (1974, 1976) developed a method for handling a non-orthogonal analysis ofvariance residual.and expectation operators. In p r a c t i c e , the general l i n e a r hypothesis of the Biomedical Computing Programs (Dixon, 1971) enjoys wide popularity. The program estimates the parameters by imposing r e s t r i c t i o n s on main e f f e c t s and i n t e r a c t i o n s . That the program handles no empty c e l l s i n subclasses i s one of i t s severe drawbacks. 2.2 Pure Versus Mixed Stands 2.2.1 D e f i n i t i o n A forest can be s i l v i c u l t u r a l l y c l a s s i f i e d as a "pure" or a "mixed" stand according to i t s species composition. The former consists of a s i n g l e tree species of s i l v i c u l t u r a l or economic importance, whereas the l a t t e r has two or more species. The t h e o r e t i c a l concept i s straightforward, but i n p r a c t i c e the d e f i n i t i o n of a " s i n g l e " species or a "pure" stand i s vague. Stands are customarily designated as "pure" when 90 percent or more of the canopy i s of a s i n g l e species (Baker, 1950). Tourney and Korstian (1947) maintained that i n estimating timber for commercial purposes a stand i s usually c l a s s i f i e d as pure when 80 - 20 -percent or more of the overstory i s of a s i n g l e species and forms p r a c t i c a l l y a l l of the commercial product. The Society of American Foresters defined a pure stand as "of a f o r e s t , crop or stand, composed p r i n c i p a l l y of one species: by convention, generally to the extent of greater than 80% of the species, based on numbers, basal area or volume", and mixed stands are "of a f o r e s t , crop'or stand, composed of two or more species by convention, generally to the extent of le s s than 20% of species other than the p r i n c i p a l one, based on numbers, basal area or volume" (Ford-Robertson, 1971). According to the B.C.Forest Service, i n pure forest types major species must be 81 percent or greater by volume of l i v e trees i n the main stand. Veterans are not included i n t h i s c a l c u l a t i o n . In mixed stand a second species must be 20 percent or more (B.C. Forest Service, Forest Inventory D i v i s i o n C l a s s i f i c a t i o n and Sampling Manual, 1972). 2.2.2 S i l v i c u l t u r a l Considerations of Pure and Mixed Stands The l i t e r a t u r e on s i l v i c u l t u r a l aspects of pure (monoculture) and mixed stands (multicultures) i s too voluminous to be thoroughly reviewed here. A comprehensive review of l i t e r a t u r e published within the l a s t two decades has been prepared by Haddock (1974). - 21 -In Europe, I t has been reported that large clearcuts and establishment of even-aged monocultures deteriorated forest habits to such a degree that the extent of changes of the tree species composition sometimes requires a r t i f i c i a l amelioration (Klinka and Scholz, 1974). Others observed that calamities experienced i n even-aged coniferous monocultures were caused by various damaging agents which would ulti m a t e l y r e s u l t i n d i s c o n t i n u i t y of wood production, losses of investment, and y i e l d s that could not be sustained. In s p i t e of calamities experienced i n some pure stands, H i l e y (1959) advocated f o r B r i t i s h f o rests the use of c e r t a i n crop tending, p r a c t i c e s used i n South A f r i c a monocultures, e s p e c i a l l y those of pines. Keets (1966) argued the need for research to provide the s o i l data required to grow successive crops of Pinus patula i n South A f r i c a . Keeves (1966) and Whyte (1973) have presented evidence of a decline i n growth of second r o t a t i o n crops of Monterey pine (Pinus  r a d i a t a D. Don) on c e r t a i n s o i l s i n South A u s t r a l i a and New Zealand, res p e c t i v e l y . Bunn (1967) reviewed the work of Keeves (1966) and others and concluded that few quantitative studies have been made. In some instances, suspected reasons f o r observed decline i n p r o d u c t i v i t y include the marginal character of shallow s o i l s , l o s s of nutrients through cropping, or l o s s of nutrients by means of slash burning. Florence (1967) provided an excellent bibliography and gave some valuable suggestions f o r research, with emphasis on edaphic f a c t o r s . He focused on the s i t u a t i o n faced i n South A u s t r a l i a where the decline - 22 -i n p r o d u c t i v i t y of Monterey pine on c e r t a i n s o i l s i s of a serious magnitude. He observed that a monoculture pla n t a t i o n i s regarded as a simple system and i n comparison to natural communities, lacks t h e i r dynamic vegetational and s o i l b i o l o g i c a l processes necessary f o r s t a b i l i t y . He expressed the need f o r greater understanding of these processes i n natural forests i n order to put the plant a t i o n problems into better perspective. He also reviewed the influence of species and species mixtures on s o i l s , l i t t e r decomposition, and nutrien t return, and changes i n the r e l a t i o n s h i p i n f o r e s t - s o i l m i c r o f l o r a . In s p i t e of the controversies about monoculture i n Europe and other countries, monocultural p r a c t i c e s are prevalent i n North America. In 1950, Baker said " i t i s only human nature to take a chance and gamble on high p r o f i t s rather than to play safe when i t i s obvious that i t w i l l cost money to do so". Smith (1962) i n discussing the concept of monocultures, pointed out that "departures from the basic idea of growing southern pines i n pure, even-aged stand are, however, l i k e l y to be uncommon. If the existence of such stands of these species was of i t s e l f i n v i t a t i o n to calamity, t h i s would have become obvious i n the e x i s t i n g natural stands of t h i s s o r t " . The point that pure stands are susceptible to attacks by insects and fungi has been argued by f o r e s t e r s . Cases can be c i t e d where losses r e s u l t i n g from insect and diseases attack have occurred to both mixed and pure stands ( B a s k e r v i l l e , 1965; McSwain, 1970). - 23 -The s i t u a t i o n regarding monocultures i n B r i t i s h Columbia i s a complicated one. In B.C. c l e a r - c u t t i n g has developed as a consequence of several converging influences. Among these are increased u t i l i z a t i o n as a r e s u l t of improved sawmilling techniques and the construction of a number of pulp m i l l s , r e s t r i c t e d road development, r i s i n g p r i c e s , increased mechanization, shortage of labor and the opinion on the part of some foresters that planting could e f f e c t i v e l y solve the regeneration problem created by large area c l e a r - c u t t i n g and }sometimes necessary tslash burning (Haddock, 1974). On the other hand, the a b i l i t y of lodgepole pine and many other species to e x i s t n a t u r a l l y i n extremely dense stands over vast areas i s w e l l known (Horton, 1956). Thousands of trees per acre may become established a f t e r f i r e and the high s u r v i v a l rate can lead to extreme stagnation. Lodgepole pine i s not exacting i n i t s s o i l requirements, grows on a wide v a r i e t y of s o i l types, and reaches i t s best developments on moist but well-drained sandy or grave l l y loam (Duffy, 1964). Natural monocultures of lodgepole pine occur widely i n the Northern I n t e r i o r on areas associated with major natural disturbance ( f i r e , wind throw, e t c . ) . In addition, lodgepole pine may occur extensively on lower q u a l i t y s i t e s , sometimes as an uneven-aged open stand as an "edaphic" climax (Haddock, 1974). One way or the other, the pure stands w i l l continue to e x i s t i n B r i t i s h Columbia as well as elsewhere i n North America. Advocates for multicultures often claim that mixed stands are superior to pure ones based on the assumptions that pure stands: - 24 -(1) may f a i l to u t i l i z e the site f u l l y ; (2) make excessively heavy demands on s o i l nutrients; (3) may cause a slow deterioration of upper s o i l layers by fostering the formation of some sort of acid raw humus; and (4) are susceptible to attacks by insects and fungi. In managing commercial forests, the productivity of stands is of major concern. The above-mentioned disadvantages w i l l reflect on the growth and yield of pure stands directly or indirectly. Therefore, to some extent the potential impact of monoculture can be quantitatively assessed by comparing natural yields of pure and mixed stand types. In his recent report, Haddock (1974) has extensively reviewed the environmental impact of monocultures in second-growth forests in the Interior of British Columbia. He defended monocultural practices by il l u s t r a t i n g nutrient cycling and insect attacks in even-aged pure stands. In his report, quantitative evidence i s lacking, however. Scattered published information on comparisons of growth and yield of pure and mixed stands can be found. But most examples are restricted to local conditions. In view of the increasing concern expressed by foresters about the impact of monoculture practices in British Columbia, there i s a pressing need to provide information on the effects of pure stands on growth and yield in comparison to that of mixed stands. - 25 -2.2.3 Comparison of Growth and Y i e l d of Pure and Mixed Stands Turnbull (1963) gave a comprehensive review of l i t e r a t u r e on growth and y i e l d , with emphasis on stand structure, for pure and mixed stands up to 1960. Recent studies on t h i s subject seems to s h i f t from comparison of y i e l d to factors contributing to the difference i n growth and y i e l d between pure and mixed stands. In Europe, Vacovski (1967) presented data showing that, on fresh or fresh to moist s i t e s with r i c h to very r i c h s o i l s , the performance of pure beech was s l i g h t l y superior to mixed stands or to pure oak. On fresh s i t e s with r i c h s o i l s , the mixtures were more productive than pure stands of e i t h e r species. On fresh s i t e s with r i c h to moderately r i c h s o i l s , stand p r o d u c t i v i t y increased with increasing proportion of oak. Hegre and Langhammer (1967) described the establishment and treatments of a mixed stand established i n 1947 (2+0 aspen and 2+2 spruce, planted as alternate t r e e s ) , and gave graphs and tables showing heights, d.b.h., volume, increment, spacing, etc. i n period up to 1966. Comparison with increments from y i e l d tables for the two species i n pure stands shows that the mixture has developed very s a t i s f a c t o r i l y , but i t seems l i k e l y that s i t e q u a l i t y i s better than usual f o r good spruce and aspen stands. In 1971, Langhammer discussed the advantages and problems associated with mixed coniferous and coniferous/broad-leaved stands, p a r t i c u l a r l y as regards height increases and forms of various species - 26 -and their relation to the success of mixtures, effect on the s o i l , and the effect on growth of thinning or of removing one of the species in a mixture. From studies on 13 t r i a l plots in Czechoslovakia over a period of 30 years, Zakopal and Mares (1968) presented evidence to support the growing of Picea abies in mixture with oak on Fageto- Quercetum sites. The admixtures of spruce not only result in increased volume production but also in a substantial improvement in oak value production. The best results are obtained with an admixture of 30% spruce in the top story. Tarrant (1961) investigated stand development and s o i l f e r t i l i t y in a Douglas-fir—red alder (Alnus rubra Bong.) plantation in the Wind River Experimental Forest in south western Washington and found that planted Douglas-fir mixed with "off site" alder in comparison with Douglas-fir growing alone showed no real difference in either height or diameter growth. Tarrant also found that when f i r and alder measurements are combined, more than twice the cubic volume of wood was produced in the mixed stand than in the pure Douglas-fir stand. The beneficial results of mixture of alder with Douglas-fir continued to 1976 when the experiment was effectively destroyed by bear damage. In natural stands, Eidmann (1952) reported some very instructive figures on the Sitka spruce (Picea sitchensis (Bong.) Carr.) — western hemlock (Tsuga heterophylla Sarg.) stands of the Cascade Head Experiment Forest. Data from four plots are given in Table 2-1. One plot had approximately the same number of trees of both Plot Table 2-1. Yields of pure hemlock and hemlock-spruce admixed stands (Eidmann, 1952). Species Number Averaqe Average Volume of Trees Height Diameter per acre m cm cu. m/ha HEMLOCK 193 45.1 m 48.8 651 SPRUCE 173 50.3 68.8 1110 Total Volume cu. m/ha 1761 HEMLOCK SPHUCE 161 289 32.9 45. 1 39.3 61.7 274 13 43 1617 HEMLOCK SPRUCE 383 77 42. 1 47.6 48.5 65.5 1212 4 18 1630 HEMLOCK 781 39.6 40.1 1686 1686 - 28 -species, a second plot had a higher percentage of spruce, a third plot was mainly hemlock and the last plot was pure hemlock. Irrespective of the individual share of each of the species, the total volume for both i s approximately the same in a l l four cases. Eidmann argued that this was one example out of thousands which have shown that one species can replace another and that they can then grow together and prosper with advantage to the whole stand. From these impressions, the author concluded that mixed conifer stands are preferable to pure stands in Europe. Several Nelder plantations for spacing t r i a l s and for assessing species competition effects have been established at the Research Forest of the University of British Columbia (Walters and Smith, 1973). Four species, Douglas-fir, western hemlock, western redcedar (Thuja plicata Donn.), and Sitka spruce were planted. Results on the species effects are not yet available because the stands are in their early developing stages, but Douglas-fir has survived and grown best to-date'. Based on data collected from forest inventories, Smith (1976) compared average yields of pure and mixed species stands for Douglas-fir, "balsam" f i r , lodgepole pine, and spruce in Interior, B.C. The averages indicate a difference in yield between pure and mixed stands in Douglas-fir and "balsam" f i r , but no differences in lodgepole pine and spruce were observed. Based on the same data, Smith (1977) developed yield estimates and preliminary guides for spacing and thinning British Columbia forests. - 29 -2.3 Forest Inventory Zones 2.3.1 D e f i n i t i o n The factors d i r e c t l y responsible f o r the d i s t r i b u t i o n of species, the forest types, and growth of forest stands are, i n the f i r s t place, the geographic l o c a t i o n and topography and the r e l a t e d climate as well as the geology and b i o t i c f a c t o r s , which are i n t e r -related with the s o i l forming processes and the r e s u l t i n g s o i l s . The c l a s s i f i c a t i o n of f o r e s t types i n B r i t i s h Columbia has been the subject of several s p e c i a l i z e d studies. Table 2.2 compares the c l a s s i f i c a t i o n of forest regions of B.C. by Whitford and Craig (1918), Rowe (1959), Chapman, Turner, Farley and Ruggles (1956), and Krajina (1965) (from Stanek, 1966). The zonal c l a s s i f i c a t i o n system i n t h i s study d i f f e r s s l i g h t l y from Krajina's which c l a s s i f i e s B.C. f o r e s t s on an e c o l o g i c a l basis. The B.C. Forest Service Inventory D i v i s i o n has grouped B.C. forest areas into twelve "forest inventory zones". The need f o r t h i s zoning system has been expressed i n the "Growth Manual" (B.C.F.S., 1966). "Because of the large number of Crown Units presently i n existence, i t i s obvious that we cannot hope, i n a reasonable period, to e s t a b l i s h enough permanent p l o t s to provide r e l i a b l e growth estimates for a l l growth types, age classes, and s i t e s i n each un i t . Therefore, we have combined a l l of the e x i s t i n g Units into twelve groups, three on the Coast and nine i n the I n t e r i o r . " Table 2-2. Classifications of forest regions of British Columbia (Stanek, 1966) . Whitford and Craig (1918) FOREST TYPES Krajina (1965) ZONES | SUBZONES Rove (1959) SECTIONS — Chapman et al. (1956) BIOTIC REGION COASTAL BELT I. PACIFIC COASTAL MESOTHERMAL FOREST REGION COAST FOREST REGION Gulf Islands and Puget Sound Lovland Douglas-fir - Western Red Cedar Coastal Douglas-fir -JSarryJDak.j:. Douglas-fir(drler) Madrono - Douglas-fir "(wetter) JeSSAJL?;!*.*!* Western Hemlock (drier) CI C2 "C3~ [ Strait of Georgia | Southern Pacific Coast Western Red Cedar - Bestern Hemlock Coastal j Northern Pacific Coast Western Hemlock - Sitka Spruce Western Hemlock - Amabilis Fir Western Hemlock Pacific Silver Fir "-Wes tern Hemlock' ' (wetter) C4 Queen Charlotte Islands Coast Forest Subalplne II. PACIFIC COASTAL SUBALPINE REGION SUBALPINE FOREST REGION Subalplne Forest Mountain | Subalpine ForeBt (lower) Hemlock PSubalpine Parkland (upper) SA3 Coastal Subalpine INTERIOR TREELESS AND SEMITREELESS IV. CORDILLERAN COLD STEPPE AND SAVANNA FOREST REGION MONTANE FOREST REGION Of lovoos Ari A Sage Brush— Artemisia Tridentata Grass and Semlopen * Agropyron Splcatum Ponderosa Pine - j Bunch Grass (drier) G Grassland Yellow Pine (or Ponderosa Pine) Bunch Grass Ponderosa Pine (wetter) Ml Ponderosa Pine - Douglas-fir Dry Forest Interior Douglas-fir III. CANADIAN CORDILLERAN FOREST REGION M2 Central Douglas-fir Interior Douglas-fir _Pine Grass (drier) False Bojcwood (wetter) Columbia Forest INTERIOR WET BELT Interior Western Hemlock Western Larch (drier) COLUMBIA FOREST REGION Douglas-fir - Western Larch j Int. West. Red Cedar - West. Hemlock West. Red Cedar - Engelmann Spruce Western Hemlock - Alpine Fir Western Hemlock (wetter) CL1 CL2 Southern Columbia Northern Columbia INTERIOR TREELESS AND SEMITREELESS Cariboo Aspen Lodgepole Pine -Douglas-fir Parkland Subzones not proposed • MONTANE FOREST REGION Cariboo Parklands Crass and Semiopen -Agropyron Splcatum M3 M4 M5 Northern Aspen Montane Transition Douglas-fir - Lodgepole Pine SPRUCE - ALPINE FIR OF THE INTERIOR PLATEAUS AND MOUNTAIN REGIONS V. CANADIAN CORDILLERAN SUBALPINE FOREST REGION SUBALPINE FOREST REGION Subalpine Forest Engelmann Spruce - Alpine Fir or Lodgepole Pine Subalplne Engelmann Spruce - j Subzones not proposed Subalpine Fir ' i SA1 | East Slope Rockies SA2 jInterior Subalpine White Spruce - Alpine Fir VI. CANADIAN BOREAL FOREST REGION BOREAL FOREST REGION Peace River Parklands and Boreal Forest Sub-boreal Spruce (mainly white spruce vlth some Engelmann spruce) Subzones not proposed B17 B18 -fil9~ ~ B23" "B24" "B2S B26'j Aspen Grove a) Mixed Wood . b) Hay River a) b") c) Lower-Nbrth-Upper Foothills a) Lower Mackenzie Boreal White and Black Spruce Upper Liard Stikine Plateau a)bawson b)Centr. Yukon c)East.Yukon Treeless (teste Land VII. ALPINE TUNDRA REGION Tundra Alpine - Arctic . a) Coastal b) Interior Subzones - 31 -Map 1 shows geographically the boundary of the twelve forest inventory zones i n B.C.; the zonal d e f i n i t i o n (Public Sustained Y i e l d Units and Tree Farm Licenses) i s given i n Appendix 1. The zonal c l a s s i f i c a t i o n i s e s s e n t i a l l y geographical. 2.3.2 Regional P r o d u c t i v i t y of Forest Stands Most forest growth studies, i n B.C. and elsewhere, have been based on materials c o l l e c t e d over r e l a t i v e l y l i m i t e d areas. Comparisons of the r e s u l t s of growth and y i e l d s i n various regions or b i o c l i m a t i c zones n a t u r a l l y show the changes i n growth on t r a n s i t i o n from favorable to l e s s favorable c l i m a t i c conditions. T r a d i t i o n a l l y the zonal differences i n growth and y i e l d have been a t t r i b u t e d by foresters to differences i n s i t e q u a l i t y . Indeed, f o r e s t s i t e has been defined by the Society of American Foresters as "an area, considered as to i t s ec o l o g i c a l factors with reference to capacity to provide forest or other vegetation", or as "the combination of b i o t i c , c l i m a t i c and s o i l conditions of an area" (Ford-Robertson, 1971). Differences i n stand y i e l d caused by various geographical locations have been w e l l recognized by f o r e s t e r s . In compiling normal y i e l d tables f o r even-aged stands of Sitka spruce and western hemlock, Meyer (1937) used the basic data c o l l e c t e d over the e n t i r e coastal range of the species, extending from Southern Oregon through Washington and B r i t i s h Columbia to Southeastern Alaska and analyzed the data without Map 1. Forest inventory zones i n B r i t i s h Columbia. Numbered Forest inventory Zones are defined i n the Appendix I. - 33 -regard to species composition or geographical l o c a t i o n . Meyer subsequently found that these factors a f f e c t both s i t e index and y i e l d and provided supplementary tables f o r correcting y i e l d s . Further analyses by Barnes (1962) on the data used by Meyer indicated that many y i e l d v a r i a b l e s d i f f e r e d markedly among regions. Stands of the same age and s i t e index had much, smaller average diameters i n Alaska and B r i t i s h Columbia than i n Oregon and Washington. The diffe r e n c e i n average diameter amounted to about 20 percent, and the corresponding diffe r e n c e i n volume was even greater. These differences are at t r i b u t e d to denser stands at early ages i n the northern l a t i t u d e s , which resulted i n more severe competition and e a r l i e r crown closure. Cooler and wetter summers to north probably contribute to e a r l i e r and more complete restocking. The average height of stands of the same age and s i t e index also d i f f e r s . Average heights i n Oregon, Washington, and B r i t i s h Columbia were about equal, but average height i n Alaska was about 15 percent l e s s . This indicates a larger number of shorter trees i n the subdominant part of the Alaska stands. Barnes (1962) concluded "some recognition of geographical l o c a t i o n , seemed necessary". Therefore, separate r e g i o n a l l y y i e l d tables for (1) Oregon—Washington, (2) B r i t i s h Columbia, and (3) Alaska were constructed for stands i n which 40 percent or more of t o t a l basal area was i n western hemlock. If there are v a r i a t i o n s i n growth and y i e l d within the Northwestern P a c i f i c Coast, they must also occur within B r i t i s h Columbia. - 34 -Andody (1968) described graphical methods for determining yield, mean annual increment, and age of culmination of mean annual increment with s p e c i a l reference to the Central I n t e r i o r region of B r i t i s h Columbia. Her analysis demonstrated average differences i n the order of 25 to 30 percent between zonal and l o c a l curve averages. She suggested the d e s i r a b i l i t y of constructing curves for twelve forest inventory zones rather than the seven inventory zones then used by the B.C. Forest Service. Many e f f o r t s have been made to estimate growth and y i e l d of forest stands across B r i t i s h Columbia. The f i r s t comprehensive empirical y i e l d tables were published i n 1958. They were developed to give an i n d i c a t i o n of the growth rate of any forest stand i n the province. The basis for the tables was 11,500 samples each c o n s i s t i n g of four or more p l o t s . Seven inventory zones, eleven type groups, four s i t e classes, four dbh l i m i t s , and gross cubic-foot volumes i n l i v i n g merchantable trees of a l l commercial species were involved. Tables were constructed by averaging volumes for each age c l a s s and p l o t t i n g them over age. Balanced freehand curves were drawn for each dbh l i m i t . The empirical y i e l d tables prepared by the B.C. Forest Service were revised l a t e r ( F l i g g , 1960). Ad d i t i o n a l samples were included, type groups were expanded to sixteen, the basis of sampling by zone and growth type and s i t e was reported, and d e t a i l s of c a l c u l a t i o n c l a r i f i e d . In addition to the y i e l d tables, the B.C. Forest Service also addressed i t s e l f to the problem of l o c a l i z i n g estimates of y i e l d . - 35 -Young (1966) I l l u s t r a t e d an approach i n which volumes were plotted over age f o r two u t i l i z a t i o n l i m i t s . Y i e l d s were expressed as average volumes per acre i n cubic feet (net allowing f o r decay only) to close u t i l i z a t i o n stands. Volume over age curves (VAC's) were numbered and revised or withdrawn as new information became a v a i l a b l e from the continuing inventory program. Each VAC was drawn to give a zonal summary f o r a type group or to represent the y i e l d that could be expected from a p a r t i c u l a r Public Sustained Y i e l d Unit. The VAC's were used to define the point at which maximum mean annual increment i n volume occurred and theyears corresponding to t h i s set the r o t a t i o n . The volume over age curves of the B.C. Forest Service were drawn freehand and the p o s s i b i l i t y of combining t h e i r curves was not being tested. However, Richmond (1969) found no differences i n many Public Sustained Y i e l d Units when the rough classes of good, medium, poor, and low were replaced with approximate s i t e index. He also found that several types could be grouped. On the other hand, Smith (1973) analyzed data on volumes by age classes from VAC summaries for type groups within seven major inventory zones and reported that highly s i g n i f i c a n t and important differences i n intercept and regression c o e f f i c i e n t s resulted i n large v a r i a t i o n s i n standard errors of estimate among the s t r a t a studied. When using only zone, type group, and s i t e c l a s s , large differences remained among zones and between coast and i n t e r i o r regions. He also showed that use of zonal curves improved estimates within species groups. - 36 -In the same study, Smith (1973) investigated volume y i e l d of I n t e r i o r lodgepole pine for 22 separate sustained y i e l d units and concluded that there were large and important differences from unit to unit i n volume per acre for a given age and s t r a t a . These r e s u l t s confirm and extend the conclusions of Andody (1968). In discussing s o i l , s i t e , and land c l a s s i f i c a t i o n , Rowe (1962) indicated that a geographic s t r a t i f i c a t i o n of land i s needed to provide a framework within which r e l i a b l e s o i l - s i t e and growth r e l a t i o n s h i p s can be established. The old z o n a l i t y concept i n s o i l c l a s s i f i c a t i o n , regardless of i t s t h e o r e t i c a l shortcomings, was a step i n the r i g h t d i r e c t i o n so f a r as s o i l use i s concerned. 2.3.3 Regionality and B i o c l i m a t i c Zone Studies i n Other Regions In Europe, r e f o r e s t a t i o n , mainly with exotic tree species, has been c a r r i e d out i n the B r i t i s h I s l e s f o r some time. To f a c i l i t a t e the s e l e c t i o n of tree species, F a i r b a i r n (1968) divided Great B r i t a i n into zones on the basis of the length of the growing season and p r e c i p i t a t i o n . Sweden has been divided into s i x growth regions, and a proposal has been made for t h e i r further d i v i s i o n into sub-regions on the basis of the r a d i a l growth revealed by the material obtained from nationa l f o r e s t inventories (Hagberg, 1959). - 37 -In Finland, f o r the purposes of forest taxation, the country was divided into 9 forest taxation regions and these again into 5 s i t e q u a l i t y or tax classes. A so-called mean volume un i t for taxation has been determined f o r each tax clas s of each forest taxation region on the basis of the national forest inventories (Koivisto, 1971). Lukkala (1938) divided Finland into 5 c l i m a t i c regions f o r f o r e s t drainage purposes. Heikurainen (1959, 1973) on the basis of the annual increment estimated from material c o l l e c t e d from swamp forests i n various parts of Finland, also divided the country into zones which d i f f e r s l i g h t l y from those proposed by Lukkala. Some e f f o r t s have been made i n Europe to predict regional forest p r o d u c t i v i t y by c l i m a t i c coefficients.. Week (1955) i n h i s book "Forest Increment and Y i e l d " proposed the composite c l i m a t i c index N n Z - 60 T + 10 x 92 x 100 i n which N represented p r e c i p i t a t i o n , T the mean temperature during the period May to July, n the number of days on which there was at le a s t 0.1 mm of r a i n , and Z the number of f r o s t - f r e e days i n the year. When average annual increment (dry-weight) for each of the 25 growth-districts was compared with the corresponding v c l i m a t i c index, a c o r r e l a t i o n of 0.743 +0.09 was obtained. - 38 -Paterson (1956) has attempted s i m i l a r c o r r e l a t i o n s on a more ambitious i n t e r c o n t i n e n t a l scale, using data on p r o d u c t i v i t y of mature, undisturbed high f o r e s t , which he c a l l e d " i d e a l s i t e c l a s s " . Paterson's c l i m a t i c index incorporated s t a t i s t i c s f o r mean temperature of the hottest and the coldest months, mean annual p r e c i p i t a t i o n , p r e c i p i t a t i o n e ffectiveness, and a s o l a r r a d i a t i o n f a c t o r . The logarithm of t h i s index p l o t t e d against " i d e a l s i t e c l a s s " was reported to be a s t r a i g h t l i n e . However, Week (1957) quite properly pointed out that p r o d u c t i v i t y should have been expressed i n terms of dry-weight production, and also that wood-volume comprised only a part of t o t a l production. K o i v i s t o (1971) derived volume increment functions and forest growth i n Finland to obtain a precise idea of the changes i n forest growth from the south coast of Finland to the Northern timber l i n e , and to d i v i d e the country on t h i s basis into homogeneous growth regions. He concluded that although the growth of forest i n the f i r s t place depends on climate and the f e r t i l i t y of s o i l , v a r i a t i o n s i n r e l a t i v e increment were better explained by more accurately measurable secondary growth f a c t o r s , the most important being the age of the stand. Depending on the species, age accounted for 69 to 75 percent of the logarithmic variance. According to K o i v i s t o (1971), the e f f e c t i v e temperature sum seemed to be the most promising c l i m a t i c f a c t o r . The length of thermal growing season, however, was the v a r i a b l e selected for the function i n stands dominated by pine or spruce, and l a t i t u d e for the function - 3 9 -i n birch-dominated stands. Spruce and b i r c h seem to react to c l i m a t i c changes more r e a d i l y than does pine, for the c l i m a t i c factor was the fourth i n sequence to be included i n the function of pine stands, but the t h i r d i n the functions of the other three species. K o i v i s t o (1971) also observed that from one growth region to another, the mean volume increment during the r o t a t i o n of maximum volume increment f a l l s by about 24 percent. If the increment of the southernmost region i s given a r a t i n g of 100, the mean wood production ca p a c i t i e s of the growth regions follow the sequence 100 — 82 — 56 — 36. The wood production capacity, therefore, does not decline r e c t i l i n e a r l y towards the northerly regions. The decline i s slow at the beginning, and i t s rate increases as i t nears the timber l i n e . 2.4 Chapter Summary In t h i s chapter publications concerned with s t a t i s t i c a l methods for the analysis of experimental data with unequal c e l l frequencies, s i l v i c u l t u r a l and mensurational problems associated with mono- and m u l t i c u l t u r a l practices i n f o r e s t r y , and regional p r o d u c t i v i t y of forest stands were separately reviewed. The l e a s t squares theory which i s widely recognized i n regression analysis has been applied to the analysis of unbalanced data since 1934. Techniques involving the employment of l e a s t squares theory i n the analysis of unbalanced data are more complicated than i n regression analysis because the matrix - 40 -formulated i n the former i s not of f u l l rank. Methods to compute unbiased estimates f o r parameters have been proposed; however, computational procedures to obtain unique estimates for data with empty c e l l s which are common i n f o r e s t inventory data are lacking. S i l v i c u l t u r a l and mensurational problems with regard to pure and mixed stands have been studied. S i l v i c u l t u r a l studies on t h i s subject are often lacking i n quantitative support while mensurational in v e s t i g a t i o n s are frequently r e s t r i c t e d to very l i m i t e d experimental p l o t s . The need to subdivide the forest area i n B.C. into "Forest Inventory Zones" was explained. L i t e r a t u r e on zonal and regional p r o d u c t i v i t y of forest stands i n B.C. as well as i n other regions were discussed. - 41 -3.0 METHODOLOGY 3.1 Linear Model The complete model f or a two-way c l a s s i f i c a t i o n with unequal observations i n the subclasses i s Y... = u + a. + B. + (aB). . + e... (3-1) xjk I J i j i j k for i = l , . . . . , r J 1,...«,s k l,....,n #^< where Y . = the k-th observation i n the i - t h a clas s and j - t h 8 c l a s s ; y = the o v e r a l l mean with equal subclass numbers; = e f f e c t of the i - t h a cl a s s ; Bj = e f f e c t of the j - t h B cl a s s ; (aB)^j = e f f e c t of the i j - t h aB class a f t e r the average e f f e c t s of a and B have been removed these are the i n d i v i d u a l i n t e r a c t i o n e f f e c t s expressed as deviations from the mean u + a, + g. i j ; 2 e.., = random errors, assumed to be N(0,a ). i j k The l i n e a r model (3-1) can be expressed : i n matrix notation: - 42 -Y = Xb + e (3-2) where Y i s Nxl vector of observations Y; b i s pxl vector of parameters; X i s an NxP matrix of known values ( i n most cases O's and l ' s ) and e i s an Nxl of random terms; e can be defined as e = y - E(y). (3-3) The vector b i n Y = Xb + e i s a vector of parameters; i t i s the vector of a l l elements of the model. In the above model the vector b w i l l have b' = {u.o^.o^,... ,a r,3 1,8 2,... ,3 s,a6 n,a6 1 2,... ,aB l g, a8 2 1,a3 2 2,. •. >a&2s' ''' > a e r i > a ^ r 2 " * * ' ^ r s * as i t s elements. It includes the termy, the terms representing species e f f e c t a,the terms representing Forest Inventory Zone e f f e c t s 3, and the terms representing i n t e r a c t i o n e f f e c t s between speies a and inventory zones 8-For r species and s zones, i t can have as many as p = l + r + s + rs elements. If there are empty subclasses, p i s equal to p = l + r + s + r s - m where m i s the number of empty c e l l s . - 43 -The matrix X i n (3-2) i s c a l l e d the incidence matrix, or more commonly, the design matrix, because the presence of the O's and l ' s throughout i t s elements representing the incidence of terms of the model among the observations. 3.2 Normal Equation The normal equation corresponding to the model (3-2) Y = Xb + e can be derived by the le a s t squares p r i n c i p l e s . The expected value 2 and variance of the e vector are 0 and a I (I i s an i d e n t i t y matrix), res p e c t i v e l y ; that i s , E(e) = 0 (3-4) Var(e) = E {e - E(e)} {e - E(e)} * = E(ee') = a 2 I . (3-5) The least squares estimate of b i s derived by minimizing the sum of squares of the observations from t h e i r expected values. The expectation of Eq. (3-2) equals E(y) = E(Xb) + E(e) = Xb, (3-6) - 44 -since E(e) = 0. The sum of squares of the e vector i s as follows, e'e = (Y - E(y))'(Y - E(y)) = (Y - Xb) ' (Y - Xb) = Y'Y - 2b'X'Y + b'X'Xb. (3-7) Choosing as the le a s t squares estimate b that value of b which minimizes e'e involves d i f f e r e n t i a t i n g e'e with respect to the elements of b. Equating 6(e'e)/6b to zero and w r i t i n g the r e s u l t i n g equations i n terms of b, the normal equations X'Xb = X'Y (3-8) are derived. The p r i n c i p l e used to derive the normal equations f o r the lea s t squares analysis i s i d e n t i c a l to that to derive the normal equations i n m ultiple regression a n a l y s i s . The differ e n c e between these two methods l i e s i n the fac t that the elements i n the X matrix are eit h e r continuous or d i s c r e t e constants i n the regression analysis while they are eit h e r 1 or 0 i n the l e a s t squares analysis. Table 3-2 shows the le a s t squares equation f o r the fo r e s t inventory data c l a s s i f i e d by type group and forest inventory zones i n Table 3-1. I t can be seen from t h i s Table 3-2 that the r e s u l t i n g matrix of X'X i s symmetrical. The diagonal elements of the matrix i n d i c a t e the numbers of observations contributing to the e f f e c t being considered. Table 3-1. Western hemlock i n v e n t o r y data c l a s s i f i e d by s p e c i e s types and inv e n t o r y zones(F. I. Z.) (Unit: c u b i c f e e t / a c r e / y e a r ) F. I. Z SPECIES TYPES a H+ F H + C H + B H+S H + h A SOBTOTAL 1. ' 2 3 4 5 6 1 NO. PLOT 8 12 7 11 6 2 46 MEAN 58.43 65. 91 81 .85 77.79 69. 73 39. 93 69. 24 2 NO. PLOT 26 16 14 18 2 3 79 MEAN 121.52 79.91 73. 35 109.73 89. 11 56. 46 98. 58 3 NO. PLOT 10 22 19 3 1 3 58 MEAN 49.27 39.01 45. 16 54.48 80. 28 29. 00 43. 79 7 NO. PLOT 26 17 35 4 5 1 88 MEAN 30. 39 22 .94 27. 9 1 17.82 42. 95 41.33 28. 23 'OTAL PLOT 70 67 75 36 14 9 271 MEAN 70. 14 49.52 45. 80 85.1 5 63. 69 41.95 42. 95 Remarks:H: pure ] tiemlock t ype; H+F: hemlock and f i r mixed type H+C: hemlock and cedar mixed type; H+B: hemlock and balsam mixed type; H+S: hemlock and spruce mixed type; H+HA: hemlock and hardwood mixed type. Table 3-2. The least squares equations fcriulated fcon the western healoci data S? 1 S? 2 SP 3 S? U SP 5 SP 6 ! 2 1 rz 2 ?Z 3 t ! I S211 SZ1 2 SZ13 SZ1« SZ21 SZ22 SZ23 SZ20 SZ31 SZ32 SZ33 SZ31 SZ41 SZ02 SZ43 SZ»» SZ51 SZ52 SZS3 SZ50 SZ61 SZ62 SZ63 S Z 6 0 7C1C-8 -,1 " 7 ' 3 6 9 \ 6 7? " 03 8 26 10 26 12 16 22 17 7 14 19 35 11 18 3 « 6 2 1 5 2 3 3 1 15557.C 26 10 26 8 26 10 26 57 67 12 16 22 17 ,, ~ ,_ " " 12 16 22 17 ,,,, f , 1* 7 3 i " 19 35 7 ,„ io « 3= 36 11 1 6 3 a. 1= 11 6 2 1 5 9 5 2 3 3 1 e6 3 12 7 11 6 2 06 8 1 2 7 . 79 26 16 18 13 2 3 7S 26 Ifi 10 22 9 126 17 35 « S 1 8 26 26 10 1C 26 26 12 12 16 16 •22 22. 19 19 35 35 11 11 13 18 3 3 " 4 6 6 2 2 1 1 S 5 2 2 3 3 3 3 1 1 30 35 le3e.E3 306?.. 55 6 2 1 5 891.525 2 3 3 1 377.578 6 2 3155.15 2 3 7737.71 1 3 2535.80 5 1 2U56.35 567.sci 3155.eS 352.72.3 7 90.2 22 7 90.8 33 1275.50 858.258 365.507 572.565 1026.93 868.071 • 976.861 355.til 71.2710 . e16.385 ! 178.210 50. 2 = 00 210.75s 79.6760 165.373 86.S960 1 S1.333C 0.100225E.07 Absorption processes Step - .+ + + + + -1 - + + + - + + + - + + + - + + + - + + + - + + + - + + + Step 2 - - - + + + + + + + + + + + + + + Remarks: SPl=species type 1; FZl=forest inventory zone 1; and SZll = i n t e r a c t i o n term f o r SP1 and FZ1. Other symbols are s i m i l a r l y defined. - 47 -At the r i g h t hand side of the l e a s t squares equations are products of XY, which are t o t a l s and subtotals of classes and subclasses. The l a s t column of the table i s the t o t a l unadjusted sum . . -of squares, that i s X'Y. A simultaneous consideration of the le a s t squares equations i n Table 3-2 w i l l produce unbiased l e a s t squares estimates b for the e f f e c t s i n question, which are free from entanglement. 3.3 Imposing R e s t r i c t i o n s Unfortunately, one cannot solve the equations formulated above without further manipulation of the matrix, because the matrix i t s e l f i s not of f u l l rank. Examining the c o e f f i c i e n t s of the equations i n Table 3-2 reveals that some equations are l i n e a r combination of the others. The general mean U, for example, i s a l i n e a r combination of the equations for species e f f e c t s , or a cl a s s ; i t also i s a l i n e a r combination f o r growth zone e f f e c t s , or 3 c l a s s . Again the equation f o r any one i n the species or growth zone e f f e c t i s also a l i n e a r combination of the i n t e r a c t i o n e f f e c t equations that contain the p a r t i c u l a r e f f e c t . The species 3 e f f e c t i n Table 3-2, f o r example, i s a l i n e a r combination of the equation SZ31, SZ32, SZ33, and SZ34. In f a c t , there are p le a s t squares equations i n the system, but a c t u a l l y only q equations are independent, that i s , q = 1 + (r - 1) + (s - 1) + (r - 1) (s - 1) - m = r s - m - 48 -where m, as previously defined, i s the number of empty c e l l s . In Table 3-2, m equals to 0. A unique s o l u t i o n to the l e a s t squares equations can be obtained only i f they are reduced i n number to the number of degrees of freedom, that i s , q. Although numerous r e s t r i c t i o n s can be imposed i n order to accomplish t h i s , the r e s t r i c t i o n that the constants for the main e f f e c t s sum to zero within a set and that the constants for the (a8).. sum to zero over each row and over each column i s the most s a t i s f a c t o r y (Harvey, 1960). The l i n e a r mathematical model (3-1) i t s e l f suggests t h i s set of r e s t r i c t i o n s since the e f f e c t s of a^ , 8..» (aB)^  and the e are expressed as deviations from the mean. 3.4 Absorption Process The method of obtaining estimates of the e f f e c t s as deviations from the mean i s to impose the r e s t r i c t i o n on the least squares equations that E a. =0, E 8. =0, E(aB).. = E(aB).. =0. Because of these i 1 j J i 1 3 j 1 J r e s t r i c t i o n s , the e f f e c t of a l e v e l within a class can be derived from the e f f e c t s of the rest of the l e v e l s within t h i s class and thus eliminated. The same p r i n c i p l e applies.to equations for i n t e r a c t i o n e f f e c t s as well as main e f f e c t s . In p r a c t i c e , the absorption process i s c a r r i e d out by a number of subtractions and additions within the c o e f f i c i e n t matrix and the r i g h t hand side members. When t h i s i s done the c o e f f i c i e n t s of one of the a^ , say a^ , must be subtracted for the c o e f f i c i e n t s of the other a.'s. The subtractions are required within the a., B.5 E(aB).. and i i J ± i l E(aB)..« A f t e r these changes have been made by column the same procedure j 1 J - 49 -i s followed by row with the modified c o e f f i c i e n t s f o r the (aB)^  equations and f o r the r i g h t hand side members. In Table 3-2, the bottom part, shows a p r a c t i c a l example of the technique of the absorption process. Since the model involves main e f f e c t s and primary i n t e r a c t i o n e f f e c t s , two absorption steps are needed. In Step 1, equations f o r SP1 and FZ1 e f f e c t s are selected to be absorbed and eliminated as indicated by a "-" sign. This i s to 6 A 4 A ^ s a t i s f y the r e s t r i c t i o n s E SP. = 0 and E FZ. = 0. Because SP1 and i-1 3=1 3 FZ1 are elected to be absorbed, t h e i r i n t e r a c t i o n s also have to absorbed by the equations f o r the e f f e c t s within the classes. The negative signs under the estimate SZ11, SZ21, SZ31, SZ41, SZ51, SZ61, SZ12, SZ13, and SZ14 i n Steps 1 and 2 of Table 3-2 demonstrate the imposition of the r e s t r i c t i o n s E (<xB) . . = E (aB) . . = 0 . 1 3 • 13 i 3 The actual absorption procedures are c a r r i e d out by sub-t r a c t i n g the c o e f f i c i e n t s of the columns with "-" sign from the c o e f f i c i e n t s of the columns with "+" sign within a c l a s s . For the species e f f e c t s , f o r example, the c o e f f i c i e n t s i n column 2 are subtracted and then added to those i n columns 3 to 7; the c o e f f i c i e n t s i n column 8 are subtracted and then added to those i n 9, 10, and 11; and column 12 from columns 13, 14 and 15, etc. The second step of the absorption process i s done by subtracting the c o e f f i c i e n t s i n column f o r the estimate SZ12 and then added to columns f o r the estimates SZ22, SZ32, SZ42, SZ52, and SZ62. S i m i l a r l y , the subtractions are c a r r i e d out for columns /v SZ13 and SZ14. - 50 -There i s no r e s t r i c t i o n on s e l e c t i n g the within a class to be absorbed and subsequently eliminated. However, i t has to be kept i n mind that e f f e c t s with empty c e l l s should not be selected. A reduced c o e f f i c i e n t matrix i s thus obtained a f t e r completing the addition and subtraction procedures on column and row and eliminating the columns and rows with negative sign i n Steps 1 and 2 of the absorption process. The f a c t that the off-diagonal elements of reduced c o e f f i c i e n t matrix must be symmetrical about the main diagonal provides a check for computational correctness. Table 3-3 shows the reduced c o e f f i c i e n t matrix derived from the least squares equations i n Table 3-2 using the absorption processes prescribed on the bottom of Table 3-2. I t shows only the t r i a n g u l a r elements. If the numbers of observations i n subclasses are a l l equal, the off-diagonal elements i n the reduced c o e f f i c i e n t matrix w i l l be a l l zero; i n other words, the experimental data are orthogonal. 3.5 Estimation of Constants From the reduced set of the l e a s t squares equations, a unique s o l u t i o n f o r the e f f e c t s can be r e a d i l y derived. Of the methods employed to solve the simultaneous equations, the matrix inversion i s most s a t i s f a c t o r y . Table 3-3. The reduced set of least squares equations for the western 1EAN SP 2 SP 3 SP 4 SP 5 SP 6 FZ 2 271 -3 137 5 70 145 -34 70 70 106 -56 70 70 70 84 -61 70 70 70 70 79 33 -14 -11 -11 -22 -17 125 12 8 10 -10 -7 -1 46 42 -13 10 -25 -19 -19 46 -14 22 18 18 13 18 -6 8 12 2 2 • 2 . 2 4 -13 23 18 18 18 18 4 -1 1 18 25 18 18 19 -13 10 2 14 2 2 2 -1 10 18 ' 46 18 18 18 -1 - 11 18 18 25 18 18 -5 -10 2 2 -6 2 2 3 -25 18 18 11 18 18 3 -22 18 18 18 14 / 18 -26 -7 2 2 2 -3 2 -2 -19 18 18 18 17 18 - 2 -17 18 18 18 18 19 -29 -1 2 2 2 2 3 -6 -19 18 18 18 18 . 17 -6 3 F2 4 SZ22 SZ23 SZ24 SZ32 SZ33 104 46 134 4 4 62 16 4 20 52 4 -5 20 20 63 -1 -1 34 8 8 55 8 -1 8 18 8 15 44 -1 a 8 8 34 15 15 3 3 34 8 8 • 34 8 -4 3 8 18 8 8 18 3 -19 8 8 34 8 8 -2 -2 34 8 8 34 8 -11 -2 8 18 8 8 18 -2 -23 8 8 34 8 8 -6 -6 34 8 8 34 8 -13 -6 e 18 8 8 18 -6 -31 8 8 34 e 8 SZ42 SZ43 SZ44 SZ52 SZ53 SZ54 76 8 8 63 19 32 34 19 19 49 8 34 8 8 42 8 8 13 8 14 25 34 8 8 34 8 8 34 8 14 34 14 8 45 8 . 8 8 18 8 8 18 8 34 3 8 34 8 8 34 SZ63 SZ64 VOLtf«E 15997.0 - 1592. 17 -1474.97 -1844.24 -4013. 17 -4532. 22 4602.56 -645.349 -70C.800 -2204. 39 42.0930 -723.797 -2233.CE 259.784 81.0750 -1572.49 -717.529 -1107. 19 -2932. 23 -363.427 -526.452 39 -2602. 55 10 23 -18.2020 10 10 37 -361. 364 U i 0.144229E+C7 Remarks: Symbols SP2. FZ2, and SZ22 etc. are defined in Table 3-2. - 52 -Equation (3-8) suggests that the solu t i o n of estimate vector b i s b = (X'X) 1 X'Y (3-9) which i s a w e l l recognized formula f o r deriving regression c o e f f i c i e n t s i n regression analysis. In Equation (3-9), X'X i s the c o e f f i c i e n t matrix, the t r i a n g u l a r portion of Table 3-3 and X'Y the r i g h t hand side vector on Table 3-3. The unique s o l u t i o n f o r b vector i s r e a d i l y obtained by mul t i p l y i n g X'Y , the r i g h t hand side member, on the l e f t by the inverse of X'X matrix. The inverse of X'X matrix i s shown i n Table 3-4 and the estimated b vector i n Table 3-5. The complete set of estimated c o e f f i c i e n t s , as shown i n Table 3-6 i s derived by reversing the r e s t r i c t i o n s on the e f f e c t s previously imposed. The estimate of SP1, fo r example, i s SP1 = - E SP. = 6.3912. i=2 The inverse covariance matrix shown i n Table 3-4 needs further explanation. The magnitudes of the off-diagonal elements i n the matrix r e f l e c t the confounding among e f f e c t s being considered. In the s p e c i a l case that a l l numbers of observation i n subclasses are equal, the off-diagonal elements w i l l be zero, then the t r a d i t i o n a l analysis of variance can be used. T a b l e 3 - 4 . H a t r l x Inverse to the complete vnrlance-covarlance matrix f o r w e s t e r n h e m l o c k d a t a . R O 1 K E A N C.97407ZE-02 R O 3 S P 7 -C.7I 2539E-C2 0.2016206-01 R O V . 4 S P 3 -0.666271E-C2 C.4057386-02 0.220)276-01 R O W ; S P 4 -0.213e6«E-C2 -C.4666706-03 -0.939330E-0.3 0.4C149CE-01 ROW 6 S P 5 / 0.97G372E-02 -0.1230916-01 -0.1278176-01 -0.I73CS86-0I 0.8791856-01 ROW 7 SP 6 0. 12E287E-01 -0.I54341E -01 -0.159067E -01 -0.2C43C66 -01 -0.322732E -01 0.1C0018 RCW 9 f ? ? -0.237C73E-C2 0.2369576 -C2 0.2268916-02 -0.291652E -02 0.3755626 -02 -0.630983E -02 0.2448076 -01 ROW 10 FZ 3 c. 32CE95F-02 -0.3920346 -02 -C.4C9396E -02 O.307787E -02 0.1901336 -01 -0.1188956 -01 -0.1057896 -01 0.356401E- 01 ROW II F M 0. 12C273E-C2 -0.1357086 -02 -0.3090266 -02 0.1611666 -02 -0.123138E -01 0.1789456 -01 -0.8572726 -02 -0.141524E- 01 0.316776F -01 ROW 17 SZ22 a. 2365576-02 -0.2375376 -C2 -0.2267136 -C2 0.29176(6 -02 -0.3736456 -02 0.631099E -02 -0.1666706 -01 0.868617E-•02 O.612290E -07 0.5?7353E-01 ROV If S223 - 0 . 3920346-02 0.3634016 -03 0.4805376-02 -0.2366496 -02 -0.IR3019E -01 0.126009E -01 0.8686176- -02 -0.292469E- 01 0.124128E -01 - 0 . 1815C1E-01 0.6121296 -01 ROW 19 SZ24 -0 . 1357CEE-02 0.5853416 -03 0.3244616 -02 -0.1457336 -02 0.1246826 -01 -0.1T740IE -01 0.612290E -02 0.124128E- 01 -O.241203F -01 - 0 . 1F372CE-CI -0.2111086 -01 0.6165686-01 RCW 2 1 J73 2 0.726R91E-C2 -0.226775E -02 -0.277799E-02 0.3018346-02 -0.3657806 -02 0.6411656 -02 -0.1545036 -01 0.8487786- 02 0.748406E -02 0. 7t3f.c6E-C2 -0.6595006 -02 -0.5034246-02 0.6060236 -01 pnw 2 2 S733 - 0 . 4C935BE-02 0.4805376 -02 -0.33I14E6 -03 -0.2192856 -02 -0.1812826 -01 0.1277456 -01 0.84877BE -02 -0.281761E- 01 0.138469E -01 - o . 6595CCE-C2 0.2I7E29F -CI -C. 1210746-01 -0.1894366- -01 0.6545606 -01 RCW 23 SZ34 - 0 . 309076E-C2 0.3244616 -02 -0.634 7406 -02 0.2756556 -03 0.1420146 -01 -0.160070E -01 0.740406E -07 0.13B46TE-•01 -0.76U.B 7E -01 - 0 . 503424E-CZ -0.121C74E -01 0.1866146 -01 -0.1292746 -01 -0.1537426 -01 0.534635E -01 RCW ?'. S742 - 0 . 7916 57F-C2 0.2917686 -02 0.3018346-02 -0.235I97E -01 0.152763E -0 2 O . l I5971E -01 -0.1224906 -01 0.1 977306- 02 0.344330F -07 o . 443515F-C2 -0.0452526 -04 -C.9934826 -03 0.3218636 -02 O. l 138616 -0 3 -0.2 35464E -02 0.734075E -01 P O 71 S743 0 . 30776 7F-C2 -0.736<49F -02 -0.2192856 -C2 0.2B3 563E -01 -0.253CC1F -01 0.5602686 -02 0.197730E -02 -0.Z60224F-•03 -0.755I09F -07 -0. B45752F-04 -0.613258F -02 0.4290696 -02 0.1 I3861F -03 -0.720375E -0? 0.2B5654E -02 -0.449855E -0 1 0.177159 " O 7 7 S74 4 0 . 16 1 1 BOE-0? -0.145753F -02 0.275655E -C3 0.12461 IE -01 0.949523E -02 -0.207091F -01 0.344330E -02 -0.755109E-•0? - 0 . 3 197776 -07 -0. 9'»3 ' in?E-C3 0.429C69E -02 -0.431507E -02 -0.2354646 -02 0.2B5654E -02 -0.226674C -02 -0.290904E -01 -0.B09664E- 01 0.145369 R C W 75 S752 0. 375<:ft2E-0? -0.375P45E -02 -C .3«578C6 -C2 0.1527636 -02 0.3I84B3E -02 0.492094E -02 0. 36f.304E -01 -0 . 324 766E-•0 1 -0.1 149-iCE -C? -0 . 0. 343694E -01 0.359932E-02 -0.4566086 -01 0.34567BE -01 0.223816F. -02 -0.48862 IE -0 1 0.4107R3E- 01 0.677B97E -07 c . 2f E<;25 3C S7! 3 0 . 190133F-01 -0.183019E -01 -0.1812826 -01 -O.253001E -01 0.920578E -01 -0.103327E -01 -0.324766E -01 0.671377E-•01 -0.164037E -01 c . 343f.94E-Cl -0.735309E -01 0 .1814776 -01 0.3456786 -01 -0.746017E -01 0.167086E -01 0.4107B3E -01 -0 . 10251 B 0.331066E -01 - c . 1B2P01 C.4467M R O W 31 5 254 - 0 . 12313PE-01 0.1246e2E -CI 0.1420146-01 0.9499236 -02 -0.432417E -01 -0.67B338E -02 -0.114950E -0? -0.164032E- 01 0.44835OE -02 0 . 359937F-0? 0.1B 1477E -01 -0.1199086 -01 0.2238I6E -02 0.167C86E -01 -0.994246E -02 0.627892E -02 0.33I066E-•01 -0.379139E -01 - 0 . 474E16F-CI -0.136375 C.176072 P O 33 S762 - 0 . - 0 . -c. 63C5e3F-C2 336FC2F-0I Bfc977cE-Cl 0.631C59F--0.3477P3E C.376649E -02 -02 -01 0.6411656-02 C.26A632E-01 C.34|35«6-CI 0.1 I597IF-01 -0.348969F-01 0.22567C 0.4920546 -0.3279456 -02 -02 -0.370929E 0.255020F -01 -01 0.75B665F -0.7809B2E--01 -01 0.537061E-0.323IO3E- 02 02 -0.744 1 3 E 0.79547 BE -01 -01 R C W 34 SJE3 - 0 . - 0 . c. ROW 11PP5SE-01 3 '.7783f-C? 37<i e<.9E-C 1 35 S764 0. 1260C9F-01 -0.211004E-01 -0.117465 C.127745E-01 0.205737F-01 C.4938936-01 0. 560266E-02 -0.327945E-02 -0.314123E-01 - 0 . 103327E--0.221711E 0.237C29 -01 -01 -0.315I3JE 0.191392E -01 -01 0.537061E-0.323103E--02 -02 0. 147077E--0.50J870E-01 01 -O.I 8833 7C 0.355377E -01 -01 0 . 0 . 0 . 17B945E-01 2tt<: 32E-C1 34l35«r-Cl -O.177401E-0.205733F-C.491B93F--01 - CI -CI -O.I6007C6--C.ei78256--C.l 1C3B6 -01 •01 -0.20709IF-01 0.25502CE-01 -0 . 14051 7 -0.67833BE-O2 0.I91392F-01 -C.I46C9 7 0. 775916E--0.797341F 0.4557 39 -01 -01 -0.244 1 34E-o.2954?nn--01 -01 -0.1B8337F-0.355377E-01 0 1 0.747757E-- 0 . 1 027 1 1 -01 Table 3- 5. Estimated constants f o r western hemlock mean annual qcowth data (Unit: c u b i c f e e t / a c r e year) SPECIES TYPES MEAN 58.51C8 F.I.Z. 2 29.8350 3 -8. 9767 7 -27.9539 SP 2 SP 3 SP 4 SP 5 SP 6 •6. 56956 -1.44166 6. 44312 1 2. 0059 -16.8290 INTERACTIONS 1.86749 -13.5517 14.9447 -11.2467 -15.0592 •3. 95092 -2. 93073 -1. 4992 18.7401 -3.70639 •1.05164 -1.20493 -19.1815 0.38802 27.6050 Remarks: SP 2: hemlock anl f i r mixed; SP 3:hemlock and cedar mixed; SP 4: hemlock and balsam mixed; SP 5: hemlock and spruce m i x e i ; SP 6: hemlock and hardwood mixed. Table 3-6. Complete set of estimated c o n s t a n t s f o r western hemlock mean annual qrowth data (Unit; c u b i c f e e t / a c r e / y e a r ) MEAN SP 1 58.5108 6.3912 F.I.Z. 1 7.0957 -13.5726 2 29.8350 26.7804 3 -8.9768 -6.6529 7-27.9539 -6.5549 SPECIES TYPES SP 2 SP 3 SP 4 SP 5 SP 6 -6.5696 -1.4417 6.4431 12.0059 -16.8290 INTERACTIONS 6.8701 17.6874 5.7359 -7.8814 -8.8394 -1.8675 -13.5517 14.9447 -11.2467 -15.0592 -3.9509 -2.9307 -1.4992 18.7401 -3.7064 -1.0514 -1.2049 -19.1815 0.3880 27.6050 Remarks: SP 1: pure hemlock type; other symbols see remarks i n Table 3-5 - 56 -3.6 P a r t i t i o n i n g the T o t a l Sum of Squares The t o t a l unadjusted sum of squares equal YY and the reduction i n the sum of squares due to f i t t i n g model (3-1) are designated as R(fi, a , B> aB) > where y, a , B> aB are the b vector as i n the previous section and, res p e c t i v e l y , the estimates of A S\ ^ y, a , B> aB i n the model. R(y, a , 3» aB) i s computed as follows, R(y, a , 3, aB) = yY.. + £a.Y..+ E3.Y.. + E (aB)..Y.. (3-10) x 1 1 j 2 2 i j 1 J 1 3 This formula i s equivalent to that i n the c a l c u l a t i o n of sum of squares due to regression i n regression analysis. I t can be shown that the Y's i n Eq. (3-10) are the r i g h t hand side members of the reduced set of l e a s t squares equations. The sum of squares f o r error i s computed by SS = ,Y'Y - R(y, a , £ , OB) (3-11) error which i s the differ e n c e between the t o t a l unadjusted sum of squares and the sum of squares due to f i t t i n g the model. Two methods may be used to compute the sum of squares f o r each source of v a r i a t i o n . An i n d i r e c t procedure involves differences i n reduction i n sums of squares due to f i t t i n g d i f f e r e n t models. The sum of squares f or a , for instance, i s computed by SS a = R(y, a , 3, aB) - R(y, aB). (3-12) - 57 -Si m i l a r l y , the sum of squares due to 8 and aB SS 0 = R (y, a, 8, aB) - R (y, a,aB) (3-13) and A A A A SS a g== R (y, a, 6, a£ - R (y, a,8) (3-14) In applied regression analysis terminology, the sum of squares f o r each source of v a r i a t i o n involves computing the differ e n c e of sums ^ /\ A of squares between a maximum model, f o r instance, R(y, a, 8J aB) and a reduced model, R(y, B, aB) which includes a l l v a r i a b l e s i n the maximum model except the one classa,whose sum of squares i s being considered. The i n d i r e c t method ,involving f i t t i n g various models, therefore, i s tedious and time consuming. The sum of squares f o r each source of v a r i a t i o n may also be computed by a d i r e c t method which employs the segments of the matrix inverse to the variance-covariance matrix and the constant estimated. The d i r e c t method computes the sums of squares as follows, SS = b'Z _ 1b, (3-15) where b' i s a row vector of the constant estimates for a given set; i s the inverse of the segment of the inverse of the variance-covariance matrix corresponding, by row and column, to t h i s set of constants; and - 58 -b i s a column vector of the set of constants. The sum of squares obtained i n t h i s manner i s equal to the reduction i n the sum of squares due to f i t t i n g a l l constants minus the reduction i n sum of squares due to f i t t i n g a l l constants except the set being considered. In the given example, the sum of squares for species types i s computed by the constant estimates f o r SP2, SP3, SP6 i n Table 3-5 m u l t i p l y i n g the matrix inverse of the submatrix with elements i n , column and row, 2 to 7 i n Table 3-4. The computed sum of squares for each source of v a r i a t i o n and the complete analysis of variance table fo r the given inventory data are shown i n Table 3-7. 3.7 Least Squares Means, Variances,and Hypothesis Tests The l e a s t squares mean for cl a s s of a a n d classes 3 are y + and y + r e s p e c t i v e l y . The variance f o r y, y + a n d y + 8.. are computed from the appropriate inverse elements and estimate of 2 a from the error l i n e of the analysis of variance. e The variance f o r SP2, for instance, i s computed as S 2 0 = ( c 1 1 + c 2 2 + 2c 1 2 ) a 2 (3-16) u+sp2 e where i s the i j "th element of the reduced (X'X) Table 3-7. A n a l y s i s of v a r i a n c e t a b l e f o r western hemlock mean annual growth data Source of V a r i a t i o n D.F. Mean Squares F-value Tabulated F-value (5% l e v e l ) Species type 5 F. I.Z. 3 I n t e r a c t i o n s 15 E r r o r 247 1853.5 16509.0 1761. 1 917.46 2. 02 17.99** 1. 92* 2. 24 2. 63 1. 70 * S i g n i f i c a n t at 5% l e v e l . * * S i g n i f i c a n t a t 1% l e v e l . - 60 -I f the i n t e r a c t i o n e f f e c t s are s i g n i f i c a n t , the investi g a t o r i s more interested i n the aB subclass means rather than the clas s means. The lea s t squares subclass mean i s aB.. = y + a. + "3. + (aB).. (3-17) i j i 3 13 and the variance may be computed from the inverse matrix and as follows: S% = { C W + C a i a i + C 6 j B j + C a 3 i j a 6 i j + 2 C U a i + aB±j 2CyB. + 2cu(aB).. + 2ca.B. + 2ca.(aB).. + 2 C 3 j ( a 3 ) i j } a2& (3-18) The l e a s t squares means for these e f f e c t s being absorbed by other e f f e c t s are computed as Eq. (3-19) i f the complete set of estimate c o e f f i c i e n t s , as i n Table 3-6, i s a v a i l a b l e . The variances are derived by recognizing the means as l i n e a r combinations of the absorbent e f f e c t s . I f , f o r instance, the a- e f f e c t has been absorbed, the class mean i s A A A 6 A y + a n ^ y + a - y - E a., (3-19) 1=2 and variance f o r the clas s mean i s 6 6 6 g2 = { c y y + E ^ . a . _ 2 E c y a . _ 2 E c a . a j } ( 3 _ 2 Q ) u _ h x l i=2 i=2 - 61 -Si m i l a r l y , the least squares means and variances for subclasses being absorbed can also be derived. Let 0^ C_ = a 2 ( C ^ ) (q x q) denote the covariance matrix of b (q x 1), where b i s the reduced vector of estimators. The absorbed e f f e c t s can be expressed as, say, y = c'b and z = d'b, where c and d are (q x 1) vectors. In the present s i t u a t i o n c and d have components -1 and 0. Var(y) = O (c'C c) = 0 q E i = l j = l E C ^ c . c . i J = o H i i 2 E C V + 21 E 1=1 1 l^i<j=q C 1 3 c . C i J (3-21) S i m i l a r l y f o r Var(z), Cov(y,z) a 2(c'C d) = a 2 q q Z E C 1 J c . d i = l j = l i 3 = a q .. . E C 1 : Lc.d. + E E C 1 J ( c . d . + c.d.) i = l 1 1 l=i<j=q 1 2 3 1 (3-22) If the assumptions f o r analysis are met, that i s , data are normally-distributed and equally-dispersed, the variance-covariance matrix of Table 3-3 provides a handy way f o r various hypothesis t e s t s . M u l t i p l e range tests can also be c a r r i e d out for the le a s t squares means (Kramer, 1956, 1957). - 62 -3.8 Computer Program The l e a s t squares technique i s very v e r s a t i l e ; i t can apply to data with equal frequencies as w e l l as unequal frequencies i n subclasses. As described i n previous sections, i t analyzes p r a c t i c a l l y a l l kinds of data structures: balanced, unbalanced, and data with empty c e l l s i n some subclasses. Computer programs which employ the le a s t squares p r i n c i p l e to analysis of variance can be found i n some s t a t i s t i c a l computing packages such as Biomedical Computer Programs Package (BMDP) (Dixon, 1971). A severe r e s t r i c t i o n for using the e x i s t i n g least squares analysis programs i s that they are incapable of analyzing data with empty c e l l s i n subclasses when in t e r a c t i o n s must be included i n the model. A computer program, i n Fortran language, to analyze generalized multiple c l a s s i f i c a t i o n unbalanced data (unequal and without observations i n some subclasses) following the procedures described above has been developed by the writer. The algorithm to compute the estimated constants and the sum of squares f o r a source of v a r i a t i o n i n the program i s e s s e n t i a l l y s i m i l a r to that of the general l i n e a r hypothesis program of the Biomedical Computer Programs Package ; however, i t d i f f e r s from the l a t t e r i n the formulation of the incidence matrix (design matrix). Table 3-8 presents two incidence matrices f o r the main f a c t s i l l u s t r a t e d i n t h i s Chapter. Two main factors are considered: species types and growth zones. As shown i n Table 3-8, the BMDP program uses l ' s and O's to indic a t e the presence and absence of the f i r s t 5 e f f e c t s Table 3-8. Comparison of the i n c i d e n c e matrix between the proqram and BMDP general l i n e a r h y p o t h e s i s proqram MAIN FACTORS MEAN A SP 1 A SP2 A SP3 A A SP4 SP5 A A SP6 FZ1 A FZ2 A FZ3 A FZ 1 1 0 0 0 0 0 1 0 0 0 1 0 1 0 0 0 0 0 1 0 0 I AN'G 1 0 0 1 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 1 0 1 0 0 1 1 0 0 0 0 1 0 0 -1 0 1 0 0 0 0 1 0 -1 0 0 1 0 0 0 0 1 -BMDP 1 0 0 0 1 0 -1 -1 -1 -1 0 0 0 0 1 1 0 0 -1 -1 -1 -1 -1 1 0 1 0 -Remarks: A SP 1 A ,SP2 A ,SP2, A A SP4,SP5, and A SP6 are s p e c i e s t y p e s ; A FZ1 A ,FZ2 A ,FZ3, , A and FZ4 are F o r e s t Inventory Zones - 64 -of the species type factor and to ind i c a t e the presence of the 6th e f f e c t (e.g. SP6). S i m i l a r l y , i t uses l ' s and O's f o r the f i r s t 3 e f f e c t s i n the zonal f a c t o r and for the presence of the 4th zonal e f f e c t . I f inte r a c t i o n s between these two main factors are to be included, the c o e f f i c i e n t s are computed from these two main fac t o r s . The c o e f f i c i e n t s f o r the i n t e r a c t i o n between SP1 and FZ1, for example, are the product of the c o e f f i c i e n t s i n SP1 and FZ1 columns. In short, the BMDP program does the absorption processes, s-1 r-1 that i s , a = - £ a. and 8 = £ 8.> i n the course of formulating the s . , I r • i J i = l 3=1 incidence matrix (X). The l a s t e f f e c t i n a main cla s s thus i s absorbed and eliminated. The product X'X formed from the incidence matrix X i s a reduced matrix equivalent to that shown i n Table 3-3. Admittedly, the procedure which combines the absorption processes with the formation of the incidence matrix e f f e c t i v e l y reduces the computational complex. However, i f there i s an empty c e l l i n the subclass and the in t e r a c t i o n s are needed to be considered i n the model, the procedure f a i l s . The BMDP program assumes that the number of degrees freedom f o r the primary i n t e r a c t i o n s i s (s - 1) (r - 1) which i s true only i f observations occur i n a l l c e l l s . In the cases where there are empty c e l l s , the number of degrees of freedom equals to (s - 1) ( r - 1 ) - m, where m i s the number of empty c e l l s . The program assigns l ' s and O's i n the incidence matrix to indicat e the presence and absence of an e f f e c t i n a y i e l d v a r i a b l e (Table 3-8). Once the l e a s t squares equations (matrix) i s formed from the incidence matrix, absorption processes appropriate to the - 65 -the e f f e c t s considered are selected to reduce the matrix to a form such that a unique s o l u t i o n f or the e f f e c t s can be computed. It goes without saying that i t takes more computer time to run t h i s program than the BMDP program; however, t h i s i s a pr i c e that one has to pay for analyzing more generalized unbalanced data. The Fortran source program i s l i s t e d i n Appendix 2. 3.9 Chapter Summary This chapter considers the computational procedures for analysis of unbalanced data with s p e c i a l reference to a two-way c l a s s i f i e d model. The procedures described i n t h i s chapter are more complicated than these commonly found i n textbooks, but they are methods appropriate to the analysis of unbalanced data with no observations i n some subclasses, such as data which are frequently observed i n forest inventories. Because no e x i s t i n g computer program handles data with empty c e l l s i n subclasses, a program i n Fortran language was developed by the writer to analyze the forest inventory data. Difference i n computational aspects between the BMDP general l i n e a r program and the program are i l l u s t r a t e d . - 66 -4.0 DATA BASE A t o t a l of 20,458 temporary pl o t s has been provided by the B.C. Forest Service Inventory D i v i s i o n f o r a n a l y s i s . The number of sample p l o t s by species i s shown i n Table 4-1. The table indicates that Douglas-fir, spruce, and lodgepole pine data which are the bulk of the data set provide excellent bases for the pursuit of the objectives set f o r t h i n Chapter 1. In addition, data of western hemlock (Tsuga heterophylla Sarg.), "balsam'fir (Abies spp.), and aspen (Populus  tremuloides Michx) can also be used i n t h i s study. However, a further grouping of these pl o t s by pure, co n i f e r mixed, and hardwood mixed types revealed that there i s no p l o t f o r the"balsam Lhardwood type and the pure aspen type. Consequently, data for"balsam"and aspen were excluded. The western hemlock plo t s which have been used to i l l u s t r a t e the analysis procedures i n Chapter 3 were also excluded because p l o t s a v a i l a b l e f o r analysis were r e s t r i c t e d to only a few inventory zones. Methods used i n c o l l e c t i n g and summarizing the data made av a i l a b l e f or t h i s study have been reported i n the Forest Survey Manual and Growth Manual published by the Forest Inventory D i v i s i o n of the B.C. Forest Service. Only the recent compilations were s u i t a b l e since early summaries of p l o t s did not include information on numbers of trees, basal area, and average stand dbh (Smith, 1973). Data have been summarized to 7.1, 9.1, 11.1, and 13.1 inches dbh to a close u t i l i z a t i o n standard. The data for 7.1-inch dbh and close u t i l i z a t i o n were analyzed throughout the study because they provided more sample plot s than the others and they f a c i l i t a t e d comparisons between Coast and I n t e r i o r Douglas-fir. - 67 -Data were made a v a i l a b l e by un i t name, type group, s i t e c l a ss, s i t e index^ age class* numbers of regions, compartment, and sample p l o t , net cubic foot volume with deduction for decay only ' to a close u t i l i z a t i o n standard, basal area, and number of trees per acre, age i n 10's and height of dominant and codominant trees i n 10's, forest inventory zone, volume over age curve number. A l l data were measured and recorded i n the Imperial system, therefore, the r e s u l t s were reported here i n the o r i g i n a l u n i t s . Should the need a r i s e , the re s u l t s can be converted i n t o metric units. In r e t r i e v i n g data, which were stored i n a magnetic tape, the values f o r overmature stands (stand age greater than 150 years) were f i r s t deleted because there i s l i k e l y to be l i t t l e i n t e r e s t i n managing stands i n t e n s i v e l y once they become older than about 130 years (Smith-, 1973). Furthermore, the overmature stands might lead to biased estimates of e f f e c t s . The p l o t s with zero values for volume, basal area, and number of trees because e i t h e r they are below the dbh l i m i t or measurements f o r t h i s dbh l i m i t and close u t i l i z a t i o n standard were not a v a i l a b l e also were eliminated. After the deletions the f i n a l number of p l o t s of these three species a v a i l a b l e f o r analyses i s l i s t e d i n Table 4-1. The fa c t that d i f f e r e n t standard volume tables were used f o r a species, growing i n d i f f e r e n t zones needs further explanation. On the Coast, a stand i s i d e n t i f i e d as mature or immature and a volume table appropriate to the stand status i s used i n volume computation. In the In t e r i o r , however, only one standard volume table f o r a species i s used T a b l e 4-1. The number o f s a m p l e p l o t s by s p e c i e s f o r d a t a p r o v i d e d by B. C. F o r e s t S e r v i c e I n v e n t o r y D i v i s i o n S p e c i e s T ype G r o u p T o t a l P l o t s 7. 1 - i n c h DBH < 150 Y e a r s Do u g l a s - f i r 1 — 8 4790 2967 Y e l l o w p i n e 27 151 --W e s t e r n w h i t e p i n e 32 21 1 L a r c h 33, 34 279 W e s t e r n h e m l o c k 12 -- 17 1914 271 B a l s a m 18 -- 20 1764 --S p r u c e 21 26 5523 2615 L o d g e p o l e p i n e 28 -- 31 5 154 4199 ' :P (opulus s p p . 21, 36 135 --Red a l d e r 43 1 --w h i t e b i r c h 40 50 --A s p e n 41, 42 1 38 5 — TOTAL 20458 10052 R e m a r k s : t y p e g r o u p c o d e s d e f i n e d by B.C. F o r e s t S e r v i c e . - 69 -throughout except i n Zone 11 and 12 where . separate volume tables are used for mature and immature spruce and lodgepole pine r e s p e c t i v e l y . To pursue the objectives set f o r t h i n Chapter 1 and to f a c i l i t a t e analyses, data were further grouped by species type (species composition) such as pure, conifers mixed (CMIX), and hardwood mixed (HMIX). A comparison of the species types and the included type groups (designated by the B.C. Forest Service) i s presented i n Table 4-2. Several a d d i t i o n a l stand parameters were computed from the o r i g i n a l data. Mean annual height, basal area, and volume increments were obtained d i r e c t l y from the plot values divided by i t s stand age. Relative stand density (RSD) for each sample plot was computed from the basal area of the plot divided by the mean basal area of the age cl a s s . The mean basal area of an age clas s i n turn i s the average basal area f o r a l l p l o t s at the age c l a s s , ignoring the species types. The r e l a t i v e stand density was computed separately for each species; for Douglas-fir i t was computed independently for the Coast and the I n t e r i o r zones. 4.1 Chapter Summary In t h i s chapter, the data source and the sample plo t s for each species a v a i l a b l e f o r analysis were i d e n t i f i e d . The reasons f o r se l e c t i n g Douglas-fir, spruce and lodgepole pine f o r subsequent analyses and choosing the 7.1-inch dbh l i m i t were explained, b r i e f l y . Table 4-2. Comparison of s p e c i e s types and f o r e s t type groups S p e c i e s D o u g l a s - f i r S pecie s Ty pe PURE CMIX HMIX Inventory Type Groups 1 IF) 5 (FP1) ,2 (FC,FCy) ,3 (FH) 4 (FS), 6 (FPy) 8 (FDecid) S pruce PURE CMI X HMIX 21 (S) 22 (SF) , 23 (SH,SC) , 24 (SB) 25 (SP1) 26 (SDecid) Lodgepole Pine PURE CMIX HMIX 26 (PI) 29 (P1F), 30 (P1S,P1B) 31 (PlDecid) Remarks: PURE: pure D o u g l a s - f i r type, etc. CMIX: D o u g l a s - f i r - c o n i f e r s mixed tjjpe; HMIX: Do u g l a s - f i r - h a rdwood mixed type. F: D o u g l a s - f i r ; P i : Lodgepole p i n e ; C:Cedar; Cy:Yellow Cedar H: Hemlock; Py:Yellow pine; Decid: Dedidous s p e c i e s ; S:Spruce; B:Balsam. - 71 -For a selected species, not a l l sample plo t s provided by the B.C. Forest Service were s u i t a b l e f o r analysis i n the pursuit of the objectives mentioned i n Chapter 1, because they are either overmature or below the dbh l i m i t . The f a c t that d i f f e r e n t volume tables were used f o r a species because of differences i n growing zones and stand age was also explained. The species types used i n the study and the f o r e s t type groups designated by the B.C. Forest Service were compared. - 72 -5.0 RESULTS AND DISCUSIONS 5.1 Occurrence of Douglas-fir-, Spruce-, and Lodgepole Pine-dominated Forest Types i n B r i t i s h Columbia Numbers of sampled plo t s f o r Douglas-fir, spruce, and lodgepole pine c l a s s i f i e d by pure type (PURE) (81% or more of the leading species by volume- per acre), pure and co n i f e r mixed type (CMIX), and pure and hardwood mixed type (HMIX) are l i s t e d i n Table 5-1. Since these p l o t s were randomly sampled from forest area across B.C., i t i s l o g i c a l to i n f e r that r e l a t i v e frequency of the pl o t s i s a useful i n d i c a t o r of the frequency with which the various types occur i n nature. Table 5-1 indicates that more than 50 percent of Douglas-fir, spruce, and lodgepole pine occur n a t u r a l l y as pure type stands. Because these p l o t s spread a l l across B.C., i t i s reasonable to say that i f there are any adverse e f f e c t s of e s t a b l i s h i n g pure forest crops i n B.C., these e f f e c t s should have been w e l l r e f l e c t e d i n these stands. On the other hand, the table shows that about 45 percent of these stands are grown n a t u r a l l y i n mixture with e i t h e r other coniferous or hardwood species. The growth behavior of mixed stands not only i s more complicated than but also d i f f e r s from pure stands (Meyer, 1937; Mulloy, 1944, 1947; Barnes, 1962; Turnbull, 1963); i t i s of mensurational and managerial s i g n i f i c a n c e to examine the extent of difference i n y i e l d between pure and mixed type stands. Table 5-1 Numbers of'sample p l o t s by sp e c i e s and f o r e s t s p e c i e s type SPECIES "YPE DF . DF SP COAST INTERIOR PL TOTAL PORE % 404 1450 1144 2356 5354 63.6 62.2 43. 7 56. 1 54 .7 CMIX % HMIX 172 798 1190 1010 3170 27.1 34.2 45.5 • 24.1 32.4 59 84 281 833 1257 9.3 3.6 10.7 19.8 12.9 TOTAL 635 2332 2615 41 99 9781 Remarks: D F - - D o u g l a s - f i r ; SP--Spruce spp; PL--Lodqepole Pine PURS--pure type stand; CMIX—pure and c o n i f e r s mixed type stand; HMIX--pure and hardwoods mixed type stand. - 74 -The degree of species mixture varies within three species studied. As shown i n Table 5-1, the r e l a t i v e frequency of the pure type pl o t s i s 62.2 to 63.6% for Douglas-fir, 56.1% for lodgepole pine and 43.7% f o r spruce; of the co n i f e r mixed type i s 24.1% for lodgepole pine, 27.1 to 34.2% for Douglas-fir and 45.5% for spruce; of the hardwood mixed type i s 3.6 to 9.3% f o r Douglas-fir, 10.7% for spruce, and 19.8% for lodgepole pine. Douglas-fir which occurs mainly as pure type (62.3 to 63.6%) mixes predominately with other conifer species (27.1 to 34.2%) and r a r e l y with hardwood species (3.6 to 9.3%). Spruce i s grown e i t h e r as pure (43.7%) or i n mixture with other coniferous species (45.5%) and hardwood species (10.7%). Lodgepole pine which frequently grows as pure types (56.1%) mixes commonly with ei t h e r coniferous species (24.1%) or deciduous species (19.8%). The frequencies are of e c o l o g i c a l s i g n i f i c a n c e which w i l l not be pursued further here. Table 5-2 shows the frequency of the sampled plo t s c l a s s i f i e d by species composition type and f o r e s t inventory zone. Douglas-fir i s d i s t r i b u t e d from Zone 2 to Zone 8. The reasons that no Douglas-fir stands grow i n Zones 9 to 12 are e i t h e r that Douglas-fir stands grown i n these zones were not sampled or that they were sampled but have been c l a s s i f i e d under other species types as they were not dominant species. The same arguments apply to spruce and lodgepole pine as w e l l . The r e l a t i v e frequencies i n Table 5-2 are generally i n agreement with Krajina's c l a s s i f i c a t i o n (Krajina, 1969). Forest Inventory Zone 2, for example, f a l l s i n the Coast Douglas-fir Zone (CDF) where the Douglas-fir type occurs predominately; Zone 4 i s the I n t e r i o r Douglas-fir T a b l e 5-2 The f i TYPE equency o f sample p l o t s by f o r e s t t y p e s and f o r a s t i n v e n t o r y zones FOREST INV2NT0BY ZONE 8 1 0 1 1 12 DF,P UK E 347 69. 2 DF+CMIX 107 DF+HMIX SP,PUHE % • SP+CHIX SP+HMIX % 21.4 47 9. 4 57 42. 5 65 48.5 12 9.0 691 74. 2 232 ' 24.9 8 0. 9 52 26. 0 144 72. 0 4 2.0 164 71.3 62 27.0 4 1.7 84 65. 1 44 34. 1 1 0.8 44 33. 8 83 63.9 3 2.3 1 6 20.3 59 74. 7 4 5. 0 320 40. 6 400 50. 7 69 8. 7 45 16. 4 223 80.3 9 3. 3 231 91.7 21 I.3 96 41.2 114 48.9 23 9. 9 292 35.0 439 52. 6 1 03 12. 4 37 46.3 24 30.0 19 23.7 522 66. 4 146 1 3. 6 118 15. 0 PL,PURE " " 642 1 5 3 1 4 2 137 964 2 13 27 - 78 * ~ 5 6 - 8 55. 0 5 1. 8 44. 8 67.2 40. 0 57. 5 - 39.6 PL+CMIX - - 250 49 132 154 138 2 17 1 - 69 22. 1 17. 6 48. 2 50. 3 9.6 40.7 2.1 35. 0 P L * H K I X " ." 238 76 - 15 332 103 1 9. - 50 % ~ 2 1 « 1 27.4 - 4.9 23. 2 1 9. 3 40. 4 - 23.4 s: 1. DF,PUEE - - D o u g l a s - f i r pure type s t a n d ; DF + CMIX— D o u q l a s - f i r - C D n i f mixed t y p e ; DF+HMIX-- D o u q l a s - f i r - h a r d w o o d mixed type-SP,PURE—pure spruce t y p e ; SP + C M I X — s p r u c e - c o n i f e r s mixed t y p e s ; PL, PURE--pure l o d g e p o l e p i n e type; PL + CHIX — l o d q e p o l e pine-conif•=» mixed t y p e ; PL+HMIX— l o d q e p o l e pine-hardwoods mixed t y p e . 2. % I n d i c a t e s the p e r c e n t a g e of a s p e c i e s type w i t h i n an i n v e n t o r y zone f o r t h i s s p e c i e s . - 76 -Zone (IDF) where Douglas-fir and ponderosa pine (Pinus ponderosa) are major species i n the d r i e r subzone (IDFa) and lodgepole pine occurs frequently i n the wetter subzone (IDFb). The Forest Inventory Zone 8 where lodgepole pine dominates f a l l s i n the Cariboo Aspen-lodgepole pine-Douglas-fir Zone (CALPDF). According to Kraji n a , the number of coniferous trees growing i n the CALPDF i s very l i m i t e d , due to the severe winter and the dry and f a i r l y warm summer. Lodgepole pine and Douglas-fir occur here most often. Spruce becomes frequent mainly on moister and cooler s i t e s , e s p e c i a l l y those flooded i n the spring. The occurrence of Populus tremuloides which i s a frequent tree i n the Zone becomes even more common i n the secondary stands e s p e c i a l l y on r i c h loamy s o i l s as i s well-shown i n Table 5-2. According to K r a j i n a (1969), the Forest Inventory Zones 9 to 12 are the Engelmann Spruce-Subalpine F i r Zone (ESSF), Boreal White and Black Spruce Zone (BWBS), and Sub-boreal Spruce Zone (SBS). The major coniferous species are Picea glauca, Picea mariana, Pinus contorta, Lar i x l a r i c i n a , and Abies lasiocarpa and deciduous angiospermous trees such as Populus tremuloides, P_. balsamifera, Betula papyrifera, J3. r e s i n i f e r a , Alnus t e n u i f o l i a and A. c r i s p a frequently occur. The r e l a t i v e frequencies i n Table 5-2 are i n l i n e with Krajina's f i e l d observations. 5.2 S i t e D e t e r i o r a t i o n and Pure Stands Some pure f o r e s t s , notably of c o n i f e r s , may cause a slow d e t e r i o r a t i o n of the upper s o i l layers by f o s t e r i n g the formation of acid and raw humus (Baker, 1950). The needles of pine and spruce i n - 77 -p a r t i c u l a r decompose very slowly and tend to form deep layers of the poorly decomposed raw humus material i n which seedlings f i n d i t d i f f i c u l t to grow. I f trees with leaves which decompose r e a d i l y — usually hardwoods — are mixed with the con i f e r s , they not only improve the humus layers d i r e c t l y , but they also tend to develop conditions i n which the decomposition of the conifer needles themselves i s considerably accelerated. The average s i t e indices f or pure, conifer mixed, and hardwood mixed type stands of Douglas-fir, spruce, and lodgepole pine i n various f o r e s t inventory zones are l i s t e d i n Table 5-3. The above statements made by Baker may gain strong support by the figures obtained from the inventory data. In most of the cases the coni f e r mixed and hardwood mixed stands have higher s i t e index than t h e i r counterpart pure type stands. The improvement i n s i t e q u a l i t y by mixing coniferous or deciduous species i s more s i g n i f i c a n t i n the zones where s o i l f e r t i l i t y i s poor, e.g. Zone 4 and the Northern I n t e r i o r Zones. However, the b e n e f i c i a l e f f e c t of mixed stands on s o i l f e r t i l i t y cannot be accepted without reasonable doubts. I t has been frequently observed by foresters that hardwood species such as Populus, Alnus, and Betula occur h a b i t u a l l y on f e r t i l e s o i l and r a r e l y on poor s i t e s while spruce and lodgepole pine pure stands are often found growing on poor s i t e s . Therefore, the higher s i t e indices shown i n Table 5-3 for coni f e r mixed and hardwood mixed stands cannot be f u l l y interpreted as the b e n e f i c i a l e f f e c t of mixed stands on s o i l improvement; Table 5-3 Avera qes of s i t e indax i n v e n t o r y by s zonas pec i e s t y p e s and f o r e s t FOHES T INVENTOfiY. ZONE TYPE 2 3 a 5 6 7 8 9 10 11 12 A veraye h e i g h t s of dominant and codominant t r e e s at aqe 100, f t . DF,PUHE 125. 2 103.0 73 . 5 82. 3 7 0 . 6 83. 4 7 3 . 1 - - - -DF+CMIX 116 .6 107. 2 75. 0 8 1 . 6 7 3 . 9 84. 7 6 9 . 3 - - -DF+KMIX 129.4 126 .7 8 8 . 1 8 0 . 0 ' 7 0 . 0 86. 0 - - - - -SP,PURE - - 80 . 9 85 . 4 8 0 . 2 86. 6 84 . 4 82 . 7 - 5 9 . 0 7 7 . 2 SP+CMIX - - 79 . 8 37 . 4 78 . 4 84. 5 7 9 . 8 8 4 . 4 - 6 1 . 3 6 7 . 0 SP+HMIX - 8 5 . 5 9 6 . 0 90 . 8 8 1 . 8 7 9 . 4 8 6 . 0 - 62 . 3 7 9 . 7 PL,PUSE - - 6 3 . 1 7 9 . 0 7 6 . 7 82. 7 7 0 . 8 86 . 2 86 . 3 - 6 0 . 3 PL+CMIX - - 6 8 . 0 8 4 . 3 7 3 . 3 86. 1 7 3 . 3 8 7 . 2 70 . 0 - 64 . 5 PL+HMIX - - 7 8 . 8 7 9 . 2 - 83 . 3 7 5 . 0 8 3 . 8 9 0 . 0 - 68 . 8 Bemarks: DF,PUS2, DF+CaiX, DF+HMIX; SP,PURE, SP+CMIX, SP+HHIX;PL,PURE, PL+CHIX, and PL+HMIX see T a b l e 5 - 2 . - 79 -the higher s i t e indices i n mixed stands might simply be explained by the fac t that these stands were established on better s i t e conditions. The i d e a l experiments to assess the e f f e c t of forest types on s o i l d e t e r i o r a t i o n are to e s t a b l i s h species with various degrees of mixture on the same s i t e condition and observe any changes of s i t e with the development of stands. U n t i l such experimental data become av a i l a b l e , foresters are forced to speculate on the e f f e c t of stand types on s o i l from data such as used i n t h i s study. 5.3 Number of Trees Per Acre One of the advantages claimed f o r e s t a b l i s h i n g mixed stands has been that the l i m i t e d forest land can be more economically u t i l i z e d , that i s , more forest trees per un i t can grow i n mixed stands than i n pure stands. Table 5-4 which shows the average number of trees per acre fo r three type groups i n B.C. forest inventory zones demonstrates c l e a r l y the advantages of mixed conifer stands over pure ones. With the exception of three cases, co n i f e r mixed stands grow considerably more trees per acre than pure stands do. However, Table 5-4 shows no d e f i n i t e advantages of hardwood mixed stands over pure type stands i n number of trees per acre. The compensative e f f e c t of d i f f e r e n t coniferous species growing on the same area has been recognized by foresters as w e l l by ec o l o g i s t s . For hardwood mixed stands, because deciduous species usually have larger crowns and more requirement f o r sunlight, the number of trees per acre i s , i n general, le s s than those of c o n i f e r mixed and pure type stands. T a b l e 5-4 Averaged numbers of t r e e s per a=re i n v a r i o u s s p e c i e s t y p e s and f o r e s t i n v e n t o r y zones ( U n i t : T r e e s / A c r e ) ( 7 . 1 i n c h e s DBH and l a r g e r ) FOREST INVENTORY ZONE TY PE 2 3 4 5 6 7 8 9 10 1 1 12 DF+PURE 197 1 85 91 115 108 137 . 89 - - - -DF+CMIX • 159 208 120 141 130 . 174 92 - - - -DF+HMIX 164 152 84 106 100 147 - -' - - -SP,PURE - - 206 172 153 150 190 154 - 177 138 SP+CMIX - - 180 190 162 159 210 177 - 215 200 SP+HMIX - - 174 288 16 4 132 184 191 - 179 200 PL, PURE - - 1 48 166 17 1 229 187 199 208 - 150 PL+CMIX - - 176 180 178 1 94 206 218 226 - 220 PL+KMIX - - 179 172 - 1 52 163 218 156 241 co o Remark: DF,PURE; DF+CMIX; and DF+HMIX ETC. see T a b l e 5-2. - 81 -The number of trees per acre v a r i e s within three species investigated. Comparing the Southern and Central I n t e r i o r Zones i n Table 5-4 where data on a l l three species are a v a i l a b l e , one can c l e a r l y recognize that the numbers of trees per acre i n Douglas-fir stands are s u b s t a n t i a l l y l e s s than those i n spruce- or lodgepole pine-dominated stands. Being a shade i n t o l e r a n t species Douglas-fir requires more growing space and thus reduces i t s number of trees per acre, or gaps i n stocking may occur. The r e s u l t s are of p r a c t i c a l i m p l i c a t i o n f o r spacing t r i a l s . The optimum spacing i s species-dependent, that i s , the most productive spacing f o r one species i s n o t n e c e s s a r i l y the one appropriate to other species. Well-planned experiments are needed to demonstrate such possible r e l a t i o n s h i p s . 5.4 Relative Stand Density Stand density i s much l e s s r e a d i l y defined and quantified (Curtis, 1967); foresters are of d i f f e r e n t opinions as to theprop'er basis for computing stand density. In t h i s study, r e l a t i v e stand density was computed for each sample p l o t by a formula described i n Chapter 4. The average r e l a t i v e stand density:£or various forest type and inventory zone combinations i s shown i n Table 5-5. I f c o n i f e r mixed type stands are capable of growing more trees per acre, i t goes without saying that the r e l a t i v e stand density i n the stand i s higher than i n pure stands. With a few exceptions i n Table 5-5, T a b l e 5-5 R e l a t i v e stand d e n s i t y f o r v a r i o u s s p e c i e s type f o r e s t i n v e n t o r y zone combinations and FOREST INVENTORY ZONE TYPE 2 3 4 5 6 7 8 9 10 11 12 DF,PURE 1.05 0. 95 0.77 1. 05 0.88 1. 22 0.77 DF+CHIX 0.87 1. 03 0. 87 1. 07 0.93 1. 41 0.67 - — DF+HMIX 0.96 1. 07 0.70 0. 83 0.71 1. 25 - - - - -SP,P0RE - - 1.12 1. 08 0.93 1.01 1.01 0. 98 0.75 0. 93 SP+CMIX - - 0.98 1. 22 0.95 1.09 1 .07 1. 08 - 1.00 0. 84 SP+HMIX - - 0.79 1. 87 0.99 1. 04 0.85 1. 13 - 0.76 0. 92 PL,PURE - - 0. 74 1. 06 0.99 1. 15 0. 95 1. 28 1.38 0. 60 PL+CMIX - - 1. 00 1. 23 1.05 1.24 1. 14 1. 36 0.75 - 1. 11 PL+HMIX - - 1. 00 0. 95 - 1. 17 0.87 1. 29 1.11 _ 1.02 Bemarks:DF, SP. PL, PURE, CMIX, and HMIX as d e f i n e d i n Table 5-2. Dimensionless u n i t ; based on the mean b a s a l area of an aqe c l a s s . - 83 -the r e l a t i v e stand density i n co n i f e r mixed stands i s co n s i s t e n t l y higher than that i n pure stands. However, the conclusion cannot be generalized to hardwood mixed stands for which no d e f i n i t e trend has been shown. 5.5 Average Stand Age, Mean Annual Height, Basal Area, and Volume Growth. 5.5.1 Average Stand Age Table 5-6 presents average stand age, mean annual increments f o r height of dominant and codominant trees, basal area per acre, and volume.per acre for Douglas-fir, spruce, and lodgepole pine. The corresponding averages for the height, basal area, and volume y i e l d are l i s t e d i n tables 5-7, 5-8, and 5-9, r e s p e c t i v e l y . The average stand age for hardwood mixed type stands i s much younger than for pure type and co n i f e r mixed type stands. Deciduous species such as aspen, red alder, cottonwood and b i r c h have been considered as "pioneer" species (Turnbull, 1963) and are ready to e s t a b l i s h immediately a f t e r areas have been logged or large scale forest f i r e s occurred. The hardwood species e s t a b l i s h r a p i d l y and i n large numbers on f r e s h l y opened areas. C h a r a c t e r i s t i c a l l y , those trees grow r a p i d l y from the time of establishment and are r e l a t i v e l y short l i v e d . Table 5-6 Aqe, mean annual heiqht, basal arsa, and volume qrowth by species types and f o r e s t inventory zor.es FOREST INVENTORY ZONE TYPE 2 3 4 5 6 7 8 9 10 11 12 DF,PURE AGE 47 80 93 95 92 77 10 1 HT 1.83 1.23 0.72 0.84 0.70 0.98 0.68 -BA 2.83 2.17 0.76 0.95 0.87 1.25 0.73 -VOL 91.00 74.33 18.56 25.64 21.23 32.79 17.58 -DF+CMIX AGE 52 91 94 94 92 80 87 HT 1.58 1.16 0.75 0.87 0.75 0.99 0.70 - - - -EA 2.27 2.41 0.88 1.11 ' 0.96 1.49 . 0.73 -VOL 73.99 83.06 22.95 29.32 25.79 40.45 17.99 -DF+HMIX AGE 45 53 68 78 78 65 HT 1.86 1.78 1.09 0.97 0.85 1.15 -3A 2.52 2.63 0.68 0.89 0.81 1.36 -VOL 79.40 90.10 15.72 23.94 18.63 35.52 -SP,PURE A G E - - 116 114 138 118 112 112 - 111 109 HT - - 0.76 0.81 0.71 0.79 0.78 0.78 - 0.57 0.74 BA - - 1.36 1.30 1.04 1.14 1.25 1.19 - 0.97 1.21 VOL - - 41.96 38.97 35.97 37.97 38.27 35.51 - 24.20 34.77 SP+CMIX AGE - - 108 103 114 109 101 103 - 116 102 HT - - 0.77 0.84 0.73 0.83 0.80 0.81 - 0.60 0.66 BA - - 1.24 1.20 1.13 1.29 1.37 1.31 - 1.22 1.12 VOL - - 35.84 34.16 32.31 38.27 39.92 39.51 - 31.29 28.73 SP+HHIX AGE - - 85 75 78 62 88 93 - 116 100 HT - - 0.94 1.33 1.03 1.03 0.84 0.92 - 0.62 0.83 BA - 1.29 2.67 1.41 1.53 1.19 1.52 - 0.97 1.25 VOL - - 36.72 73.40 36.50 38.17 30.96 43.19 - 23.40 32.89 PL,PURE AGE - - 90 KT - - 0.67 BA - - 0.81 VOL - - 21.50 PL+CMIX AGE - - 91 HT - - 0.73 E A - - 1.11 VOL - - 29.63 FL+HMIX AGE - - 73 HT - - 0.91 3A - - 1.11 VOL - - 32.96 87 73 76 92 0.88 0.91 0.98 0.75 1.15 1.15 1.31 1.06 34.06 32.7£ 39.17 30.56 85 87 76 102 0.93 0.79 1.01 0.74 1.28 1.14 1.41 1.21 35.95 31.25 40.74 35.20 78 - 63 33 0.92 - 1.07 0.83 1.09 - 1.14 0.99 33.05 - 29.63 28.76 90 100 - 92 0. 88 0. 90 - 0. 66 1.36 1.53 - 0.73 41. 54 48. 53 - 18. 35 104 145 - 95 0. 86 0. 55 - 0. 69 1 . 44 0. 69 - 1. 24 45.93 18.88 - 30.94 81 67 - 99 0.94 1.15 - 0.73 1 . 40 1. 29 - 1. 17 42.10 35.91 - 31.82 Remarks: 1. DF, SP, PL, PURE, CMIX, and HMIX as defined i n Table 5-2. 2. HT:heiqht ( f t / y e a r ) : BA:basal area (sq. ft/acre/year) VOL: volume (cubic f t /acre/year). Table 5-7 Mean._height of dominant and condominant t r e e s by types and i n v e n t o r y zones. (Unit: Foot) FOREST INVENTORY ZONE TYPE 2 3 4 5 6 7 8 9 10 1 1 DF,PURE 83.7 98. 8 67 . 6 79 . 5 64. 3 76 .0 67. 9 - - -DF+CJilX 82. 1 105.4 70 .7 81 . 9 68. 6 79 .2 6 1. 0 - - -DF+HMIX 83.6 95.0 73 . 8 75 . 0 66. 7 75 .2 • • - -SP,PURE - - 87 . 5 92 . 1 98. 1 93 .3 88. 0 86.8 - 63. 5 SP+CMIX - - 83 . 3 90 . 0 82. 7 90 .4 80. 8 87.0 - 70.0 SP+HMIX - - 80 . 0 100 . 0 80. 0 66 .7 73. 9 85.3 - 71. 6 PL,PURE - - 60 . 5 76 . 3 65. 8 75 .0 68. 5 79.8 90. 0 — PL+CMIX - - 66 , . 6 79, . 4 68. 6 76 .4 74. 8 88.9 80. 0 -PL+HMIX - - 71 , . 0 71 , .7 - 67. . 3 68. 3 76. 6 77. 4 12 81.2 6 7. 4 79. 9 60.7 65.2 71.5 Remarks: DF, SP, PL, PURE, CMIX, and HI* IX as d e f i n e d i n Table. 5-2. ' a b l e 5- Hean b a s a l a r e a p e r a c r e of D o u q l a s - f i r , s p r u c e , and l o d q e p o l e p i n e by s p e c i e s t y p e s a n d i n v e n t o r y z o n e s ( U n i t : Sq. f t . / a c r e ) FOSSST INVENTORY ZONE TYPE 2 3 4 5 6 7 8 9 10 1 1 12 DF,PUHE 129. 2 174.7 7 1 . 2 90. 2 80 .7 96.5 72. , 9 - - - -DF+CBIX 118. 0 218.1 83. 2 104. 6 88 , .2 119.3 63. 6 - - - -DF+HMIX 113. 2 140. 5 45. 9 69. 0 63, .3 89.0 - - - • - -SP,PURE - - 156. 9 148. 0 1 42 , 6 134. 1 1 40. 3 133.2 - 107. 7 132. 1 SP+CMIX - - 134. 1 160. 1 128. , 5 140. 9 138. 6 140. 9 - 141. 5 115. 1 SP+HMIX - - 109. 5 200. 0 109, .5 97,3 1 04. 4 140.42 - 11 1. 5 125. 3 PL,PURE - - 72. 9 100. 4 83. .2 100. 2 96. 8 12 3.0 152. 8 - 66. 7 PL+CMIX - - 101. 3 10 8. 9 98. 6 106. 3 1 22. 8 149. 4 100. 0 - 117. 7 PL+HMIX - - 87. 3 84.6 - 72. 1 81. 3 114. 1 86. 4 _ 115.5 oo R e m a r k s : DF, SP, PL, PURE, CH I X , and HMIX a s d e f i n e d i n T a b l e 5-2. Table 5-9. Mean net volume by s p e c i e s (Unit: FOREST TYPE 2 3 4 5 6 DF,PURE 4158 5983 1731 2430 1959 DF+CMIX 3855 7520 216 1 2762 2363 DF+HMIX 3573 4805 106 1 1855 1459 SP,PUR E - - 48 50 4426 4946 SP+CMIX - - 3880 4772 3672 SP+HMIX - - 3121 5880 2829 PL,PUfiE - - 1940 2971 2381 PL+CMIX - - 2701 3056 2704 PL+HMIX - - 2583 2566 _ of D o u q l a s - f i r , spruce, and lodqepole pine types and i n v e n t o r y zones cubic f o o t / a c r e ) INVENTORY ZONE 7 8 9 10 1 1 12 2537 1767 - - - -3238 1572 - - - -2319 - - - - -4468 43 02 3 97 7 - 2679 3793 4 189 4041 4250 - 3638 2940 2354 2726 3998 - 2703 3300 2989 2796 3757 4834 - 1733 3 077 3576 4757 2737 - 2930 .1 86 5 23 73 3419 2410 _ 3137 Remarks: DF, SP, PL, PURE, CMIX, and HMIX as d e f i n e d i n Table 5-2. - 88 -The future composition of any stand i s one of the prime concerns i n forest management. The hardwood species are, for a large part, i n t o l e r a n t species which can be expected to die out long before the admixed coniferous species have reached maturity. Mulloy (1947) i l l u s t r a t e d i n d e t a i l the change of hardwood composition with respect to age. In preparing y i e l d tables, he indicated that a stand with 90% hardwoods at age 30 reduces i n hardwood composition to 17% at age 80 and 9% at age 100. The average stand age l i s t e d i n Table 5-6 bears no d i r e c t evidence on the dynamic change i n species composition of hardwood stands; however, i t s lower age than those of pure type and coni f e r mixed type stands suggests the dynamic nature of hardwood mixed stand structure. 5.5.2 Mean Annual Height Growth Table 5-6 and Figures 5-1, 5-2, and 5-3 show that the mean annual height increment of dominant and codominant trees i s higher i n hardwood mixed type stands than i n pure and conifer mixed type stands. The reason f o r the higher mean annual height growth of hardwood mixed type stands can be explained i n part by the fac t that they are younger i n age than the other stands. Among the three species studied, the s u p e r i o r i t y of hardwood stands i n height growth i s more conspicuous i n Douglas-fir than i n spruce and lodgepole pine. Douglas-fir i s int o l e r a n t with other coniferous species, but i n mixture with deciduous species the s p a t i a l configuration of stands d i f f e r s completely from those of pure or conifer mixed type stands. Fiqure 5-1. Comparison of the mean heiqht increments of dominant and codominant trees of Douqlas-fir stands by types and zones . 2.0 + 1.0 0.0 +• H I H H I H H I: PORE I + H H +: CMIX I + H H H: HMIX I+H H I + H H I + H I + H H I + H I + H H H I+H I + H • H Ii I + H I + H I+H H + H H I + H I+H I + H + H + H + H I + H I + H I + H I+H I + H I + H I + H 1 + I + H I + H I + H I + H I + H I + H 1 + I + H I + H I+H I + H I + H I + H 1 + I + H I + H I + H I + H I + H I + H 1 + I + H I + H I + H I + H I + H I + H 1 + I+H I + H I + H I + H I + H I + H 1 + I + H I + H I + H I + H I + H I + H 1 + ONE ZONE ZONE ZONE ZONE ZONE ZONE ZONE 2 3 U 5 6 7 8 9 oo Vi5 ZONE 10 ZONE 11 ZONE 12 Remarks: PURE--pure type;CMIX--conifer mixed type;HMIX—hardwood mixed type. Figure 5-2. Comparison of the mean annual heiqht increments of dominant and codominant trees by types and zones of Interior spruce 2.0 + H O M W 1.0 I: PUKE + ; CMIX H: HMIX 0.0 ZONE 2 ZONE 3 H H H H H H H H H H H H I + H I + H H I + H I + H H I + H I + H I + H I + H I + H I+H H I + H I + H I + H I + H I + H I+H I + H I + H I + H I + H I + H I + H I + H I+H I + H I + H I + H I + H I + H I+H I + H I+H I + H I+H I + H I + H I + H I + H I + H I+H I + H I + H I + H I + H I + H I + H I + H I+H I + H I + H I + H I+H I + H I + H I + H I+H I + H I + H ZONE ZONE ZONE ZONE ZONE ZONE ZONE ZONE ZONE 4 5 6 7 8 9 10 11 12 o Remarks: PURE—pure type;CMIX--conifer mixed type;HMIX—hardwood mixed type. F i g u r e 5 -3. Comparison of the mean annual height increments of dominant and codominant trees of lodgepole pine' in Interior zones I: PURE • : CMIX H: HMIX a H H I + H H H I+li I I + H I+H I H B I + H 1 + I+H H I + H I H I + H I+H 1 + I + H I + H 1+ H I H I + H I + II I + H 1 + I + H I + H I + H I + H I + H I + H I + H 1 + I + H I + H I+H I + H I + H I + H I + H 1 + I + H I + H I+H I + H I + H I+H I+H 1 + I+H I + H I+H I + H I + H I+H I+H 1 + I + H I + H I+H I + H I + H I+U I+H 1 + I + H I + H I+H I + H I + H ZONE ZONE ZONE ZONE ZONE ZG NE ZONE ZONE ZONE ZONE ZONE 2 3 4 5 6 7 8 9 10 11 12 Remarks: PURE—pure type;CMIX--conifer mixed type;HMIX—hardwood mixed type. - 92 -Variations i n the mean annual height increment of dominant and codominant trees among the inventory zones i s expected and apparent. For Douglas-fir, the height growth i n Coast zones (Zones 2 and 3) d i f f e r s d r a s t i c a l l y from that i n the I n t e r i o r zones (Zones 4 to 8). Among the I n t e r i o r zones, the height growth i n the I n t e r i o r Wet Belt (Zone 7) exceeds that i n other zones. For spruce and lodgepole pine i n I n t e r i o r B.C., i t i s apparent that trees i n the Southern and Central I n t e r i o r zones grow f a s t e r i n height than i n the Northern I n t e r i o r Zones (Zones 11 and 12), because the l a t t e r are shorter i n growing season and severe i n c l i m a t i c conditions. 5.5.3 Mean Annual Basal Area Growth The mean annual basal area increments for Douglas-fir, spruce, and lodgepole pine stands are l i s t e d i n Table 5-5 and diagrammed i n Figures 5-4, 5-5, and 5-6. On the Coast, pure Douglas-fir stands i n Zone 2 (the Southern Coast Region) grow f a s t e r than co n i f e r mixed type or hardwood mixed type stands. However, on medium s i t e zones such as Zone 3 (the South Coast T r a n s i t i o n Belt) the reverse i s true. In the I n t e r i o r , mean annual increment i n Douglas-fir conifer mixed type stands i s c o n s i s t e n t l y higher than i n pure Douglas-fir and Douglas-fir hardwood mixed type stands, because Douglas-fir c o n i f e r mixed stands are capable of growing more trees per acre than the l a t t e r (Table 5-4). Zonal v a r i a t i o n s i n mean annual basal area growth of Douglas-f stands are also apparent (Figure 5-4). The best zones for Douglas-fir stands are i n descending order Zone 2, Zone 3, Zone 7, Zone 5, Zone 6, 3.0 + Fi g u r e 5-4. Comparison of the mean b a s a l area increments of D o u g l a s - f i r stands by types and zones 0.0 I I I H I H H I H + H I + H + H I+H I + H I + H I + H I + H I + H I + H I+H I + H I + H I + H I+H I + H I + H I + H I + H + I+H I + H + I + H I + H 1 + I+H I + H I + H I + H I + H + I + H I + H I + H 1 + + I + H I + H I + H + I + H 1 + I + H I + H I + H 1 + I + H I + H I + H I + H I + H I + H I + H I + H I + H 1 + I+H I + H I + H I + H I + H I + H 1 + I+H I + H I+H I + H I + H I+H 1 + I+H I + H I + H I+H I + H I + H 1 + I + H I + H I + H I+H I + H I + H 1 + I+H I+H I+H I+H I + H I + H 1 + I + H I + H I + H I + H I + H I+H 1 + I: PURE + : CM IX H: HMIX ZONE ZONE ZONE ZONE ZONE ZONE ZONE ZONE 2 3 4 5 6 7 8 9 ZONE 10 ZONE 1 1 ZONE 12 Remarks: PURE--pure type;CMIX--conifer mixed type;HMIX— hardwood mixed type. Figure 5-5. Comparison of the mean b a s a l area incremants of spruce stands by types and f o r e s t i n v e n t o r y zones ZONE ZONE 2 3 H I : PORE H CMIX H H: HMIX H H H H H I H H H + H I H I H H + H 1 + * H H I + H I+H H + H I + H I+H I H I+H I+H + H I + H I + H I + H +• I + H I + H I + H I + H I + H I + H I+H 4- I + H I+H I + H I + H I + H I + H I+H I + H I + H I + H I+H I+H I + H I + H I+H I + H I + H I + H I+H I + H I + H I + H I+H I + H I + H I + H I + H I + H I + H I + H I+H I + H I + H I + H I + H I + H I + H I + H I+H I + H I + H I + H I + H I + H I+H I + H I+H I + H I + H I + H I + H I + H I + H I + H I + H I + H I + H I + H I + H I + H I + H I + H I+H I + H I + H I+H I+H I + H I + H I + H I+H I + H I+H JONE ZONE ZONE ZONE ZONE ZONE ZONE ZONE ZONE 4 5 6 7 8 9 10 1 1 12 Remarks: P0RS--pure type;CMIX--conifer mixed type;HMIX— hardwood mixed type. Figure 5-6. Comparison of the mean annual basal area increments of lodgepole pine stands by types and inventory zones 2.0 + < Pi o < H < < 00 < 1.0 0.0 + • I: H: PURE CM IX HMIX ZONE 2 ZONE 3 + I + I+H I + 1 + 1+ H I H 1 + 1 + 1 + + I+H I H + H + H I+H 1 + I + H 1 + I+H I H + H + H I + H 1 + I + H I + H I + H I H + H + H I + H 1 + I + H I + H 1+ H I H • H I+K I + H 1 + I + H I + H I+H I H + H I + H I + H 1 + I + H I + H I+H I+H I + H I + H I + H 1 + I + H I + H I + H I + H I + H I + H I + H 1 + I+H I + H I+H I+H I + H I + H I + H 1 + I + H I + H I+H I + H I + H I + H I + H 1 + I + H I + H I+H I + H I+H I + H I + H 1 + I + H I + H I+H I + H I + H I+H I + H 1 + I + H I + H I+H I + H I + H ONE ZONE ZONE ZONE ZONE ZONE ZONE ZONE ZONE 4 5 6 7 8 9 10 11 12 Remarks: PURE--pure type;CMIX--conifer mixed type;HMIX—hardwood mixed type. - 96 -Zone 4, and Zone 8. The sequence confirms the f i e l d observations by Krajina (1969). In Zone 4 (the I n t e r i o r Dry Belt) and Zone 8 (the Nechako-Fraser Plateau Region) the numbers of trees per acre are s i g n i f i c a n t l y l e s s than those i n other I n t e r i o r zones (Table 5-4). A comparison of the mean basal area growth of spruce stands by species composition types and inventory zones indicates that mixed type stands y i e l d more than pure spruce type stands i n basal area growth. Inspection of Tables 5-4 and 5-6 reveals that the mean basal area growth i s highly correlated with number of trees per acre. Variations i n mean basal area increment of spruce stands i n the I n t e r i o r zones are much l e s s than those of Douglas-fir stands, although Zone 11 (the Northern Central Plateau Region) shows more reduction i n mean basal area growth than do the other zones. The advantages of c o n i f e r mixed stands i n term of basal area growth are shown i n lodgepole pine. Lodgepole pine-conifer mixed stand grow co n s i s t e n t l y f a s t e r i n mean annual basal area increment per acre than do pure lodgepole pine or lodgepole pine-hardwood mixed type stand Table 5-6 and Figure 5-6 i n d i c a t e , excepting i n Zone 10 where sample plot s f o r lodgepole pine-conifer mixed type stands were too few to ascertain (Table 5-2), lodgepole pine-conifer mixed stands grow f a s t e r than pure lodgepole pine stands i n mean annual basal area. The co n i f e r mixed stands are not only capable of growing more trees per acre but also capable of producing trees larger i n DBH than pure type stands as evidenced i n Table 5-4. - 97 -5.5.4'' Mean Annual Volume Growth Mean annual volume increments f o r the pure and mixed stands of Douglas-fir show a s i m i l a r pattern to the mean annual basal area growth (Table 5-6 and Figure 5-7). For I n t e r i o r stands, the trend i s complicated by the fact that pure stands grow f a s t e r i n some zones and slower i n the others (Table 5-6, Figure 5-8). Figure 5-9 and Table 5-6 show that lodgepole pine-conifer mixed type stands grow uniformly f a s t e r than pure type stands i n the mean annual volume growth with an exception of Zone 10 where pure stands grow e x c e l l e n t l y . 5.6 Difference i n Growth and Y i e l d Between Coast and I n t e r i o r Douglas-fir Stands Forest trees grown on the Coast have been observed to be d i f f e r e n t from those grown i n the I n t e r i o r (B.C. Forest Service, 1976); however, to what extent they d i f f e r has not yet been reported. The Douglas-fir inventory data provide a sound basis f o r a comparison of growth and y i e l d between these two geographical regions. Table 5-10 presents some stand parameters for Douglas-fir stands on the Coast and i n the I n t e r i o r . The Coast Douglas-fir stands are p r i m a r i l y second growth, therefore, the average stand age i s much younger than that i n the I n t e r i o r . Because of the difference i n stand age (55 and 89, r e s p e c t i v e l y ) , a d i r e c t comparison cannot be made; Figure 5-7. Comparison of the mean annual volume increments of Douqlas-fir by types and zones 100 + 80.0< 60. 0< w oi o < H 3 U o > 40. 0< 20. Q< 0. 0+-I + H I + H I + H I + H I+H I + H I + H. I + H I+H I + H I + H I + H I+H I + H I+H I + H I+H I + H I + H I + H I + H I + H I + H I + H I+H I + H I + H I + H I+H I + H I + H I + H I + H I + H I+K I + H I + H H H H + H + H + H + H I + H I + H I + H I + H I + H I + H I + H I + H I + H I + H I + H I + H I + H I + H I + H I + H I + H I + H I + H I + H I + H I + H I + H I + H I + H I+H I + H I + H I + H I + Ii I + H I + H I + H I + H I + H I + H I + H ZONE 2 ZONE 3 + + + H I + H I + H + I + H + I + H 1 + + I + K + I + H + I + H + I + H 1 + I + H + I + H 1 + I + H + 1 + I+H I+H I + H 1 + I + H I + H I + H I + H 1 + I + H I + H I + H I + H 1 + I + H I+H I + H I + H 1 + I + H I+H I + H I + H 1 + I + H I + H I + H I + H 1 + I + H I + H I + H I + H 1 + I + H I + H I + H I + H 1 + I + H I + H I + H I + K 1 + ZONE ZON E ZONE ZONE ZONE U 5 6 7 8 I: PU8E +: CMIX H: HMIX oo ZONE 9 ZONE 10 ZONE 1 1 ZONE 12 Remarks: PURr>-pure type;CMIX--coni£er mixed type;HMIX—hardwood mixed typa. Figure 5-8. Comparison of the mean annual volume increments of spruce by types and zones 50. 0 + 40. O-i 30. 0< at < 3 s o > 20. 0 + 10. O-i 0.0 ZCNE 2 ZCNI 3 H H H H H I: PURE H + : CMIX H H H: H MI X I H H I H H I H + + H I I H + • H I I H I + H 1 + + H I H I H H I + H 1 + + H I + H I H I H I + H 1 + I+H I + H I + H I H I + H 1 + I + H I I+H I + H I H I + H 1 + I+H I I + H I + H I H I + H 1 + I+H I H I + H I + H I + H I + H 1 + I+H I H I + H I + K I + H I + H I+'H I+H + I H I + H I + H I + H I + H I + H I+H + I H I + H I + H I + H I + H I + H I+H + I + H I+H I + H I + H I + H I + H I+H + I + H I + H I + H I + H I + H I + H I+H • I + H I + H I + H I + H I + H I + H I+H + I+H I + H I + H I + H I + H I + H I+H + I + H I + H I + H I + H I + H I + H I+H 1 + I + H I + H I + H I + H I + H I + H I+H I + H I + H I+H I + H I + H I + H I + H I+H I + H I + H I + H I+H I + H I + H I + H I+H I + H I + H I + H I + H I + H I + H I + H I + H I+H I + H I + H I + H I + H I + H • I + H I+H I+H I+H I+H I + H I + H I + H I + H 1+ H I + H I + H I + U I + H I + H I + H I + H I+H I + H I + H I + H I + U I + H I + H I + H I+H I + H I+H I + H I + H I + H I + H I + H I+H I+H I + H I + H I + H I + H I + H I + H I+H I+H I + H I + H I + H I + H I + H I + H I+H I + H I + H I + H I + H I + U I + H I + H I + H I + H I+H I + H I + H I + H I + H I + H I+H I + H I + H I + H I+H I+H I + H I + H I+H I + H I + H I + H I + H I + H I + H I + H I+H I + H I + H I + H I + H I + H I + H I + H I+H I + H I + H I + H I + H • I + H I + H I + H I+H I + H I + H I + H I + H I + H I + H I + H I+H I + H I+H I + H I+H I + H I + H I + H I+H I + H I + H I + H ' I + H I + H I + H I + H I + H I + H I+H I + H I + H I + H I + H I + H 1+H I + H I + H I+H I + H I + H I + il I + H I+H I + H I+H I + H I + H I + H I + H I + H I + H I + H I + H ZONE ZONE ZONE ZON E ZONE ZONE ZONE ZONE ZONE 4 5 6 7 8 9 10 11 12 vo Remarks: PURE--pure type;CMIX--conifer mixed type;HMIX—hardwood nixed type. Figure 5-9. Comparison of the mean annual volume increments of lodgepole pine by types and zones 50. 0 + 40. 0< 30.0* < 1=3 ix W OA o > 20. O-i 10. 0< 0.0 ZONE 2 ZONE 3 + I + I I: PORE + I f ; CMIX + I ii : HMIX + I+H I + I + H I 1 + I+H I 1 + I+H I 1 + I+H I + 1 + I+H I + 1 + + I + H I H 1 + 1 + + I+H I H H I + H + 1 + + I+H I H H I+H + 1 + + I+H I H H H I + H 1 + 1 + 1 + I+H I H + H + H I + H 1 + I + H 1 + I+H I H + H + H I + H 1 + I+H I + H 1+ H I H + K + H I + H 1 + I + H I + H • I+H I H + H + H I + H 1 + I + H I + H I + H I H + H + H I + H 1 + I + H I + H I+H I H + H + H I + H 1 + I+H I + H I+H I H + H + H I + H 1 + I + H I + H 1+ H I H + H + H I + H 1 + I + H I + H I+H I H • H + H I + H 1 + I + H I + H. 1+ H I H • H I + H I + H 1 + I + H I + H I+H I H • H I + H I + H 1 + I+H I + H I + H I H + H I+H I + H 1 + I + H I + H I+H I + H I + H I + H I + H 1 + I + H I + H 1+ H I + H I+H I + H I + H 1 + I + H I + H I+H I + H I + H I + H I + H I* I+H I + H I+H I + H I + H I + H I + H 1 + I + H I + H I + H I + H I + H I + H I + H 1 + I + H I + H I+H I + H I + H I + H I + H 1 + I + H I + H I+H I + H I + H I+H I + H 1 + I + H I + H I+H I + H I + H I + H I + H 1 + 1 + H I + H I + H I + H I + H I+H I + H 1 + I + H I + H I+H I + H I + H I + H I + H 1 + 1 + H I + H I+H I + H I + H I+H I+H 1 + I + H I + H I+H I + H I+H I + H I + H 1 + I + H I + H I+H I + H I + H I+H I + H 1 + I + H I + H I+H I + H I+H I + H I + H 1 + I + H I + H I+H I + H I + H I + H I + H 1 + I + H I + H I+H I + H I+H I + H I + H 1+ • I + H I + H I + H I + H I + H I+H I+H 1 + I + H I + H I + H I + H I + H I + H I+H 1 + I + H I + H I+H I + H I + H ONE ZONE ZONE ZONE ZONE ZONE ZONE ZONE ZONE 4 5 6 7 8 9 10 11 12 o o Remarks: PURE—pure type;CMIX--conifer mixed type;HMIX—hardwood nixed type. - 101 -however, the s t a t i s t i c s l i s t e d i n Table 5-10 w i l l shed some l i g h t on the diffe r e n c e i n growth and y i e l d between these two regions. The d i f f e r e n c e i n growth and y i e l d between Coast and I n t e r i o r can be explained i n part by the diffe r e n c e i n s i t e index which i s 120 fo r the Coast and 78 for the I n t e r i o r stands. Because of the diffe r e n c e i n s i t e q u a l i t y , the average number of trees per acre i s 188 f o r the Coast stands and 121 for the I n t e r i o r stands. The number of trees per acre has two implications f o r : (1) the rate of basal area growth, and (2) the measurement of tolerance to crowding. The higher number of trees per acre on the Coast seems to suggest that Douglas-fir i s f a i r l y p l a s t i c i n response to growing space. On poor s i t e s Douglas-fir requires more space to grow while on good s i t e s i t reduces the requirements of growing space to some extent. At average age 55, the Coast Douglas-fir stands are characterized by stand parameters as following: average DBH i s 11.38 inches, average height of dominant and codominant trees 87 feet, average basal area 140 square feet per acre, and y i e l d 4584 cubic feet per acre. The same parameters f o r the I n t e r i o r stands at age 89 are 11.51 inches, 73 feet , 88 square feet , and 2264 cubic feet per acre for average DBH, height, basal area, and volume y i e l d r e s p e c t i v e l y . Comparing the mean annual increments of height, basal area, and volume between these two geographic areas, the difference becomes more evident. In mean annual height increment, the Coast stands outgrow the I n t e r i o r by 2 times; the mean annual basal area by 2.6 times; and by 3.3 times on the mean annual volume growth (84.00 vs. 25.53 cubic feet per acre per year). T a b l e 5-10. C o m p a r i s o n o f D o u g l a s - f i r s t a n d s grown on t h e C o a s t and i n t h e I n t e r i o r COAST INTERIOR Plf HE CM IX HMIX TOTAL PURE CMIX HMIX TOTAL No. P l o t s 404 172 59 635 1450 798 84 2332 S t a n d Age (Ye a r ) 51 • 67 47 55 9 1 87 67 39 S i t e I n d e x 122 113 129 120 77 80 85 78 No. T r e e s p e r a c r e 196 178 161 188 1 04 149 138 121 R e l a t i v e S t a n d D e n s i t y 1. 03 0.93 0. 98 1.00 0.90 1.16 1.15 1 . 00 A v e r a g e DBH ( i n c h ) .1 1 .03 12.19 1 1.43 11.38 11. 77 11.14 .10.35 11.51 H e i g h t , f t 86 91 86 87 71 75 75 73 B a s a l A r e a , ( s q . f t . / a c r e ) 136 156 119 140 79 103 83 88 V o l u m e , ( c u . f t / a c r e ) 4416 5240 3823 4584 200 1 2753 2146 2264 M.A.I.—HT ( f t / y r . ) 1.70 1. 36 1. 84 1 .60 0.78 0.87 1.12 0. 82 M. A. I . — BA i sq . f t / a c r e / y r . 2. 68 2. 34 2. 54 2. 56 0. 87 1.19 1.25 0. 99 M. A. I . — V O L ( c u . f t / a c r e / y r . ) 87. 29 78.64 81. 88 84.00 21.97 3 1.77 32.25 25. 53 R e m a r k s : PURE, CMIX„ and HMIX as d e f i n e d i n T a b l e 5-2. HT-- h e i g h t ; B A — b a s a l a r e a ; VOL-- n e t v o l u m e . - 103 -Among the three species composition types investigated, pure Douglas-fir type stands seem to be the best type on the Coast while Douglas-fir conifer mixed type grows the best i n the I n t e r i o r . Coastal pure Douglas-fir type stands have more trees (196 per acre), higher r e l a t i v e stand density (1.03), higher mean annual increment i n basal area (2.68 s q . f t . per acre/year) and volume (87.29 c u . f t . per acre/year) than stands of mixed types. I n t e r i o r Douglas-fir con i f e r mixed type stands have more trees per acre and higher r e l a t i v e stand density than pure, and Douglas-fir hardwood mixed type stands. 5.7 Comparison of Volume Y i e l d by Species Composition Types i n y i e l d between species composition types and among forest inventory zones: To accomplish these ends, the inventory data were c l a s s i f i e d by species composition types and inventory zones and analyzed by the fi x e d e f f e c t l i n e a r model: and Forest Inventory Zones The main objectives of t h i s study were to assess the difference Y i j k y + a. + 3. + ( a g ) . . i .1 xi + yX i j k + e I j k (5-1) where = general mean; l = i - t h species composition type; 3. J = j - t h Forest Inventory Zone; = i n t e r a c t i o n term for species type and Forest Inventory Zone - 104 -Y = a c o e f f i c i e n t f o r age, covariable, age i n natural logarithmic scale; e . = unexplained residues. 13k Because of the unbalanced nature of the inventory data used (Table 5-2), analyses followed the procedures described i n Chapter 3 by using the Fortran program developed by the w r i t e r . The l e a s t squares f o r these analyses are i n Appendix 3. The estimated constants are l i s t e d i n Table 5-11, 5-12, 5-16, and 5-19 f o r Coast Douglas-fir, I n t e r i o r Douglas-fir, I n t e r i o r spruce, and lodgepole pine volume data, respectively. The corresponding analysis of variance tables are presented i n Tables 5-13, 5-14, 5-17, and 5-20. To f a c i l i t a t e comparison among species types and inventory zones, p o t e n t i a l y i e l d s at age 100 were calculated based on the estimated constants l i s t e d i n Tables 5-11, 5-12, 5-16, and 5-19 and were presented i n Tables 5-15, 5-18, and 5-21 for Douglas-fir, spruce, and lodgepole pine, r e s p e c t i v e l y . In a l l four analysis of variance tables,'-.the covariate factor, stand age, accounts for more v a r i a t i o n than do the species types, inventory zones, and the i n t e r a c t i o n s . 5.7.1 Douglas-fir For the Coast Douglas-fir stand data, the species types and inventory zones are not s i g n i f i c a n t ; however, the i n t e r a c t i o n s for species types and zones are s i g n i f i c a n t at the 1% l e v e l (Table 5-13). T a b l e 5-11. E s t i m a t e d c o n s t a n t s f o r C o a s t D o u q l a s - f i r v olume y i e l d ( U n i t ; c u . f t . per a c r e ) TYPE PUEE CMIX HMIX -71.20 -89. m 160.61 INVENTORY ZONE 2 3 5.50 -5.50 INTERACTIONS 5.20 -5.20 -2.93 2.93 -2. 27 2. 27 o MEAN = -16176.0 LOG AGE = 121.51 Remark: PURE, CMIX, and HMIX as d e f i n e d i n T a b l e 5-2. Table 5-12 Estimated constants f o r I n t e r i o r D o u q l a s - f i r volume y i e l d (Unit: c u . f t / a c r e ) TYPE PURE -93.02 CM IX 270. 83 H MIX -177.81 GENERAL MEAN = INVENTORY ZONE .4 5 6 7 -389. 62 278. 35 -171. 52 931. 35 INTERACTIONS 2.43 16.13 4. 97 -136. 71 -4.89 -61.93 80.00 100.00 2. 45 45. 80 -84. 97 36. 71 -7079.17 -648.55 113.18 -113. 18 o ON LOG AGS = 4794.69 Remark: PURE, CMIX, and HMIX as defined i n Table 5-2. Table 5-13. A n a l y s i s of v a r i a n c e f o r the Coast D o u q l a s - f i r net volume y i e l d b y s p e c i e s types and i n v e n t o r y zones. Source_of D.F. Mean F-Value Tabulated V a r i a t i o n Squares F-Value (5%) Species type 2 976320.0 0.13 3.01 F.I.2. 1 6518.0 0.00 3.86 Type X F.I.Z. 2 31763000.0 4.19** 3.01 5 Log. Age 1 22745X10 300.30** 3.86 Er r o r 628 7575800 ** S i g n i f i c a n t at 1% l e v e l . h-1 O Table 5-14. A n a l y s i s o f v a r i a n c e f o r the I n t e r i o r D o u q l a s - f i r net volume y i e l d by s p e c i e s types and i n v e n t o r y zones. Source of D.F. V a r i a t i o n Species type 2 F.I.Z. 4 Type X F.I.Z. 7 C o v a r i a t e - - 1 Log. age E r r o r 2317 Mean S g uares 1.45038X10 •7 4.50359X10 1 . 24560X10 6 1 . 3999 1X105 2.34 107X106 F-Value 6. 19** 19.24** 0. 53 597.98** Tabulated F-value {5%) 3. 00 2.38 2. 02 3.00 o oo * S i g n i f i c a n t at 5% l e v e l ; ** S i g n i f i c a n t at 1% l e v e l . ' a b l e 5-15. P o t e n t i a l y i e l d o f D o u q l a s - f i r a t aqe 100 i n B.C. F o r e s t I n v e n t o r y Zones TYPE COAST 2 3 PURE MEAN 8579 7540 FOREST INVENTORY ZONES INTERIOR ALL 4 5 6 7 8 8433 2216 2898 2437 3398 2068 ALL 2582 CMIX MEAN 7748 8335 7970 2139 2997 2687 3812 2019 3114 fiflIX MEAN 8065 8518 8157 1945 2656 2076 3300 3097 o VO WEIGHTED 8354 8013 MEAN 8232 2194 2920 2590 3599 2064 2793 R e m a r k s : PURE, CMIX, and HMIX as d e f i n e d i n T a b l e 5-2. U n i t : c u . f t . p e r a c r e . ALL a r e w e i g h t e d means f o r s p e c i e s c o m p o s i t i o n . T a b l e 5-16. E s t i m a t e d c o n s t a n t s f o r I n t e r i o r s p r u c e net volume y i e l d ( U n i t . c u . f t . / a c r e ) MAIN EFFECTS TYPE PURE -13.74 CMIX -51.23 H MIX 64.97 GENERAL MEAN 10 FOREST INVENTORY ZONE 4 5 6 . 7 8 9 136.08 1330.26 -164.79 195.60 -47.91 266.24 -1174.91 INTERACTIONS 519.70 -1033.73 298.51 109.93 278.15 -307.11 -179.68 -212.58 -438.58-190.61 74.65 345.51 100.34 638.33 -307.12 1472.31 -107.90 -134.58 -623.60 206.27 -458.64 = -9876. 69 LOG AGE = 6873.34 12 -540.56 3 14.23 •317.56 3. 33 I-1 o Remark: PURE, CMIX, and HMIX as d e f i n e d i n T a b l e 5-2. Table 5-17. A n a l y s i s of variance f o r net volume y i e l d s of I n t e r i o r Spruce stands by s p e c i e s types and i n v e n t o r y zones. Source of D, F„ V a r i a t i o n Species type 2 F.I.Z. 7 Type x F. I. Z. 14 C o v a r i a t e — Loq. Age 1 E r r o r 2594 Mean Squares 3. 4836X10 5 4. 1757X10 7 ,7 1.1101X10' 2. 3211X10 3.2018X101 10 F-Value 0.11 13.04** 3.47** 724.94** Tabultaed F-Value (5%) 3.00 2.02 1.70 3.00 ** S i g n i f i c a n t a t 155 l e v e l . Table 5-18. P o t e n t i a l y i e l d of I n t e r i o r spruce stands at age 100 by f o r e s t i n v e n t o r y zones and types TYPE PURE, MEAN CMIX, MEAN HMIX, MEAN WEIGHTED MEAN FOREST INVENTORY ZONE 4 5 6 7 8 9 4513 4154 3991 4297 4221 3950 3743 4711 3464 4090 4117 4187 3765 6739 3663 3947 3264 4408 3944 4364 3581 4119 4076 4131 11 2636 3283 2302 2751 12 3765 2 96 2 3399 3561 MEAN 3901 3924 3722 i i - 1 i - 1 I Remarks: PURE, CMIX, and HMIX as defined i n Ta b l e 5-2. Weighted mean: weighted by number of p l o t s . Table 5-19. Estimated constants f o r lodqepole pine net volume y i e l d by s p e c i e s types and i n v e n t o r y zones TYPE PORE 9.06 CMIX 1.68 H MIX -10.74 GENERAL MEAN FOREST INVETOHY ZONE -506.26 5 104.00 6 99.73 -590.48 -21.30 50.93 148. 63 154.70 -50. 93 441.85 -133.40 -8459. 30 LOG AGE = 5929.69 7 8 234.07 -137.85 INTERACTIONS 110.76 -123.20 280. 40 369. 56 -391.15 -237.35 984.27 -133.27 396.15 -262.88 Remarks: PURE, CMIX, and HMIX as d e f i n e d i n Table 5-2. U n i t : cu. f t . per acre. 10 12 15.34 -593.84 i i— 1 H to 1498.15 -782.59 • 1637.15 338.65 138.99 443.95 Table 5-20. A n a l y s i s of v a r i a n c e f o r lodqepole pine net volume y i e l d by s p e c i e s types and i n v e n t o r y zones Source of V a r i a t i o n D. F. M e an Squares F-Value Tabulated F 7alues 15%) Species type 2 F . I . z . 7 Type x F . I . Z . 13 C o v a r i a t e - -Log Age 1 Error 4175 41288.4 8 1. 1084X10' 2.10421xl0 7; 2.9628X10 2.3502X10 10 0.02 3.00 47.18** 2.02 8.69** 1.73 1260. 63** 3.00 ** S i g n i f i c a n t at 1% l e v e l . Table 5-21. P o t e n t i a l y i e l d s of I n t e r i o r l o d q e p o l e p i n e at aqe , 100 by s p e c i e s types and f o r e s t i n v e n t o r y zones (Uni t : cu. f t . per acre) TYPE FOREST INVENTORY ZONE 4 5 6 7 8 9 10 12 MEAN PURE, MEAN 3583 3629 3720 3657 3845 3881 4480 3496 3786 CMIX, MEAN 3955 3429 . 3802 3583 4123 4263 2379 4294 3921 HMIX, MEAN 3577 3519 — 3167 3457 3926 2967 4061 3525 WEIGHTED 3664 3563 3760 3598 3782 4045 3824 3919 MEAN Femarks: PURE, CMIX, and HMIX as defined i n Table 5-2. Weighted means are weighted by number of p l o t s . - 116 -In the I n t e r i o r , species types and inventory zones are s i g n i f i c a n t at the 1% l e v e l while the-interactions are no n - s i g n i f i c a n t . Comparing the p o t e n t i a l y i e l d of Douglas-fir stands at age 100 i n Table 5-15 and Figure 5-10, one observes Douglas-fir stands i n Zone 2 (the South Coast Region) y i e l d 251 cubic feet per acre more than i n Zone 3 (the South Coast T r a n s i t i o n B e l t ) . This i s i n agreement with general observations which f i n d Zone 2 i s more productive than Zone 3. In the I n t e r i o r , the most productive zone i s 7 (the I n t e r i o r Wet Belt) with a p o t e n t i a l y i e l d of 3599 cubic feet per acre at age 100. The next most productive zone i s 5 (the West Kootenay Region) with a p o t e n t i a l y i e l d of 2920 cubic feet per acre. Zone 5 i s followed i n pr o d u c t i v i t y by Zone 6 (the East Kootenay Region) with a p o t e n t i a l y i e l d of 2590 cubic feet per acre and Zone 4 (the I n t e r i o r Dry Belt) with 2194 cubic fe e t . The l e a s t productive zone i s 8 (the Nechako-Fraser Plateau Region) with 2064 cubic feet per acre at age 100. The dry and severe weather conditions i n the l a t t e r two zones probably i n h i b i t the growth of Douglas-fir stands. The p o t e n t i a l y i e l d i s 8282 cubic feet per acre for Coast Douglas-fir and 2793 cubic feet for the I n t e r i o r stands at age 100 (Table 5-15). In other words, the Coast Douglas-fir stands o u t y i e l d the"-Interior ones by as much as 3 times. Among the species composition types, pure Douglas-fir type stands y i e l d the best among three types investigated i n Zone 2 with a p o t e n t i a l y i e l d of 8579 cubic feet per acre which i s 831 cubic feet more than that of Douglas-fir mixed type stands at age 100. However, Figure 5 - 1 0 . P o t e n t i a l y i e l d of Do u q l a s - f i r at age 1 0 0 by types and inventory zones 9 0 0 0 I H I + H 8 0 0 0 + I H + H I H + H I + H I + H I + H I+H I+H I + H I + H I+H I + H I + H I + H I+H I + H I + H I + H I + H 5 0 0 0 + I + H I + H I + H I + H I + H I + H I + H I + H I + H I + H I + H I + H I+H I + H I + H I + H I+H I + H • I + H I + H 4 0 0 0 + I+H I + H I + H I + H + 1 + H I+H + I + H •I + H I + H I+H I + H I + H I + H I + H 1 + I + H I+H I + H I + H I + H I + H I + H I + H + I + H I + H I + H I + H 1 + I + H I + H I + H 1+ I + H 1 + I + H 2 0 0 0 + I + H I + H I+H I+H I + H I + H 1 + I + H I + H I+H I+H I + H I + H 1 + I+H I + H I+H I+H I + H I + H 1 + I + H I + H I+H I+H I + H I + H 1 + I + H I + H I+H I+H I + H I + H 1 + I + H I + H I+H I+H I + H I + H 1 + I+H I+H I+H I+H I + H I + H 1 + I + H I+H I+H I+H I + H I + H 1 + I+H I + H I+H I+H I + H I + H 1 + I + H I + H I+H I+H I + H I + H 1 + ZONE ZONE ZONE ZONE ZONE ZONE ZONE 2 3 4 5 6 7 8 Remarks: PURE--pure type;CMIX - - c o n i f e r mixed HMIx--hardvood mixed type. I : PORE +: C M I X K : H MIX ZONE 9 ZONE 10 ZONE 1 1 ZONE 12 - 118 -i n Zone 3, the reverse i s true; Douglas-fir hardwood mixed type stands produce 8518 and Douglas-fir c o n i f e r mixed type 8335 cubic feet per acre which are s u b s t a n t i a l l y higher than the p o t e n t i a l y i e l d of pure Douglas-fir type stands (7540 cubic feet per acre) at age 100. The r e s u l t s suggest that on a very f e r t i l e s o i l , such as that i n the South Coast Region, pure Douglas-fir type stands tend to grow e x c e l l e n t l y ; however, on a medium s o i l , i t i s advantageous to e s t a b l i s h Douglas-fir hardwood mixed type or Douglas-fir conifer mixed type stands. In the I n t e r i o r , d i f f e r e n c e i n y i e l d among these three type stands i s not as contrasting as that on the Coast, because the l e s s productive s i t e e f f e c t i v e l y reduces the d i f f e r e n c e . The establishment of Douglas-fir hardwood type stands i n the I n t e r i o r i s l e s s desirable than on the Coast, because the stands y i e l d the l e a s t among three types i n a l l zones (Table 5-15). Again Zone 7 shows the advantages of es-t a b l i s h i n g Douglas-fir conifer mixed stands on the medium s o i l where Douglas-fir c o n i f e r mixed stands produce 3812 cubic feet per acre compared to 3398 for pure Douglas-fir stands at age 100. In general, the p o t e n t i a l y i e l d i s 3114 f o r Douglas-fir conifer mixed type stands and 2582 cubic feet per acre f o r pure Douglas-fir type stands. In other words, the establishment of mixed stands e f f e c t i v e l y increases f o r e s t p r o d u c t i v i t y by 21%. - 119 -5.7.2 Spruce The analysis of variance f o r volume y i e l d of I n t e r i o r spruce stands indicates that species composition type e f f e c t i s not s i g n i f i c a n t while the difference i n volume y i e l d among 8 inventory zones i s s i g n i f i c a n t . Interactions for species types and zones are also s i g n i f i c a n t which indicate that y i e l d of three species composition types changes from zone to zone. On zonal p r o d u c t i v i t y of spruce stands i n the I n t e r i o r , Zone 5 (the WestKootenay Region) with a p o t e n t i a l y i e l d of 4364 cubic feet per acre at age 100 i s the most productive zone. Zone 9 (the Central I n t e r i o r Region), Zone 7 (the I n t e r i o r Wet B e l t ) , and Zone 8 (the Nechako-Fraser Plateau Region) are equally productive with 4131, 4119, and 4076 cubic feet per acre at age 100. The p o t e n t i a l y i e l d of spruce stands i n Zone 4 (the I n t e r i o r Dry Belt) i s s l i g h t l y i n f e r i o r to the above three zones with a y i e l d of 3944 cubic feet per acre. The p r o d u c t i v i t y of spruce stands i n Zone 6 (the East Kootenay Region) and Zone 12 (the North Eastern P l a i n s Region) i s 3581 and 3561 cubic feet per acre, r e s p e c t i v e l y . Zone 11 i s , therefore, the l e a s t productive zone f o r spruce stands i n I n t e r i o r (Figure 5-11). The change i n l a t i t u d e does not account for the v a r i a t i o n i n y i e l d capacity of spruce stands i n I n t e r i o r . K o i v i s t o (1971) i n Finland observed that the mean wood production c a p a c i t i e s of the growth regions from south to north follow the sequence 100 — 82 — 56 — 36 based on the increment of southernmost region as 100. If the y i e l d capacity of Figure 5-11. P o t e n t i a l y i e l d s of I n t e r i o r spruce at aqe 100 by species types and inventory zones 4900 + 4000-I 3000< DA < H 3 o > 2000-i 1000< 0.0 ZONE 2 H I : PURE + H + : CMIX + H H : HMIX + H I + H I + H I + H I I + H I I *H I I + H 1 + 1 + + H I I + H I I + H 1 + I + H I I + H I I + H 1 + I+H I H I + H I I + H 1 + I+H I I + H I+H I H 1 + H 1 + I+H I I + H I + H I . H I + H 1 + I+H I I + H I + H I + H I + H 1 + I + H I I + H I + H I + H I + H 1 + I+H I H I + H I + H I + H I+H I + H I+H + I H I + H I + H I + H I + H I + H I+H + I H I + H I + H I + H I+H I + H I+H + I H I+H I + H I+H I + H I + H I+H + I H I + H I + H I + H I + H I + H I + H + I + H I+H I + H I + H I + H I + H I+H + I + H I + H I + H I + H I + H I+H I+H + I + H I + H I + H I + H I + H I + H I+H 1 + I + H I + H I + H I + H I + H I + H I+H 1 + I + H I + H 'I + H I + H I + H I + H 1+ H 1 + I + H I + H I + H I + H I + H I+H I+H I + H I + H I + H I + H I + H I + H I + H I+H I + H I + H I + H I + H I + H I + U I + H I+H I+H I + H I + H I + H I + H I + H I + H I+H I + H I + H I + H I + H I + H I + H I + H I+H I + H I + H I + H I + H I+H I + H I + H I+H I + H I + H I + H I + H I + H I + H I + H I+H I + H I + H I + H I + H I + H I + H I + H I+H I + H I + H I + H I + H I + H I + H I + H I+H I + H I + H I+H I + H I + H I + H I + H 1+ H I+H I + H I + H I + H I + H I + H I + H I + H I + H I + H I + H I+H I + H ' I + H I + K I+H I + H I + H I + H I + H I + H I + H I + H I+H I + H I + H I + H I + H I + H I + H I + H I + H I + H I + H I + H I + H I + H I+H 1 + H I + H I + H I + H I+H I + H I + H I + H I + H I+H I + H I + H I + H I + H I + H I + H I + H I + H I+H I + H I + H I + H I + H I + H I + H I+H I + H I + H I + H I + H I + H I + H I + H I + H I + H I + H I + H I + H I + H I + H I + H I+H I + H I + H I + H I + H I+H I + H I + H I+H I + H I + H I + H I + H I + H I + H I + H I+H I + H I + H I + H I + H I+H I + H I + H I+U I + H I + H ONE ZONE ZONE ZONE ZONE ZONE ZONE ZONE ZONE 4 5 6 7 8 9 10 11 12 ho o Remarks: PURE—pure type;CMIX-hardwood mixed type. -c o n i f e r mixed type,HMIX__ - 121 -spruce stands i n the southernmost zone which i s Zone 5 i s 100, the p r o d u c t i v i t y of spruce stands i n the I n t e r i o r i s 92, 100, 82, 94, 93, 95, 63, and 82 r e s p e c t i v e l y f o r Zone 4 to Zone 12. Zones such as the Central I n t e r i o r Region, the I n t e r i o r Wet Belt and the Nechako-Fraser Plateau are higher i n l a t i t u d e than the I n t e r i o r Dry Be l t , and the East. Kootenay Region; nevertheless, the former are more productive than the l a t t e r (Map 1). There i s no d i f f e r e n c e i n y i e l d among three spruce species types studied as indicated by the non-significant F-value i n Table 5-17. The p o t e n t i a l y i e l d s f o r pure spruce, spruce-conifer mixed, and spruce-hardwood mixed type stand are 3901, 3922, and 3722 cubic feet per acre, res p e c t i v e l y , at age 100 (Table 5-18). The fact that i n t e r a c t i o n s between species types and inventory zones are s i g n i f i c a n t at the 1% l e v e l deserves a further examination of the performance of i n d i v i d u a l types from zone to zone. As shown i n Table 5-18, pure spruce type stands ou t y i e l d spruce-conifer mixed type stands by 770 cubic feet per acre i n Zone 4. In other words, the f o r e s t p r o d u c t i v i t y w i l l increase by 20.6% by e s t a b l i s h i n g pure spruce type stands instead of spruce-conifer mixed type stands. S i m i l a r l y , the p r o d u c t i v i t y of spruce stands w i l l gain by 13.2% i n Zone 6 and 27.1% i n Zone 12 (the North-eastern Pla i n s Region) by forming pure spruce stands rather than spruce-conifer mixed type stands. In Zones 7 and 8 the advantages of pure spruce type stands i n y i e l d over spruce-conifer mixed type are n e g l i g i b l e (5.1 and 2.5%, r e s p e c t i v e l y ) . - 122 -In Zone 5, spruce-hardwood mixed type stands p o t e n t i a l l y y i e l d 6739 cubic feet per acre at age 100; however, the estimate was based on 1 sample pl o t (Table 5-2). Spruce-conifer mixed type stands which y i e l d 4711 cubic feet per acre are about 13.4% more productive than pure spruce type stands i n t h i s zone. The figures i n Table 5-8 strongly support the formation of spruce-hardwood mixed type stands i n Inventory Zone 9 (the Central I n t e r i o r Region) where the estimates of p o t e n t i a l y i e l d were based on f a i r l y large number of samples (Table 5-2). Spruce-hardwood mixed type stands i n t h i s zone p o t e n t i a l l y y i e l d 4408 cubic feet per acre which i s 453 cubic feet more than pure spruce stands. In other words, the stand p r o d u c t i v i t y w i l l increase by 11.5% by forming spruce-hardwood mixed type stands instead of pure spruce ones. 5.7.3 Lodgepole Pine The analysis of variance table i n Table 5-20 suggests that d i f f e r e n c e i n species composition type has no bearing on the volume y i e l d of I n t e r i o r lodgepole pine stands. Volume y i e l d d i f f e r s among inventory zones as shown by the F-value which i s s i g n i f i c a n t at the 1% l e v e l . The i n t e r a c t i o n s f o r species composition types and inventory zones are also s i g n i f i c a n t at 1% l e v e l . To f a c i l i t a t e the i n t e r p r e t a t i o n of the analysis of variance table, p o t e n t i a l y i e l d of lodgepole pine stands at age 100 was calculated based on the estimated constants i n Table 5-19 and presented i n Table 5-21 and Figure 5-12. Fiqure .5- 12. P o t e n t i a l y i e l d s of I n t e r i o r lodqepole pine stands at aqe 100 4900 + I: PURE +: cnix H: HMIX 4000H 300G< 2000< S o > 1000^ 0.0 ZONE 2 ZONi 3 + I I I + + I + + + I + + + + I + H + 1 + I+H I + H + + 1 + I+H I + H + 1 + I 1 + I+H I + H I + H I 1 + 1 + 1 + I + H I + H I + H I + H 1 + 1 + I + H I+H I I + H I + H I + H 1 + 1 + I + H I+H I I + H I + H I + H 1 + 1 + I + H' I+H I I+H I + H I + H I + I + H I + H I+H I I + H I + H I + H 1 + I + H I + H I+H I I + H I + H I + H 1 + I+H I+H I+H I I + H I+H I + H 1 + I + H I + H I+H I H I + H . I+H I + H 1 + I + H I + H I+H I H I + H I + II I + H 1 + T + H I + H 1+H I H I + H I + H I+H 1 + I + H I + H I+H I H I + H I + H I + H I* I + H I + H . I+H I H I + H I + H I + H 1 + I + H I+H I+H I + H I + K I + H I + H 1 + I + H ' I + H I+H I+H I+H I + H I + H 1 + I+H I+H I+H I + H I + H I + H I + H 1 + I + H I + H 1+ H I + H I + H I + H I + H 1 + I + H I + H I + H I + H I + H I + H I + H 1 + 1+H I + H 1+ H I + H I+H I + H I + H 1 + I + H I + H I + H I + H I + H I + H I+H 1 + I + H I + H I+H I + H I + H I + H I + H 1 + I + H I + H I+H I + H I + H I + H I + U 1 + I + H I + H I+H I+H I + H I + H I + H 1 + I+H I + H I+H I + K I + H I + H I + H 1 + I + H I + H I+H I + H I + H I + H I + H 1 + I + H I + H I+H I + H I + H I+H I + H 1 + I + H I + H I+H I + H I+H I + H I + H 1 + I + H I + H I + H I + H I + H I + H I + H 1 + I + H I + H I+U I + H I + H I + H I + H 1 + I + H I + H I+H I + H I + H I + H I + H 1 + I + H I + H I+H I + H I + H I + H I + H 1 + I + H I + H I+H 1 + H I + H I + H I + H 1 + I + H I + H I+H I + H I + H I + K I + H 1 + I + H I + H I + H I + H I + H I+H I + H 1 + I + H I + H I+H I + H I + H I + H I + H 1 + I + U I + H I+H I + H I + H I + H I + H 1 + I + H I + H I+H I + H I + H ONE ZONE ZONE ZONE ZONE ZONE ZONE ZONE ZONE 4 5 6 7 8 9 10 11 12 Remarks:PURE--pure type;CMIX--conifer mixed typejHMIX— hardwood mixed type. N5 - 124 -The zonal productivity v a r i e s among In t e r i o r lodgepole pine stands as indicated by the analysis of variance table (Table 5-20). The p o t e n t i a l p r o d u c t i v i t y of I n t e r i o r lodgepole pine can be ranked i n descending order as: the Central I n t e r i o r Region (Zone 9), the North-eastern P l a i n s Region (Zone 12), the North Western Plateau Region (Zone 10), the Nechako-Fraser Plateau Region (Zone 8), the East Kootenay Region (Zone 6), the I n t e r i o r Dry Belt (Zone 4), the I n t e r i o r Wet Belt (Zone 7), and the West Kootenay Region (Zone 5). Difference i n p o t e n t i a l y i e l d between the best zone (Zone 9) and the l e a s t productive zone (Zone 5) i s smaller i n lodgepole pine than i n spruce. The high F-value (47.18) for forest inventory zone seems to be a t t r i b u t e d to the f a i r l y large number of sample plots used. According to K r a j i n a (1969), general c l i m a t i c requirements of lodgepole pine are the widest for any coniferous tree i n B r i t i s h Columbia. The i n s e n s i t i v i t y of lodgepole pine toward changes of c l i m a t i c conditions probably explains i t s r e l a t i v e l y homogeneous production capacity among inventory zones (Table 5-21 and Figure 5-12) . Taking the p o t e n t i a l production capacity of Zone 5 as 100, the r e l a t i v e production c a p a c i t i e s f o r zones 4 to 12 are 103, 100, 106, 101, 106, 114, 107, and 110%, r e s p e c t i v e l y . The change i n l a t i t u d e , and therefore c l i m a t i c conditions, has l i t t l e bearing on the y i e l d capacity of lodgepole pine stands. Species composition types do not s i g n i f i c a n t l y a f f e c t volume y i e l d of lodgepole pine stands; however, the in t e r a c t i o n s f or types and inventory zones need further examination because they are s i g n i f i c a n t at the 1% l e v e l (Table 5-20). - 125 -Pure lodgepole pine type stands i n the North Western Plateau Region y i e l d s u b s t a n t i a l l y higher than lodgepole pine mixed type stands (Table 5-21, Figure 5-12). Comparing the estimates f o r pure lodgepole pine type stands with those of lodgepole pine-conifer mixed type stands, one can be misled by the figures because the l a t t e r was based on only 1 sample p l o t . The p o t e n t i a l y i e l d of pure lodgepole pine stands outweighs that of lodgepole pine-hardwood mixed type stands by as much as 1513 cubic feet per acre at age 100; a magnitude of such high order should not be overlooked by managing f o r e s t e r s . The pr o d u c t i v i t y increases by 51% by e s t a b l i s h i n g pure lodgepole pine stands instead of lodgepole pine-hardwood mixed stands i n Zone 10. The y i e l d s of pure lodgepole pine stands i n Zones 5 and 7 are better than the corresponding y i e l d s of lodgepole pine-conifer mixed type stands; but, the increased magnitudes are of minor order (6% and 2%, r e s p e c t i v e l y ) . Lodgepole pine-conifer mixed type stands are considerably more productive than pure lodgepole pine type stands i n Zones 4, 8, 9, and 12. The p r o d u c t i v i t y of lodgepole pine stands increases by l l v 4 , 7.2, 9.8, and.22.8%in Zones 4, 8, 9, and 12, resp e c t i v e l y , by establishment of lodgepole pine-conifer mixed type stands. 5.7.4 Summary of Volume Y i e l d by Species Composition and Inventory Zones From the above r e s u l t s and discussions, i t i s reasonable to conclude that - 126 -(1) on the Coast pure Douglas-fir stand i s more productive on very f e r t i l e s i t e s , while Douglas-fir hardwood mixed type stands are recommended on medium s i t e s i f f o r e s t e r s are a f t e r more wood production per unit area. (2) In I n t e r i o r B.C., Douglas-fir mixed type stands c o n s i s t e n t l y y i e l d more i n volume than do pure Douglas-fir stands. (3) For spruce and lodgepole pine stands i n the I n t e r i o r , the e f f e c t of species composition types on volume y i e l d i s non-significant while i n t e r a c t i o n s f o r species types and inventory zones are s i g n i f i c a n t . Interpretations of these i n t e r a c t i o n s suggested that the advantages of monocultural or m u l t i c u l t u r a l p r a c t i c e s should not be over-generalized. Pure stands are more productive i n some zones but le s s i n the others. The same argument applies equally well to advocates f o r m u l t i c u l t u r a l p r a c t i c e s . Growth of forest trees i s e s s e n t i a l l y site-dependent; over-generalization of experimental r e s u l t s i s often hazardous. Before a decision can be reached on what species composition type to e s t a b l i s h , f o r e s t e r s should c a r e f u l l y i n v e s t i g a t e l o c a l s i t e q u a l i t y and past y i e l d h i s t o r y of various f o r e s t stands. (4) The p r o d u c t i v i t y of forest land can be amplified by establishment of a stand appropriate to the area. This study i d e n t i f i e d theooptimum species composition types for Douglas-fir, spruce, and lodgepole pine i n various inventory zones. Because of lack of edaphic, c l i m a t i c and b i o t i c information, i t i s impossible to explain the reasons f o r a species type y i e l d i n g more than others i n a p a r t i c u l a r zone. Further studies along t h i s l i n e are of p r a c t i c a l value to f o r e s t e r s i n s e l e c t i o n of an optimum species composition type. - 127 -5.8 Influence of Species Composition Types on Volume Y i e l d The above analyses compared net volume y i e l d s of three species composition types i n inventory zones f o r three major commercial species i n B.C. The comparison was r e a l i n that i t involved forest stands which are presently growing i n B.C., with adjustment made only to a common age. Difference i n y i e l d among these three composition types might be a t t r i b u t e d to differe n c e i n s i t e q u a l i t y . Consequently, the influence of species composition types on volume y i e l d can be better perceived by taking s i t e index and stand age as covariates and adjusting them to a common basis . By adjusting data to a common s i t e index, the analyses serve to answer a hypothetical question:. To what extent do they d i f f e r i n y i e l d among three species composition types i f stands of the three composition types are growing on the same sit e ? Analysis of variance tables f o r net volume y i e l d f o r t h i s purpose are-shown i n Tables 5-22, 5-24, 5-26 and 5-28 for Coast Douglas-fir, I n t e r i o r Douglas-fir, spruce, and lodgepole pine, r e s p e c t i v e l y . The estimated constants are also presented i n Tables 5-23, 5-25, 5-27, and 5-29, correspondingly. The analysis of variance tables show that s i t e and age resp e c t i v e l y account f o r a large portion of the v a r i a t i o n i n volume y i e l d for a l l three species. With the exception of Coast Douglas-fir stands, s i t e accounts f o r more v a r i a t i o n than does stand age. Table 5-22. A n a l y s i s of variance f o r net volume y i e l d of the Coast D o u q l a s - f i r stands a d j u s t e d f o r s i t e index and stand age. Source of D.F. V a r i a t i o n Species type 2 F.I.Z. 1 Type X F. I. Z. 2 C o v a r i a t e s — S i t e 1 Log Age 1 E r r o r 627 Mean Squares 2.3710X10 6 1.0323X10 6 6. 5791X 10 6 2.4070X10 9 3.4257X10 9 3.7470X10 6 F-Value 0. 63 0.27 1.76 642. 40** 91 4.26** Tabulated F-Value (5%) 3.01 3.86 3.01 3.86 3.86 • • S i g n i f i c a n t at 1% l e v e l . Table 5-23. Estimated c o n s t a n t s from volume y i e l d data of Coast D o u q l a s - f i r stands (Unit: Cubic f e e t per acre) TYPE MAIN EFFECTS PURE 145.0 CMIX 98.0 HMIX -296.0 GENERAL MEAN INVENTOBY ZONE 2 3 -70.0 70.0 INTERACTIONS 248.0 -248.0 -114.0 114.0 -134.0 134.0 -29642.0 SITE = 66.0 S3 VO LOG AGE = 15411.0 Remarks: PURE, CMIX, and HMIX as de f i n e d i n Table 5-2. Table 5-24. Analysis of variance for net volume y i e l d of the Inte r i o r Douglas-fir stands adjusted for s i t e index and stand age. Source of Variation D.F. Mean Squares F-Value Tabulated F-Value (5%) Species type 2 F.I.Z. 4 Type x F.I.Z. 7 Covariates--Site 1 Loq Age 1 1.48306X10 1.14227X 10 2.31154X10 2.01114X10 1.09590X10 10.06** 7.75** 1.57 1364.67** 734.63** 3. 00 2. 38 2.02 3. 85 3. 85 Error 2316 1. 47371X 10 ** S i q n i f i c a n t at 1% l e v e l . Table 5-25. Estimated c o n s t a n t s f o r net volume y i e l d s o f the I n t e r i o r D o u q l a s - f i r stands TYPE MAIN EFFECTS PURE 16.92 CMIX 350.78 HMIX -367.70 FOREST INVENTORY ZONE 4 5 6 7 8 •480. 70 -2. 98 339.69 338. 94 -194. 95 INTERACTIONS 334.42 -154.13 -16.91 -134.82 -38.56 253.95 -145.19 -185.62 38.30 38.56 •598.36 299.32 202.52 96.52 GENERAL MEAN = -12329.0 SITE = 80.59 LOG AGE = 4260.53 Remarks: PURE, CMIX, and HMIX as d e f i n e d i n T a b l e 5-2. Unit: Cubic f e e t per a c r e . Table 5-26. Analysis of variance f o r net volume y i e l d s of the I n t e r i o r spruce stands adjusted for s i t e index and stand age. Source of D.F, Variation Species type 2 F.I.Z. 7 Type x F. I. z . 14 Covariates— Site 1 Log Age 1 Error 2589 Mean Sguares 1.5483X10 2. 5496X10* 5. 5028X10* 2. 8543X10' 2. 2287X109 2. 1005X106 F-Value 0.74 1.21 2.62* 1 358.88** 1061.02** Tabulated F-Value (5X) 3.00 2.02 1.70 3. 8 3.85 * Signifi c a n t at 1% l e v e l . ** Significant at 5% l e v e l . Table 5-27. Estimated constants from volume data of the I n t e r i o r spruce stands TYPE HAlN EFFECTS PURE 26.85 CMIX 107.55 HMIX - 134.40 GENERAL MEAN FOREST INVENTORY ZONE 4 5 6 7 8 9 -9.53 646.66 -371.47 -114.81 -113.57 -42.57 INTERACTIONS 568.94 -768.04 481.62 -77.71 18.64 -228.08 •205.28 -436. 63 -11.86 -91 .08 289. 15 -56.80 •363.66 1204.68. -469.76 168.79 -307.79 284.87 •15278.77 SITE = 70.93 LOG AGE = 6736.73 11 12 193.32 -188.03 -90.04 94.68 449.91 62.59 •359.86 -157. 27 UJ Remarks: PURE, CMIX, and HMIX as d e f i n e d i n Table 5-2. U n i t : Cubic f e e t per acre. Table 5-28. Analysis of variance for net volume y i e l d s of Interior lodqepole pine stands adjusted for s i t e index and stand age. Source of Variation D.F Mean Squares F-Value Tabulated F-Value (555) Species type F.I.Z. Type x F.I.Z. Covariates--Site Loq Age Error 2 7.53593X101 7 1.01808X10 13 6.14726X10 1 3.74457X10 1 3.72277X10^ 4174 1.45367X10 5. 18** 7. 00** 4.23** 2575. 94** 2560.94** 3. 00 2.02 1.73 3.86 3. 86 * * S i g n i f i c a n t at 1% l e v e l . Table 5-29. Estimated constants f o r net volume y i e l d s of I n t e r i o r lodgepole pine stands TYPE MAIN EFFECTS 4 5 4.41 -151.08 -207. 39 -5.81 221.93 -148.84 -14. 54 154. 64 FOREST INVENTORY ZONE 6 7 8 9 10 3.66 -207.56 131.64 346.69 -401.40 INTERACTIONS -70.26 78.30 -73.43 -252.30 1094.62 70.26 62.32 263.00 187.97 -948.11 PORE 109.66 CMIX 51.89 H MIX -161.55 GENERAL MEAN = -15477.60 SITE = 72.18 LOG AGE = 6689.94 12 273.65 -563. 74 291.48 -140.61 -189.57 64.32 -146.51 272.26 Remarks: PURE, CMIX, and HMIX as de f i n e d i n Table 5-2. U n i t : Cubic f e e t per acre. - 136 -5.8.1 Douglas-fir The e f f e c t s of species composition types on volume y i e l d are nonsignificant f o r Coast Douglas-fir stands a f t e r the adjustment has been made for s i t e i n d e x (Tables 5-22 and 5-23). Therefore, differences i n y i e l d among three stand types f o r Coast Douglas-fir stands can be a t t r i b u t e d completely to the diff e r e n c e i n s i t e q u a l i t y . In I n t e r i o r Douglas-fir stands, however, the species composition e f f e c t i s s i g n i f i c a n t at the 1% l e v e l (Table 5-24). Because the i n t e r -actions f o r species types and inventory zone are no n s i g n i f i c a n t , one can d i r e c t l y i n t e r p r e t the species type e f f e c t s shown i n Table 5-25. The estimated e f f e c t s are 16.92, 350.78, and -367.70 cubic feet per acre for pure Douglas-fir, D o u g l a s - f i r - c o n i f e r mixed type, and Douglas-fir-hardwood mixed type stands, r e s p e c t i v e l y . Interpretations of these e f f e c t s are: with the same s i t e q u a l i t y and age, pure Douglas-fir stands can be expected to produce 16.92 cubic feet per acre more than general average y i e l d of a l l Douglas-fir stands and Douglas-fir mixed type stands 350.78 cubic feet per acre more; on the other hand the Douglas-fir hardwood mixed type w i l l reduce the general average y i e l d by 367.70 cubic feet per acre. 5.8.2 Spruce The analysis of variance for volume y i e l d of I n t e r i o r spruce stands shows that at the 5% l e v e l e f f e c t s of species composition types are not s i g n i f i c a n t l y d i f f e r e n t , although i n t e r a c t i o n s between species composition types and inventory zones are s i g n i f i c a n t l y d i f f e r e n t . - 137 -The estimated e f f e c t s are 26.85, 107.35, and -134.40 cubic feet per acre for pure spruce type, spruce conifer mixed type, and spruce hardwood mixed type, r e s p e c t i v e l y (Table 5-27). 5.8.3 Lodgepole Pine In I n t e r i o r lodgepole pine stands, Table 5-28 shows that e f f e c t s of species composition types, inventory zones, and i n t e r a c t i o n s between these two factors d i f f e r s i g n i f i c a n t l y . The estimated e f f e c t s are 109.66, 51.89, and -161.55 cubic feet per acre for pure lodgepole pine type, lodgepole pine co n i f e r mixed type, and lodgepole pine hardwood mixed type stands, r e s p e c t i v e l y . The difference i n estimated e f f e c t s between pure lodgepole pine type and lodgepole pine-conifer mixed type i s n e g l i g i b l e from the p r a c t i c a l point of view. However, pure lodgepole pine stands outyield lodgepole pine hardwood type stands by 271 cubic feet per acre on average. The differences i n e f f e c t s among these three composition types are not very s i g n i f i c a n t i n a p r a c t i c a l sense; the high F value shown i n Table 5-28 for t h i s source of v a r i a t i o n probably i s due to the large number of sample plo t s used which e f f e c t i v e l y reduces the variance. In addition, the i n t e r a c t i o n term i s s i g n i f i c a n t , therefore, i t i s r i s k y to over-emphasize the species composition type e f f e c t . The e f f e c t of hardwood mixed type shows a negative value c o n s i s t e n t l y i n the above four analyses; f o r t h i s reason, i t i s l o g i c a l to conclude that hardwood mixed stands i s the l e a s t desirable composition structure f or Douglas-fir, spruce, and lodgepole pine i n B.C. - 138 -Tables 5-24 and 5-28 show that the e f f e c t of forest inventory zone i s s i g n i f i c a n t at the 1% l e v e l f or volume y i e l d of I n t e r i o r Douglas-fir, and lodgepole pine stands a f t e r data have been adjusted for s i t e index. Interpretations for the r e s u l t s are as follow: zonal v a r i a t i o n i n volume y i e l d cannot be completely a t t r i b u t e d to difference i n s i t e q u a l i t y alone; factors such as decay, form, taper, and bark thickness contribute s i g n i f i c a n t l y to volume y i e l d as w e l l . 5.9 Y i e l d s of Douglas-fir and Conifer Mixed Type Stands i n the I n t e r i o r The previous sections have demonstrated that Douglas-fir conifer mixed type stands produce more than pure Douglas-fir ones i n the I n t e r i o r . The reasons why mixed c o n i f e r stands grow more merit further i n v e s t i g a t i o n s . The average height, number of trees, basal area,and volume, per acre f o r I n t e r i o r pure and mixed Douglas-fir stand at various age class are i l l u s t r a t e d i n Tables 5-30, 5-31, and 5-32 for s i t e c l a s s I, I I , and I I I , r e s p e c t i v e l y . Inspection of dominant and codominant tree height i n Tables 5-30, 5-31, and :5-32 for three s i t e classes suggests no di f f e r e n c e i n height growth between pure and mixed con i f e r types. The same conclusion also can be r e a d i l y drawn from Figure 5-13, 5-14, 5-15 where the means at various age classes have been smoothed with aWeibull-type growth function (Yang et a l . , 1978) by weighted l e a s t squares methods. T a b l e 5 - 3 0 . Comparison o f h e i q h t , number of t r e e s , b a s a l a r e a , and volume per a c r a f o r pure D o u q l a s - f i r and D o u q l a s - f i r - c o n i f e r m i x t u r e s t a n d s qrown on I n t e r i o r s i t e c l a s s I . TYPE AGE CLASS 15 25 35 45 55 65 7 5 85 95 1 0 5 115 1 2 5 135 1 4 5 NO. PLOT - 3 6 9 27 17 20 31 36 2 3 11 9 19 13 HEIGHT 50 58 6 9 81 88 95 94 104 1 0 9 107 1 12 114 1 1 5 PURE NO. TREE -. 55 151 13 3 161 220 190 176 174 143 122 1 3 5 96 131 BA 29 76 74 105 144 143 136 158 142 145 1 3 7 149 1 6 9 VOLUME 5 3 6 1 6 5 0 1 8 4 5 2 8 4 5 4 0 8 9 4 1 6 2 3 9 9 0 4 9 4 5 4 7 6 2 4 4 1 7 4 2 1 8 4 8 7 7 5 5 4 5 NO. PLOT 2 - 1 5 15 14 3 2 43 16 22 14 15 6 8 HEIGHT 30 - 60 80 82 93 9 3 97 102 1 0 7 107 1 0 7 112 1 1 8 C H I X NO. TREE 5 0 - 38 124 1 9 9 2 4 0 2 0 0 2 1 8 2 0 2 196 224 181 145 153 BA 18 - 14 84 128 161 139 164 177 180 201 1 7 9 138 198 VOLUME 2 2 0 - 2 6 5 2 2 3 6 3 3 9 8 4 581 4 1 8 7 4 8 3 4 5 6 7 6 5 6 1 6 6 1 1 9 5 4 1 4 3 7 5 3 6 6 0 9 LO VO Remarks: J J " " J J " D « > « « l l » - J « t y p e ; C M I X - - D o u q l a s - f i r - c o n i f e r mixed t y p e . B A - B a s a l Area, u n i t . H e i q h t , f e e t ; B a s a l Area, squared f e e t ; Volume, c u b i c f e e t . T a b l e 5-31. Comparison o f h e i g h t , number of t r e e s , b a s a l a r e a , and volume p-r a - r -f o r pure D o u g l a s - f i r and D o u g l a s - f i r - c o n i f e r m i x t u r e s t a n d s grown on I n t e r i o r s i t e c l a s s I I . TYPE AGE CLASS 15 25 35 45 55 65 75 85 95 105 115 125 135 145 NO. PLOT - 1 32 53 34 61 72 62 39 4 1 26 76 55 46 HEIGHT - 30 43 53 64 70 74 75 84 84 84 86 91 94 PUKE NO. TREE - 70 49 85 123 126 124 126 122 126 131 96 103 98 I BA - 53 23 43 71 74 79 94 86 101 100 100 105 115 I-1 p VOLUME — 1327 397 897 1689 1792 198 8 2233 2189 2613 2510 2545 2824 3232 I NO. PLOT - 1 6 35 29 42 55 54 37 36 13 14 23 13 HEIGHT - 30 47 56 66 75 , 76 77 84 83 88 89 87 95 CMIX NO. TREE - 12 74 112 1 18 179 163 174 176 159 166 186 149 136 BA - 4 35 56 66 103 102 106 1 12 123 124 146 128 141 VOLUME - 138 666 1252 1656 2603 275 1 2675 3021 2951 3458 3613 3456 4152 Remarks: PURE- pure D o u g l a s - f i r t y p e ; CMIX-- D o u g l a s - f i r - c o n i f e r mixed type. B A - B a s a l Area U n i t ; H e i g h t , f e e t ; B a s a l Area, squared f e e t ; Volume, c u b i c f e e t . T a b l e 5-32. Comparison o f h e i q h t , number of t r e e s , b a s a l a r e a , and volume per aara f o r pure D o u q l a s - f i r and D o u q l a s - f i r - c o n i f e r m i x t u r e s t a n d s qrown on I n t e r i o r s i t e c l a s s I I I . TYPE AGE CLASS 15 25 35 45 55 65 75 85 95 105 115 125 135 145 NO. PLOT - 10 10 41 53 94 85 41 65 35 19 41 61 73 HEIGHT -• 27 30 37 46 50 53 55 62 63 64 65 70 71 PURE NO. TREE - 38 42 43 61 79 80 79 105 90 105 82 91 85 I BA - 16 22 21 32 39 44 47 62 66 73 74 84 90 h-1 K1 VOLUME - 214 378 295 579 729 871 956 1332 1449 1399 152 3 1832 1945 1 NO. PLOT 1 1 1 16 25 47 39 20 24 19 7 13 17 20 HEIGHT 20 20 30 36 46 53 56 56 64 64 61 64 69 68 CMIX NO. TREE 5 12 10 41 70 99 112 92 137 116 91 153 126 90 BA 2 7 3 17 35 54 59 50 79 63 91 103 99 80 VOLUME 15 38 42 279 670 1 184 1300 1 041 1724 1383 2143 2305 2194 1780 Remarks: PURE— pure D o u q l a s - f i r t y p e ; CMIX— D o u q l a s - f i r - c o n i f e r mixed t y p e . BA-- B a s a l Area. U n i t : Heiqht, f e e t ; B a s a l Aara, squared f e e t ; Volume, c u b i c f e e t . - 142 -- 143 -140 120 I 0 0 r -- 8 0 h -X x 6 0 4 0 20 h • PURE O CMIX 50 75 100 AGE (years) 50 Figure 5-14. Height growth of I n t e r i o r D o u g l a s - f i r on s i t e c l a s s I I - 144 -6 0 h 4 0 K • PURE O CMIX 50 75 AGE (years) 150 Figure 5-15. Height growth of I n t e r i o r Douglas-fir on s i t e c l a s s I II - 145 -Average basal area per acre of pure and mixed stands of various age classes i n the I n t e r i o r s i t e class I, I I , and III are l i s t e d i n Tables 5-30, 5-31, 5-32. The averages were f i t t e d to the Weibull-type growth function with a weighted l e a s t squares method. The f i t t e d curves and the means are i l l u s t r a t e d i n Figures 5-16, 5-17, and 5-18 f o r s i t e c l a s s I, I I , and III r e s p e c t i v e l y . Figure 5-16 indicates that the diff e r e n c e i n basal area per acre between pure and mixed Douglas-fir stands on good s i t e i s not s i g n i f i c a n t u n t i l stand age 85 years. Examining the number of trees per acre i n Table 5-30, 5-31, and 5-32 reveals that the diff e r e n c e i n trees per acre between pure and mixed stands i s increased at t h i s stage. Evidence i n the above Tables indicates that Douglas-fir mixed stands are capable of growing more trees per acre than pure stands are. These r e s u l t s provide support for the mixed stand advocates who argue that mixed stands can better u t i l i z e s o i l f e r t i l i t y . On medium s i t e s , the s u p e r i o r i t y of Douglas-fir mixed stands overrpure ones remains (Figure 5-17). Table 5-31 d i s c l o s e s that numbers of trees per acre remain r e l a t i v e l y constant i n both pure and mixed stands a f t e r stand age 65 and that there are co n s i s t e n t l y more stems i n mixed stands than i n pure stands. On the poor s i t e ( s i t e class I I I ) , the diff e r e n c e i n basal area per acre between pure stands and mixed stands becomes i n d i s t i n c t (Figure 5-18). While there are more trees per acre i n the mixed stands, the basal area i n pure stands i s not s i g n i f i c a n t l y d i f f e r e n t from that i n mixed stands. This could be a t t r i b u t e d to the f a c t that under a poor - 146 -AGE (years) Figure 5-16. Basal area growth of I n t e r i o r Douglas-fir on s i t e c l a s s I - 147 -Figure 5-17. Basal area growth of I n t e r i o r Douglas-fir on s i t e c l a s s II - 148 -210 180 ~ 150 OJ o o « 120 < cr < _ J 9 0 < CO < CQ • PURE O CMIX 0 50 75 100 AGE (years) 150 Figure 5-18. Basal area growth of Interior Douglas-fir on site class III - 149 -7 0 0 0 6 0 0 0 5 0 0 0 • PURE O CMIX 50 75 100 AGE (years) Figure 5-19. Volume growth of I n t e r i o r Douglas-fir on s i t e class I - 150 -70001 6 0 0 0 • PURE O CMIX 5 0 0 0 03 ? 4 0 0 0 ! 3 O > 3 0 0 0 2 0 0 0 1000 75 100 AGE (years) 150 Figure 5-20. Volume growth of I n t e r i o r Douglas-fir on s i t e c l a s s II - 151 -7 0 0 0 6 0 0 0 5 0 0 0 0> o o 3 o o > 4 0 0 0 3 0 0 0 2 0 0 0 1000 • PURE O CMIX 50 75 AGE (years ) 150 Figure 5-21. Volume growth of I n t e r i o r Douglas-fir on s i t e c l a s s I I I - 152 ~ s o i l condition, the lack of f e r t i l i t y or moisture constrains the growth of trees to such an extent that increase i n number of trees per acre does not increase the basal area per acre. The diffe r e n c e i n volume y i e l d between pure and mixed stands i s e s s e n t i a l l y i d e n t i c a l to that i n basal area (Figures 5-19, 5-20, and 5-21). The trend i s expected, since basal area (or DBH) and height are two components of the volume function. I t has been found that no diffe r e n c e i n dominant and codominant height growth ex i s t s between these two f o r e s t types. The volume y i e l d changes pr i m a r i l y according to the basal area (Figures 5-16, 5-17, and 5-18). 5.10 Y i e l d Table Construction One of the main purposes i n y i e l d studies i s to estimate forest y i e l d p r e c i s e l y at various stages of stand development. Many y i e l d functions and tables (for example, McArdle and Meyer, 1930; Vuokila, 1966; C u r t i s , 1967; Smith, 1973, 1977; Johnstone, 1976) have been prepared. In y i e l d functions the three components most often recognized are age, s i t e , and basal area expressed f o r stand density. However, i f the y i e l d tables are going to apply to a large geographic area, the question of whether or not the species composition types and inventory zones should be considered becomes important. Studies by Mulloy (1944', 1947) and Turnbull (1963) showed that mixed species are d i f f e r e n t i n y i e l d from pure stands. Previous discussions have shown that the d i f f e r e n c e i s mainly r e f l e c t e d by basal area growth. I f - 153 -v a r i a t i o n i n basal area i s being properly taken care of by a y i e l d function, i s i t necessary to consider the species composition and inventory zone effect? The l e a s t squares analysis developed i n t h i s study o f f e r s a unique feature to consider simultaneously q u a l i t a t i v e and quantitative v a r i a b l e s i n y i e l d a n a l y s i s . Tables 5-33, 5-34, 5-35, and 5-36 show that the r e l a t i v e importance, as indicated by F-values, of species composition type, f o r e s t inventory zones, i n t e r a c t i o n s f o r type and zone, s i t e , age, age i n logarithmic scale, height, number of trees per acre, average DBH, basal area per acre, r e l a t i v e stand density, and height x basal area (HT x BA) to volume y i e l d functions f or Coast Douglas-fir, I n t e r i o r Douglas-fir, I n t e r i o r spruce, and I n t e r i o r lodgepole pine, r e s p e c t i v e l y . In a l l four sets of inventory data analyzed, the factors HT x BA and basal area are the most important v a r i a b l e s among a l l investigated. The r e s u l t s are i n close agreement with the study of Smith (1973) who investigated the f e a s i b i l i t y of preparing v a r i a b l e density y i e l d tables. A stepwise elimination of these q u a l i t a t i v e and quantitative variables provides an excellent method to asce r t a i n the importance of a v a r i a b l e i n r e l a t i o n to volume y i e l d . Tables 5-37, 5-38, 5-39, and 5-40 i l l u s t r a t e the stepwise elimination procedures and F-values of va r i a b l e s at various stages f o r Coast Douglas-fir, I n t e r i o r Douglas-fir, spruce, and lodgepole pine volume y i e l d data r e s p e c t i v e l y . Table 5-33. Analysis of variance for Coast Douqlas-fir stands Source of D.F. Variation Species type 2 F.I.Z. 1 Type x F.I.Z. 2 C o v a r i a t e s — Site 1 Aqe 1 Log Age 1 Height 1 No. Trees 1 Avg. DBH 1 Basal Area 1 Relative Stand 1 Density Height X Basal A 1 Error 620 •Si g n i f i c a n t at 1% l e v e l . * * S i g n i f i c a n t at 5% l e v e l . Basal A: Basal area. Mean Squares 5 5.78580X10 2.22209X10 5 4.86580X10 4.21824X106 6 4.99141 X10 2.22381X106 5 6.15142X10 5 6.78487X10 2.73457X10 7 3.59477X10 6 4.79853X10 7 3.98991X10 4.06263X105 net volume y i e l d of F-Value 1.40 5.47** 1. 20 10.38** 12. 28** 5.51* 1.51 1.67 0.67 88.48** 11.81** 98.21** Tabulated F-Value (555) 3.01 3.86 3. 01 3.86 3. 86 3. 86 3. 86 3. 86 3.86 3. 86 3.86 3. 86 Table 5-34. Analysis of variance for net volume yi e l d s of Interior Douqlas-fir stands • Source of Variation D. F. Mean Squares F-Value Tabulated F-Value (5%) Species type 2 2. 04948X 105 1.24 3.00 F.I.Z. 4 1. 44 153X105 0.87 2. 38 Type x F. I. Z. 7 2. 64390X105 1.6 1 2.02 C o v a r i a t e s — Site 1 6. 06500X 103 . 0.04 3. 85 Aqe 4. 01504X106 24.4 1** 3. 85 Log Age v 1. 83111X 106 11.13** 3.85 Height 1. 01642X 106 6.18* 3.85 No. Trees 1 4. 46446X 106 28.23** 3.85 Avg. DBH 1 3. 70467X106 22.52** 3. 85 Basal Area 1 8. 96057X 106 54.47** 3.85 Relative Stand 1 3. 19160X 106 19.40** 3.85 Density Height X Basal A 1 6.66622X107 404. 98** 3.85 Error 2309 164597.0 •S i g n i f i c a n t at 5% l e v e l ; * * S i g n i f i c a n t at 1% l e v e l . Basal A: Basal Area. Table 5-35. Analysis of variance for Interior spruce stands Source of Variation Species type 2 F.I.Z. 7 Type x F.I.Z. 14 C o v a r i a t e s — Site 1 Aqe 1 Loq Age 1 Heiqht 1 No. Trees 1 Avq. DBH 1 Basal Area 1 Relative Stand 1 Density Heiqht x Basal A 1 Error 2582 Mean Squares 4. 4777X1 06 4. 1875X105 1 . 1289X106 44401.0 1 .3566X106 1.6891X105 1.3306X106 9.0562X105 69011.0 2.1745X107 2.3866X 106 7.7008X107 3.1098X105 * S i q n i f i c a n t at 5% l e v e l ; * * S i q n i f i c a n t at 15? l e v e l . Basal A = Basal Area. net volume y i e l d data of F-Value 14. 40** 1.35 3. 63** 0. 14 4. 36* 0.54 4. 28* 2.91 0. 22 69. 92** 7.67** 247. 63** Tabula ted F-Value (555) 3.00 2.02 1.70 3. 85 3.85 3. 85 3. 85 3. 85 3. 85 3. 85 3.85 3. 85 Table 5-36. Analysis of variance for net volume y i e l d of Inte r i o r lodqepcle pine stands Source of Variation D. F. Mean Squares F- Value Tabulated F-Value Species type 2 F.I.Z. 7 Type x F.I.Z. 13 C o v a r i a t e s — Site Aqe Loq Aqe Heiqht No. Trees Avq. DBH Basal Area Relative Stand Density Height x Basal A Error 4167 341511.0 6 1.35337X10 227514. 0 362497.0 350023.0 592440.0 175710.0 18426.3 15104.3 5. 96779X106 809279.0 1.31858X10 163838. 0 8 2.08 8.26** 1.39 2. 21 2. 14 3.62 1.07 0.1 1 0.09 36.42** 4. 94* 804.81** 3.00 2.02 1.73 3.85 3. 85 3.35 3. 85 3.85 3. 85 3.85 3.85 3.85 •Si g n i f i c a n t at 5% l e v e l ; * * S i g n i f i c a n t at 1% l e v e l . Basal A = Basal Area. Table 5-37. Comparison of F-values in Coast Douqlas-fir y i e l d analysis QUALITATIVE QUANTITATIVE VARIABLES SP . TYPE F.I.Z. TYPE X F. I.Z SIT S AGE LOG AGE HE IGUT NO. OF TREES AVG. DBH BASAL AREA R. S.D. HT X 3 D. F. 2 1 2 1 1 1 1 1 1 1 1 1 STEP 1 1. 40 5. 47 1.20 10.38 12.28 5.51 1.51 1.67 0. 67 88. 48 11.81 93.21 2 1. 53 5. 22 1.26 10. 37 12.33 5. 93 1 .12 10.17 — 129.21 11.18 9 9.88 3 1. 08 3. 94 1.44 10. 64 8.75 4.2 3 -- 1. 74 — 139. 64 11.79 180.82 4 -- 6. 70 -- 9. 90 10.45 4. 82 1 .42 -- 137.70 10.74 135.57 5 — 6. 35 — 9. 83 9.21 3.60 -- — — 186.51 14. 24 301.54 6 -- 8. 63 — 7.63 8. 1 8 260. 85 21. 35 301.96 7* -- 10.06 -- 16.18 -- — — 277. 19 14.01 572.54 8 -- 1 1. 93 -- 4.50 335.68 • -- 636.40 9 — 27. 69 329. 64 — 631.67 10 -- 14. 79 Remarks: R.S.D.-- Relative Stand Density; HT X BA — Heiqht X B a s a l Area. D.F.—Deqree of Freedom. •Estimated c o e f f i c i e n t s qiven in Table 5-41. Table 5-33. Comparison of F-values i n I n t e r i o r D o u q l a s - f i r y i e l d a n a l y s i s QUALITATIVE D.F. STEP 1 2 3 4 5 6 7* 8 9 10 11 SP.TYPE F.I.Z. TYPE X F.I.Z 2 4 1.24 0.87 1.25 0.88 3. 90 — 7 1. 61 1. 60 QUANTITATIVE VARIABLES SITE AGE 1 1 0.04 14.41 LOG HEIGHT NO. OF AVG. BASAL AGE TREES D3H AREA 1 1 1 1 1 11.13 6.18 - - 24.59 12.08 9.03 - - 25.21 12.87 8.10 - - 26.30 15.44 4.14 - - 35.85 27.56 - - 25. 59 16.28 - - 29.19 16.93 19.87 70. 00 74.07 6.45 --28.23 22.52 54.47 28.27 22.53 55.04 26.10 22.57 53.56 26,17 21.38 52.11 22.04 17. 34 60. 02 0.38 37.83 38.95 — 222.22 — 233.15 R. S. D. 1 19. 40 19.41 19.51 18. 40 HT X BA 1 404.98 407.41 412. 92 423.37 26.45 1234.57 15.06 1552.64 14.76 1568.88 1703.11 1692.70 — 33404. 25 — 36945. 74 U l V O Remarks: R . S . D — R e l a t i v e Stand Density; HT X BA --D.F.—Deqree of Freedom. •Estimated c o e f f i c i e n t s qiven i n Table 5-Heiqht x Bas a l Area; 41. Table 5-39. Comparison of F-values in I n t e r i o r spruce y i e l d analysis QUALITATIVE QUANTITATIVE VARIABLES STEP SP. TYPE F. I.Z. TYPE X FIZ SITE AGE LOG AGE HEIGHT NO. OF TREES AVG. DBH B A SAL R. AREA S.D HT X 3A D. F. 2 7 14 1 1 1 1 1 1 1 1 1 1 14.39 1. 35 3. 63 0.14 4. 36 0.54 4 .38 2. 91 0.2 6 69. 92 7. 67 247.63 2 14.44 1.34 3.62 -- 4. 23 0.41 10.71 3. 06 0.26 77. 88 9. 26 256.30 3 14.45' 1. 37 3. 63 — 4. 38 0.43 13.07 7. 55 140.63 9. 94 357.27 4 14.36 1. 36 3.61 -- 33.99 — 16. 76 7. 28 — 156.61 11 .12 385.57 5* 29.47 -- -- 26. 96 -- 16.79 — 224. 19 10.74 519. 90 6 29. 13 -- 21. 85 — 22.67 -- -- 216. 88 -- 527.84 7 24. 57 — — — — 9.80 — — 194.95 -- 596.31 8 25.57 — — — — — 245.97 -- 2250. 64 9 24. 37 — — -- — — — — — — 28359. 78 Remarks: R. S. D.—Relative Stand Density; HT X BA --Heiqht x Basal Area; D.F.--Deqree of Freedom. •Estimated c o e f f i c i e n t s qiven in Table 5-41. Table 5.40. Comparison of F-values in Interi o r lodqepole pine yield analysis QUALITATIVE QUANTITATIVE VARIABLES SP. TYPE F. I. Z. TYPE X FIZ SITE AGE LOG AGE D. F. 2 7 14 1 1 1 1 2. 08 8.26 1.39 2.2 1 2. 14 3.62 2 1. 92 8. 03 1.31 3.76 2. 06 2.23 3 1.94 8. 02 1.33 3.77 2. 16 1.29 4 1. 99 8.13 1.33 4. 13 1. 57 1.68 5 26. 77 9. 96 3.96 2. 02 2. 10 6 29. 19 10.02 2.66 1.84 1.45 7 29. 60 10. 24 2.28 0. 47 — 8 29. 76 10.20 4. 75 — 9 32. 48 9. 95 _ _ — mm 10* 40.92 11 31.62 IIGHT NO. OF TREES AVG. DBH BASAL R. AREA S. D HT X B.A 1 1 1 1 1 1 1 .07 0.11 0.09 36. 42 4.94 804.81 0.65 0. 05 — 94. 91 2.00 927.82 0.63 -- 141.13 2.00 1040.04 -- — 229. 14 1. 56 1860.23 ' — — -- 241.63 1. 54 1945. 19 — — -- 386.46 2054.06 — 389.46 — 2057.06 — — 411.11 2427.70 420.94 3056.58 • • — — — 485.61 - - 3185.28 72989.68 Remarks: 3 .S.D.--Relative Stand Density; HT X BA --Height x Basal Area; D.F.—Degree of Freedom. •Estimated c o e f f i c i e n t s given in Table 5-4 1. - 162 -5.10.1 Douglas-fir The F-value i n Table 5-37 indicates the average DBH and height are the l e a s t s i g n i f i c a n t v a r i a b l e to Coast Douglas-fir volume y i e l d ; consequently they were eliminated i n the f i r s t two steps. Species composition types and i n t e r a c t i o n s f o r types and zones were discarded i n the t h i r d step. These two q u a l i t a t i v e v a r i a b l e s were eliminated simultaneously because i n t e r a c t i o n s do not e x i s t once species type i s eliminated. Number of trees per acre and logarithmic age (log age) contribute n o n s i g n i f i c a n t l y to the volume y i e l d function. Steps 6 and 7 of Table 5-37 demonstrated c l e a r l y the importance of r e l a t i v e stand density i n constructing a y i e l d table f o r Douglas-fir. The r e l a t i v e stand density i s next to HT x BA, basal area, and age i n contributing to the v a r i a t i o n of Coast Douglas-fir volume y i e l d . The F-value of inventory zone i s next to the r e l a t i v e stand density with F-value 10.06 which i s s i g n i f i c a n t at 0.1% l e v e l with 2 and 626 degrees of freedom. The analysis provides d e f i n i t e evidence that subdivision of the Coast region for Douglas-fir i s j u s t i f i e d . The estimated constants shown i n Table 5-40 display that with the same age, basal area per acre, r e l a t i v e stand density, and HT x BA, Zone 2 (the South Coast Region) w i l l y i e l d 248 cubic feet per acre more than Zone 3 (the South Coast T r a n s i t i o n B e l t ) . Some of the difference i n net volume between these two inventory zones can be explained by b i o t i c f a c t o rs such as decay. - 163 -Among a l l v a r i a b l e s considered, s i t e contributes the l e a s t to the v a r i a t i o n of net volume y i e l d of I n t e r i o r Douglas-fir stands as i l l u s t r a t e d i n Table 5-34 and thus was f i r s t eliminated (Table 5-38). Two q u a l i t a t i v e f a c t o r s , inventory zones and i n t e r a c t i o n s f o r zones and types, are non-significant and consequently discarded. A f t e r the elimination of these two q u a l i t a t i v e v a r i a b l e s , the f a c t o r species-composition-types becomes s i g n i f i c a n t ; however, i t s F-value i s the l e a s t among a l l v a r i a b l e s i n Step 3 of Table 5-38 and hence was eliminated i n Step 4. A f t e r elimination of a l l q u a l i t a t i v e v a r i a b l e s , the analysis i n Step 4 i s reduced p r a c t i c a l l y to ordinary multiple regression analysis. Variables eliminated i n sequence were height, number of trees per acre, average DBH, r e l a t i v e stand density, log age, basal area, and f i n a l l y stand age. Relative stand density i s a v a r i a b l e which contributes to the v a r i a t i o n i n volume y i e l d of I n t e r i o r Douglas-fir next to HT x BA, basal area, and age. The nonsignificance of species composition type and inventory zone shown i n Table 5-38 i s i n no c o n t r a d i c t i o n with the r e s u l t s of previous analyses (Tables 4-14, 4-24) where these two variables were found s i g n i f i c a n t . The r e s u l t s i n Table 5-38 simply imply that these two v a r i a b l e s have been w e l l taken care of by other v a r i a b l e s such as basal area, or HT x BA. As discussed i n a previous section, species composition types cause the volume y i e l d of I n t e r i o r Douglas-fir to d i f f e r ; the r e s u l t s i n Table 5-38 suggest that i f basal area per acre and HT x BA are equal, then no d i f f e r e n c e i n y i e l d among species types or inventory zones i s expected. - 164 -5.10.2 Spruce Table 5-39 presents the stepwise elimination procedure f o r a l l v a r i a b l e s considered i n an I n t e r i o r spruce y i e l d table analysis. S i t e , average DBH, and stand age i n logarithmic scale (log age) are variables which . contribute i n s i g n i f i c a n t l y to the volume y i e l d were eliminated f i r s t l y , while the q u a l i t a t i v e v a r i a b l e species composition types i s next to basal area, and HT x BA i n importance i n y i e l d table construction. In addition, v a r i a b l e s such as stand age, height, and r e l a t i v e stand density are of s i g n i f i c a n t importance. The estimated c o e f f i c i e n t s for v a r i a b l e s i n Step 7 are l i s t e d i n Table 5-41 where i t shows a l l quantitative v a r i a b l e s (age, basal area, r e l a t i v e stand density, HT x BA) being equal, pure spruce type stands ou t y i e l d spruce hardwood mixed type stands by 286 cubic feet per acre and spruce conifer mixed type ones by 23 cubic fee t . The r e s u l t s provide, f o r the f i r s t time, good evidence that establishment of pure spruce type stands i s more desirable than spruce hardwood mixed type stands i n the I n t e r i o r . Because the same volume equation has been used i n computation of volume fo r a l l sampled p l o t s , the gain of 286 cubic feet per acre of pure spruce type stands over spruce hardwood mixed type stands could be interpreted as pure spruce stands produce better q u a l i t y or l e s s decay i n logs than do spruce hardwood mixed type stands. The i n t e r p r e t a t i o n i s well-grounded i n view of recent reports which i n d i c a t e deciduous species such as aspen, poplar, red alder are of l i t t l e u t i l i z a t i o n value i n mature stands because of poor q u a l i t y (Young, 1974). T a b l e 5-41. Y i e l d f u n c t i o n s f o r D o u q l a s - f i r , spruce and l o d q e p o l e p i n e SPECIES INT. DF,COAST 21.34 21.34 DF,INTEEIOB 1548.15 VARIABLE SPRUCE SP. TYPE F.I.Z. AGE LOG AGE HEIGHT 2. 123.91 -8.053 3. -123.91 -8.053 LODGEPOLE -66.55 P 103.34 -66.55 P+C 79.74 -66.55 .P + H -183.08 -15.39 P 26.46 -15.39 P+C -93.80 -15.39 P+H -67.34 -6.965 1111.65 -2.525 -- 55.037 -2.525 -- 55.037 -2.525 -- 55.037 BASAL 9.446 9.446 9.446 R. S. D. HT X BA S.E. E. 21.109 -378.931 166.936 21.109 -378.931 166.939 5. 613 167. 958 229.523 11.621 -152.769 193.349 11.621 -152.759 193.349 11.621 -152.769 193.349 253.400 253.400 253.400 646 646 4 0 9 563 563 563 408 408 408 ON Remarks: R . S . D . - - R e l a t i v e Stand D e n s i t y ; HT X BA--Heiqht x B a s a l A r e a ; I N T . - - I n t e r c e p t -P—PURE; P+C--CHIX, C o n f i e r s Mixed t y p e ; P+H--HMIX, Hardwood Mixed t y p e . U n i t : c u b i c f e e t per a c r e . - 166 -S i m i l a r l y , pure spruce type stands produce 24 cubic feet per acre more than spruce c o n i f e r mixed type stands (Table 5-41). The amount may be of no p r a c t i c a l meaning; nevertheless, i t indicates c l e a r l y that pure spruce type stands produce better q u a l i t y wood than do spruce co n i f e r mixed type stands i n the i n t e r i o r . Further studies to elucidate i n f u l l the reasons why pure spruce type stands y i e l d more than mixed types i n stands with equal height, basal area, r e l a t i v e stand density, and HT x BA are needed. 5.10.3 Lodgepole Pine Table 5-40 presents an elimination procedure f o r I n t e r i o r lodgepole pine f o r a l l v a r i a b l e s considered i n y i e l d table construction. Average DBH, number of trees per acre, height, r e l a t i v e stand density, log age, and stand age are of no s i g n i f i c a n c e i n contributing v a r i a t i o n s to net volume y i e l d . In other words, the e f f e c t s of these v a r i a b l e s have been w e l l represented by the others. S i t e with an F value 4.75 i s s i g n i f i c a n t at the 5% l e v e l (1 and 4186 degree of freedom). The two q u a l i t a t i v e v a r i a b l e s — species composition type and inventory zone are only next to basal area and HT x BA i n importance i n the y i e l d table construction of I n t e r i o r lodgepole pine. The high s i g n i f i c a n c e of inventory zone suggests that a separate y i e l d table f o r lodgepole pine i n each zone i s warranted. The estimated c o e f f i c i e n t s i n Table 5-41 display the fac t that pure lodgepole pine outweigh lodgepole pine co n i f e r mixed type stands by 120 cubic feet per acre and lodgepole pine hardwood mixed stands by 94 cubic feet per acre on stands with the same basal area - 167 -per acre and HT x BA. The establishment of lodgepole pine stands i s , therefore, more desirable than of lodgepole pine mixed types i n the I n t e r i o r . As discussed previously, the r e s u l t s might suggest that pure lodgepole pine type stands produce logs with better q u a l i t y (less decay) than do mixed types. 5.11 Test of Homogeneity of Variance The standard deviations of net volume y i e l d by species composition types and inventory zones are shown i n Tables 5-42, 5-43, and 5-44 f o r Douglas-fir, spruce, and lodgepole pine. As shown i n these tables, variances d i f f e r from zone to zone and from stand type to stand type. In fact the assumption of homogeneous variances of the data was rejected by B a r t l e t t ' s t e s t . A c l o s e r examination of the data w i l l f i n d that the sample plots i n the inventory zone and species composition types were measured at various stand ages. In some inventory zones, f o r instance, the Southern Coast Region (Zone 2) where most plots observed were second growth stands, the v a r i a t i o n i n net volume y i e l d i s much l e s s than the South Coast T r a n s i t i o n Belt where a wider age range of sample p l o t s was surveyed. Since stand y i e l d s were measured at d i f f e r i n g age ranges, the standard errors shown i n Tables 5-42, 5-43, 5-44 do not r e f l e c t true v a r i a t i o n i n each species composition type and inventory zone combination. Therefore, the assumption of homogeneous variance i s not rejected on t h i s ground. 5-42. Standard deviations of Douqlas-fir volume y i e l d data by species types and forest inventory zones (Unit: cubic feet per acre) FOREST INVENTORY ZONE TYPE COAST INTERIOR 2 3 ALL 4 5 6 7 8 ALL PURE 301 5 4379 3298 1590 1673 1638 1786 1344 1745 CMIX 354 1 4169 4178 1848 1801 2235 191 5 1487 1979 HMIX 2297 3389 2572 430 1458 1186 1545 -- 1497 ALL 3078 4273 3523 16 62 1707 2035 1869 1354 1798 : PURE—pure Douqlas-fir type;CMIX—Douqlas-fir-conifer mixed type; HMIX—Douglas-fir-hardwood mixed type. Table 5-43. Standard deviations of spruce volume y i e l d data by species composition types and inventory zone (Unit: cubic feet per acre) TYPE FOREST INVENTORY ZONE 4 5 6 7 8 9 11 12 ALL PURE 2171 2006 1569 2607 1893 2102 1929 2121 2135 CMIX 2236 2136 2188 2040 1681 1 911 1398 1835 2007 HMIX 2502 -- 2399 2214 1571 2096 1340 1876 1979 ALL 22 83 2045 2139 2168 1812 2005 1693 2062 2067 Remarks: PURE—pure spruce type;CMIX—spruce-conifer mixed type; HMIX—spruce-hardwood mixed type. Table 5-44. Standard d e v i a t i o n s of lo d q e p o l e pine volume y i e l d data by s p e c i e s composition types and i n v e n t o r y zones (Unit: cubic f e e t per acre) TYPE FOREST INVENTORY ZONE 4 5 6 7 8 9 10 12 ALL PURE 1585 20 18 2054 1417 1868 19 87 1992 1573 1892 i i—1 CMIX 1737 1511 1950 1456 1639 1775 — 1 54 3 1868 o i HMIX 1631 1960 — 718 1616 18 56 1598 1 480 1700 ALL 1665 1925 2007 1431 1817 19 56 2166 1658 1874 Remarks: PURE—pure lodqepole pine type; C M I X — l o d q e p o l e p i n e - c o n i f e r s mixed type; HMIX—lodqepole pine-hardwood mixed type. - 171 -5.12 Least Squares Analysis In discussing the l e a s t squares analysis, Searle (1971) pointed out that "The c a l c u l a t i o n s involved i n t h i s method of analysis are, f o r unbalanced data, usually more complicated than those of t r a d i t i o n a l analysis of variance for balanced data, so that p r i o r to the present era of computers there has been l i m i t e d demand for analyzing unbalanced data. Nowadays, however, i n view of the a v a i l a b i l i t y of vast computer storage and e d i t i n g of data we are witnessing a great increase i n the demand for analysis of unbalanced data, analysis which cannot be made merely by means of minor adjustments to t r a d i t i o n a l analyses of variance of balanced data." Indeed, Searle continued: "The s i t u a t i o n i s j u s t the opposite: unbalanced data have t h e i r own analysis of variance technique, and those for balanced data are merely s p e c i a l cases of the techniques f o r unbalanced data. The p o s i t i o n i s that unbalanced data can be couched i n matrix expressions, many of which s i m p l i f y very l i t t l e i n terms of summation formulas. In contrast, when the numbers, ofobservation-.in the subclasses are a l l the same, these matrix expressions s i m p l i f y considerably. They reduce, i n f a c t , to the well-known summation formulae of t r a d i t i o n a l analysis of variance of designed experiments, such as randomized complete blocks, f a c t o r i a l experiments designs and others." Therefore, one can think of such analyses simply as s p e c i a l cases of the more basic analyses of variance for unbalanced data. - 172 -Although the le a s t squares analysis handles unbalanced data with high v a r i a t i o n i n numbers of observation i n subclasses, a few observations with unusually high or low value may influence the r e s u l t s . This i s exemplified by the spruce data i n previous sections where one sample pl o t with extremely high volume and basal area was observed. For t h i s reason, i t i s e s s e n t i a l to screen the subclass data with few observations and decide whether or not these observations should be included i n the analysis. A further study on the e f f e c t of the v a r i a t i o n i n c e l l frequencies on the power of hypothesis t e s t should be undertaken. The imposition of constraints on parameters to be estimated i n order to obtain unique solutions f o r the normal equations derived by the l e a s t squares p r i n c i p l e i s not accepted by s t a t i s t i c i a n s without objections. Kempthorne (1952), Federer (1955), Steel and T o r r i e (1960), and Harvey (1960) considered that the r e s t r i c t i o n s E a . = I 6. = £ ( a B ) . . = I (a$).. = 0 (5-2) i . j • i l 13 i 3 i J 3 are s a t i s f a c t o r y while Scheffe (1959) considered the systems of weights forlthese equations do not matter too much. However, Searle (1971) argued that (1) the constraints are generally not the simplest, (2) such constraints are not necessary f o r solving normal equations; they are only s u f f i c i e n t ; and (3) they can be used whether or not a s i m i l a r r e l a t i o n s h i p holds f o r the elements - 173 -of the model; and only i f i t does with enough such r e l a t i o n s h i p s i n the model to make i t a f u l l rank model, w i l l the solutions of the normal equations then be estimates of the parameters of the model. According to Searle (1971), the constraints easiest to use with unbalanced data are the simplest ones of putting p - r elements of b vector equation to zero. They cannot be ju s t any p - r elements ' x ' x c " of course, f o r they must be j u d i c i o u s l y chosen so as to make I C 0 non-singular. However, Harvey (1960) found t h i s constraint un-s a t i s f a c t o r y i n p r a c t i c e . The l e a s t squares analysis o f f e r s a unique s t a t i s t i c a l method to analyze forest inventory data. As demonstrated i n previous sections, the technique i s powerful i n extracting information out of i r r e g u l a r -structured forest inventory data. I t also provides a valuable method to incorporate q u a l i t a t i v e and quantitative v a r i a b l e s i n volume y i e l d a n a lysis, a method, to the best knowledge of the wri t e r , which has not been attempted i n forest y i e l d study. Foresters have long been searching for a technique to quantify q u a l i t a t i v e v a r i a b l e s such as species type group and inventory zones which are frequently encountered i n pra c t i c e s and researches. The le a s t squares technique as used here f u l f i l l s the need s a t i s f a c t o r i l y . Methodology i s e s s e n t i a l to the exploration of new t e r r i t o r y of knowledge; progress i n science r e l i e s p r i m a r i l y on the a v a i l a b i l i t y of appropriate techniques e i t h e r i n c o l l e c t i n g or analyzing information (data). In forest science, and i n the study of growth and y i e l d i n p a r t i c u l a r , a huge amount of data has been c o l l e c t e d annually by means - 174 -of f o r e s t inventory surveys or other methods. Those data can provide a sound basis f o r various s p e c i f i e d studies i f a n a l y t i c a l procedures appropriate to the data become a v a i l a b l e . Therefore, to make f u l l use of those data, there i s a pressing need to develop a sound s t a t i s t i c a l procedure. As demonstrated i n previous sections, the l e a s t squares analysis i s very powerful and sui t a b l e for analyses of f o r e s t inventory data. 5.13 Chapter Summary In t h i s chapter r e s u l t s from analyses of Coast Douglas-fir, I n t e r i o r Douglas-fir, I n t e r i o r spruce, and lodgepole pine inventory data by the least squares technique developed i n Chapter 3 were presented and discussed. Because sampled p l o t s were randomly selected, i t was possible to i n f e r the r e l a t i v e frequencies of species types i s a us e f u l i n d i c a t o r of the frequency with which the various types occur n a t u r a l l y . It was found that more than 50% of Douglas-fir, spruce, and lodgepole pine stands currently exist as pure type f o r e s t s . S i t e d e t e r i o r a t i o n r e l a t e d to pure type stands was discussed. Although s i t e indices f o r stands of mixed types were shown to be higher than those of pure type stands, the data were inadequate - to demonstrate conclusively whether or not the higher index has resulted from b e n e f i c i a l e f f e c t s of mixed stands on s o i l or simply from the better s i t e conditions when stands were o r i g i n a l l y established. - 175 -Analyses of the numbers of trees per acre, show that conifer mixed type stands are capable of growing more trees per acre than pure or hardwood mixed type stands. Among three species, Douglas-fir requires more growing space thus reducing the number of trees per acre. The r e l a t i v e stand density calculated from basal area has a s i m i l a r pattern to number of trees per acre. Stand age, mean annual increments of height, basal area, and volume were discussed. In general, stand age i s younger i n hardwood mixed stands than i n pure or co n i f e r mixed stands because of the dynamic nature of hardwood mixed type stands. The mean annual height increment of three species composition types investigated indicates that hardwood"mixed type stands grow f a s t e r than pure, or conifer mixed type stands. The f a s t e r growth i n height can probably be explained i n part by younger age of hardwood mixed type stands. The mean annual increments for basal area are generally higher i n c o n i f e r mixed type stands than i n pure or hardwood mixed type stands while the mean annual increments f o r volume show a s i m i l a r pattern to that of basal area. Differences i n growth and y i e l d between Coast and I n t e r i o r Douglas-fir stands were also i l l u s t r a t e d . Net volume y i e l d s of Douglas-fir, spruce, and lodgepole pine inventory data were further analyzed and p o t e n t i a l production capacity of various species types on B.C. inventory zones was compared and i d e n t i f i e d . The volume y i e l d data were i n turn adjusted for age and s i t e index to t e s t the hypothesis that there .was no diffe r e n c e i n y i e l d among species composition types f o r stands of same s i t e q u a l i t y . The - 176 -d i s p a r i t y i n y i e l d between pure Douglas-fir type and Douglas-fir conifer mixed type i n the I n t e r i o r was elucidated. F i n a l l y , a l l stand parameters were used i n y i e l d table analysis, and a stepwise elimination procedure was used to r a t i f y the r e l a t i v e importance of a stand parameter i n the construction of a stand y i e l d table. Standard deviation i n y i e l d f o r species types and inventory zone combinations was i l l u s t r a t e d and discussed. The advantages of using the revised l e a s t squares method i n the analysis of inventory data were discussed. - 177 -6.0 SUMMARY AND CONCLUSIONS Methodology i s e s s e n t i a l to the exploration of new t e r r i t o r y of knowledge; progress i n science depends p r i m a r i l y on the a v a i l a b i l i t y of appropriate techniques e i t h e r i n c o l l e c t i n g or analyzing information (data). In forest science, i n the study of growth and y i e l d i n p a r t i c u l a r , a huge amount of data has been c o l l e c t e d annually by means of forest inventory surveys or other methods. Those data can provide a sound basis for various s p e c i f i e d studies i f a n a l y t i c a l procedures appropriate to the data become a v a i l a b l e . A tremendous amount of data has been accumulated from past forest inventory surveys, but the use of s t a t i s t i c a l methods other than ordinary regression analysis to analyze those data has seldom been attempted. To make f u l l use of those data provided by inventory surveys, there i s a pressing need to develop sound s t a t i s t i c a l procedures for the analysis of those data. Contributions of the study to the spectrum of forest science can, therefore, be summarized i n methodological and s u b s t a n t i a l connotations. Methodologically, the study developed a s t a t i s t i c a l procedure to analyze i r r e g u l a r , unbalanced inventory data by l e a s t squares p r i n c i p l e . Admittedly, the l e a s t squares analysis was employed by s t a t i s t i c i a n s to analyze unbalanced data f o r t y years ago; however, i n forest d i s c i p l i n e s , where data c o l l e c t e d are often unbalanced i n nature because :the experimental p l o t s are usually subjected to uncontrollable environmental factors and experiments are often long i n duration, the methods have unfortunately been neglected. - 178 -Furthermore, the study developed a computing algorithm which e f f e c t i v e l y analyzes unbalanced data with some subclasses without observations, when in t e r a c t i o n s are to be included i n the model. Computer programs adapting the le a s t squares p r i n c i p l e f o r the analysis of unbalanced data do e x i s t ; but they f a i l when there are empty subclasses and in t e r a c t i o n s are to be considered. The program which considers more generalized unbalanced data has progressed one step farther i n s t a t i s t i c a l computation technique. In addition, the developed method provides a unique means to incorporate q u a l i t a t i v e as w e l l as quantitative v a r i a b l e s i n forest y i e l d analysis ( y i e l d table construction). Such analyses have never been attempted i n y i e l d studies because of lack of su i t a b l e methods. The method as demonstrated i n the study, o f f e r s s a t i s f a c t o r y alternatives to quantifying q u a l i t a t i v e v a r i a b l e s which are often encountered by foresters i n research as well as i n p r a c t i c e . Some valuable information has been induced by applying the developing l e a s t squares analysis to Douglas-fir, spruce, and lodgepole pine inventory data provided by the B.C. Forest Service Inventory D i v i s i o n , i n connection with the study of growth and y i e l d of pure and mixed stands i n B.C. forest inventory zones. From the r e s u l t s and discussions presented i n the previous chapters, the following summaries and^conclusions can be drawn. The r e l a t i v e frequency of the various kinds of inventory' pl o t s i s a useful i n d i c a t o r of the r e l a t i v e frequency with which the various types occur i n nature. More than 50% of Douglas-fir, spruce, and lodgepole pine stands occur n a t u r a l l y as pure types; i f there are any - 179 -adverse e f f e c t s on establishment of pure type stands, these e f f e c t s should have been well r e f l e c t e d q u a n t i t a t i v e l y i n those stands. A main c r i t i c i s m of the establishment of pure stands has been that pure type stands usually deteriorate s o i l conditions. Estimates of s i t e index from the inventory data analyzed seem to support the argument; however, i t was pointed out that the higher s i t e indices i n mixed type stands may be a t t r i b u t e d to the better s i t e conditions when the stands were o r i g i n a l l y established. Comparing the number of trees per acre among three species composition types, one sees c l e a r l y that conifer mixed stands are capable of growing more trees per acre than pure or hardwood mixed type stands. Among three species investigated, Douglas-fir requires more growing space thus reducing the number of trees per acre. A l t e r n a t i v e l y , stocking may be l e s s complete because of mortality and/or incomplete regeneration. The r e l a t i v e stand density based on basal area per acre also indicates that stand density i s higher i n co n i f e r mixed type stands than i n pure type or hardwood mixed type stands f o r a l l three species. The average stand age i n hardwood mixed type stands i s much younger than i n pure or co n i f e r mixed type stands. The r e s u l t s suggest that hardwoods are "pioneer" species i n B r i t i s h Columbia forests and that the composition of hardwood stands i s gradually changing from hardwood mixed type to co n i f e r mixed type or to pure type with the increase of stand age. - 180 -The figures show that the mean annual height increment i s higher i n hardwood mixed type stands than i n pure or conifer mixed type ones. The r e s u l t s can be interpreted by the fact that stand age i n hardwood mixed type stands i s much younger than i n pure or co n i f e r mixed type stands. Variations i n mean annual height increments among forest inventory zones are noted. The annual basal increment of co n i f e r mixed type stands i s co n s i s t e n t l y higher than those of other two species composition types. Zonal v a r i a t i o n s i n the mean annual basal area growth are also apparent. The mean annual volume increment follows a trend s i m i l a r to that of the mean annual basal area increment. Differences i n growth and y i e l d between Coast and I n t e r i o r Douglas-fir stands were compared. It was found that Douglas-fir stands can grow more trees per acre on the Coast (188 trees per acre) than i n the I n t e r i o r (121 t r e e s ) . The mean annual volume growth i s 84.00 cubic feet per acre f o r the Coast stands and 25.53 cubic feet f o r the I n t e r i o r stands; the former outgrow the l a t t e r by 3.3 times. Difference i n growth between these two geographic areas can be accounted for i n part by differences i n s i t e index (120 and 78 for the Coast and the I n t e r i o r , r e s p e c t i v e l y ) . On the Coast f e r t i l e s i t e s (the Southern Coast Region), pure Douglas-fir type stands y i e l d more than Douglas-fir c o n i f e r or hardwood mixed stands. However, on medium s i t e s (the South Coast T r a n s i t i o n B e l t ) , Douglas-fir hardwood mixed type stands ou t y i e l d pure type stands. In the I n t e r i o r Douglas-fir c o n i f e r mixed type stands co n s i s t e n t l y produce more wood than the other two composition types. - 181 -Analyses of the net volume y i e l d data of Coast Douglas-fir, I n t e r i o r Douglas-fir, spruce, and lodgepole pine with stand age as a covariable provide comparisons i n volume y i e l d f o r various species composition types i n B.C. inventory zones. E f f e c t s of species composition types and forest inventory zones are not s i g n i f i c a n t i n Coast Douglas-fir stands, but in t e r a c t i o n s for types and zones are. Based on the estimated constants, the p o t e n t i a l y i e l d f o r three species composition types i n two Coast zones were tabulated at stand age 100. On very f e r t i l e s o i l such as that i n the Southern Coast Region, pure Douglas-fir type tends to grow e x c e l l e n t l y ; however, on medium s o i l , i t i s advantageous to e s t a b l i s h Douglas-fir mixed type stands. In I n t e r i o r Douglas-fir stands, the species composition types and inventory zones e f f e c t s are s i g n i f i c a n t while i n t e r a c t i o n s thereof are not. In general, the p o t e n t i a l y i e l d i s 3114 cubic feet f or Douglas-f i r c onifer mixed type stands and 2582 cubic feet f or pure Douglas-fir type at age 100. The establishment of Douglas-fir co n i f e r mixed type stands e f f e c t i v e l y increases forest p r o d u c t i v i t y by 21%. The most productive Douglas-fir stands are found i n Zone 7 while the poorest are i n Zone 8. The p o t e n t i a l y i e l d s of Douglas-fir co n i f e r mixed stands are i n v a r i a b l y higher than the other two composition types i n a l l zones. For the I n t e r i o r spruce data, the e f f e c t s of species composition type on net volume y i e l d are not s i g n i f i c a n t while zonal e f f e c t s and i n t e r a c t i o n s f or zones and types are. The most productive spruce stands are i n Zone 5 with a p o t e n t i a l y i e l d of 4264 cubic feet per acre - 182 -at age 100; Zones 7, 8 and 9 are equally productive while Zone 11 i s the le a s t productive zone for spruce i n the I n t e r i o r . From the zonal production capacity, i t was observed that changes i n l a t i t u d e do not account f o r the v a r i a t i o n i n volume y i e l d f o r spruce i n the I n t e r i o r . Inspection of the int e r a c t i o n s suggests that pure spruce stands ou t y i e l d mixed type stands i n Zones 4, 6, and 12 while spruce-conifer mixed type stands i n Zone 5 and spruce-hardwood mixed type stands i n Zone 9 produce more volume than do pure spruce stands. For the I n t e r i o r lodgepole pine data, the e f f e c t s of species composition types are not s i g n i f i c a n t while zonal e f f e c t s and i n t e r -actions are. The best zone f or lodgepole pine y i e l d i s Zone 9 and the poorest Zone 5. Zonal v a r i a t i o n s i n volumes are smaller i n lodgepole pine than i n Douglas-fir and spruce because the general c l i m a t i c requirements of lodgepole pine are wider than the l a t t e r . Consequently, the changes i n l a t i t u d e have l i t t l e bearing on the productive capacity of lodgepole pine i n the I n t e r i o r . Pure lodgepole pine type stands i n Zone 10 outweigh stands of mixed types by 51% i n volume. On the other hand lodgepole pine con i f e r mixed type stands are more productive than pure lodgepole pine type stands by more than 10% i n Zones 4 and 12. Therefore, the advantages of monocultural or m u l t i c u l t u r a l practices can not be over-generalized. Pure type stands are more productive i n some zones but l e s s i n the others. The same i s true as to the advantages and disadvantages of m u l t i c u l t u r a l p r a c t i c e s with regard to forest crops. - 183 -Growth of f o r e s t trees i s e s s e n t i a l l y site-dependent. Before a decision i s reached on what species composition type to e s t a b l i s h , foresters should c a r e f u l l y i n v e s t i g a t e the l o c a l s i t e q u a l i t y and past y i e l d h i s t o r y of various forest types to ensure the maximum pot e n t i a l p r o d u c t i v i t y of a p a r t i c u l a r s i t e can be realized.., It goes without saying that the p r o d u c t i v i t y of forest land can be amplified by e s t a b l i s h i n g a forest type appropriate to the area. The study i d e n t i f i e d the optimum species composition types for Douglas-f i r , spruce, and lodgepole pine i n B.C. forest inventory zones. Further studies to elucidate the reasons for a species type to y i e l d more than others i n a p a r t i c u l a r zone are needed i n order to help foresters i n t h e i r s e l e c t i o n of an optimum species composition type. The volume y i e l d data were further analyzed for the species composition e f f e c t s by taking s i t e index and stand age as covariables to test the hypothesis that no differences i n volume y i e l d e x i s t among three species composition types for stands growing on same s i t e conditions. The r e s u l t s i n d i c a t e that f o r Coast Douglas-fir the e f f e c t s for species types and inventory zones as well as i n t e r a c t i o n s thereof are not s i g n i f i c a n t . Therefore, the diff e r e n c e i n y i e l d among species composition types can be interpreted i n f u l l by dif f e r e n c e i n s i t e index for the Coast Douglas-fir stands. In I n t e r i o r Douglas-fir stands, however, the species composition types are s i g n i f i c a n t at the 1% l e v e l , the estimated e f f e c t i s 16.92, 350.78, and -367.70 cubic feet per acre f o r pure Douglas-fir, Douglas-fir conifer mixed, and Douglas-fir hardwood mixed types r e s p e c t i v e l y . - 184 -In other words, on same s i t e conditions, Douglas-fir conifer mixed type stands y i e l d 334 cubic feet more than pure type stands and 718 cubic feet per acre than Douglas-fir hardwood mixed type stands i n the I n t e r i o r . On equal s i t e conditions, there i s no differ e n c e i n volume y i e l d among three species composition types i n I n t e r i o r spruce stands while zonal e f f e c t s and i n t e r a c t i o n s for types and zones are s i g n i f i c a n t . In I n t e r i o r lodgepole pine stands, e f f e c t s of species composition types, zones, and in t e r a c t i o n s thereof d i f f e r s i g n i f i c a n t l y . On same s i t e conditions, pure lodgepole pine type stands y i e l d 48 cubic feet per acre more than lodgepole pine co n i f e r mixed type stands and 272 cubic feet more than lodgepole pine hardwood mixed type stands. In a l l three species — Douglas-fir, spruce, and lodgepole pine investigated, the e f f e c t s of hardwood mixed type show co n s i s t e n t l y a negative value. Consequently, hardwood mixed type stands are the l e a s t desirable stand composition structure f o r these species i n the I n t e r i o r . Because the I n t e r i o r Douglas-fir stands show differences i n volume y i e l d between pure and coni f e r mixed type stands a f t e r s i t e index and stand age have been adjusted, the data were further analyzed by f i t t i n g the average height, basal area, and volume i n various age classes on three s i t e classes using Weibull-type growth functions. It was found that there was no differ e n c e i n height growth between pure and conifer mixed types. The d i s p a r i t y i n volume between pure and mixed type stands has resulted from the ine q u a l i t y i n basal area per acre. Douglas-fir conifer mixed type stands are capable of growing - 185 -more trees per acre, thus more basal area per acre, and consequently more volume than do pure Douglas-fir type stands i n the I n t e r i o r . Analyses were f i n a l l y c a r r i e d out for y i e l d table construction by incorporating a l l stand parameters, q u a l i t a t i v e and quantitative v a r i a b l e s , e.g. species composition type, forest inventory zones, i n t e r a c t i o n s f o r types and zones, s i t e , age, log age, height, number of trees per acre, average DBH, basal area per acre, r e l a t i v e stand density, and height x basal area. A stepwise elimination procedure was used to ascertain the importance of these v a r i a b l e s i n y i e l d table construction. In a l l four sets of inventory data analyzed, the v a r i a b l e s , height x basal area and basal area are most important. In addition, stand age, r e l a t i v e basal area, and forest inventory zone are a l l highly s i g n i f i c a n t i n contributing to the v a r i a t i o n s i n volume y i e l d of the Coast Douglas-fir stands. The high F-value for forest inventory zone suggests that the subdivision of the Coast-area for Douglas-fir i s j u s t i f i e d . Estimated constants for inventory zones i n d i c a t e that on the same age, basal area, r e l a t i v e stand density, and height x basal area, Zone 2 w i l l y i e l d 248 cubic feet per acre more than Zone 3. The d i s p a r i t y i n y i e l d between these two inventory zones can be explained by b i o t i c factors such as decay. For I n t e r i o r Douglas-fir, the most s i g n i f i c a n t v a r i a b l e s are height x basal area, basal area, stand age, log age, and r e l a t i v e stand density. E f f e c t s of species composition type and f o r e s t inventory zones are non-significant i n comparison with the above 5 v a r i a b l e s . Accordingly, i f height x basal area, basal area, stand age, r e l a t i v e - 186 -stand density are equal, no v a r i a t i o n i n volume y i e l d among species composition types and forest inventory zones i n the I n t e r i o r i s expected. For I n t e r i o r spruce, the prominant variables i n y i e l d table analysis are height x basal area, basal area, species composition types, stand age, height, and r e l a t i v e stand density. Inspection of the estimated e f f e c t s f o r three species composition types suggests that a l l v a r i a b l e s , basal area, height, stand age, and r e l a t i v e stand density being equal, pure spruce type stands outyield stands of spruce-hardwood mixed type by 286 cubic feet per acre and spruce-conifer mixed type by 23 cubic feet. The r e s u l t s provide good evidence that establishment of pure spruce type stands i s more desirable than of spruce hardwood mixed type stands. Pure spruce stands produce better q u a l i t y (less decay) logs than do spruce-hardwood mixed type stands. For lodgepole pine, the most s i g n i f i c a n t v a r i a b l e s i n y i e l d table analysis are height x basal area, species composition types, and forest inventory zones. The high s i g n i f i c a n c e of the forest inventory zones suggests that a separate y i e l d table f o r lodgepole pine i n each zone i s warranted. The estimated e f f e c t s f o r three species composition types ind i c a t e that pure lodgepole pine type stands outweigh lodgepole pine c o n i f e r mixed type by 120 cubic feet per acre and lodgepole pine hardwood mixed type by 94 cubic feet per acre. In other words, the establishment of pure lodgepole pine type stands i s preferred to the mixed types since on the same basal area and height x basal area bases, pure type stands tend to produce, more net volume per acre than do mixed type stands. - 187 -A p p l i c a t i o n of these methods to the temporary sample plo t data provided by the B.C. Forest Service has c l e a r l y demonstrated the widespread d i s t r i b u t i o n of pure stands and lack of s u b s t a n t i a l e f f e c t s of monocultures on y i e l d . Nevertheless, the fact that higher y i e l d s may r e s u l t from some multicultures should encourage establishment of long term studies of spacing and mixtures of species. Only time and careful'observation iof f i e l d t r i a l s can resolve the issues that cannot be c l a r i f i e d from analyses of inventory data by l e a s t squares methods such as those developed and applied i n t h i s t h e s i s . - 188 -7.0 LITERATURE CITED Andody, E. 1968. 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Central European experience as background for guidelines f o r development of Central B r i t i s h Columbia p o l i c i e s . Unpubl. Report. Faculty of Forestry, UBC, Vancouver, B.C. 62p. - 192 -Koi'visto,P. 1971. Reglonality of forest growth i n Finland. Institutum F o r e s t a l l s Fenniae Communicat. 71: 1-70. Krajina, V.J. 1965. Biogeoclimatic zones and c l a s s i f i c a t i o n of B r i t i s h Columbia. In Ecology of Western North America. 1: 1-17. _. 1969. Ecology of forest trees i n B r i t i s h Columbia. Ecology of Western North America 2: 1-146. Kramer, C.Y. 1955. On the analysis of variance of a two-way c l a s s i f i c a t i o n with unequal subclass numbers. Biometrics 11: 441-452. ' . 1956. Extension of multiple range tests to group means with unequal numbers of r e p l i c a t i o n . Biometrics 12: 307-310. »1957. Extension of multiple range tests to group correlated adjusted means. Biometrics 13: 13-18. Kurkjian, B. and M. Zelen. 1962. A calculus for f a c t o r i a l arrangements. Annals Math. Stat. 33: 609-619. Langhammer, A. 1971. A look at mixed f o r e s t s . Tidsskr Skogbr 79: 312-413. ( o r i g i n a l not seen, c i t e d from Forestry Abstract 2:276). Lukkala, O.J. 1938. Suomen jakaminen i l m a s t o l l i s i i n . Mersajitusu-yohykkeisiin. Helsink. ( o r i g i n a l not seen, c i t e d from Kovisto, 1971). McArdle, R.E. and W.H. Meyer. 1930. The y i e l d of Douglas-fir i n the P a c i f i c Northwest. Tech. B u l l . No. 201 USDA 64p. McSwain, G.A. 1970. Engelmann spruce (Picea engelmanii (Parry) USDA For. Serv. American Wood L e a f l e t FS-264. Meyer, W.H. 1937. Y i e l d of even-aged stands of s i t k a spruce and western hemlock. USDA Tech. B u l l . No. 544. 88p. Mullet, G.M. and V.J. Walsh. 1974. A generalized inverse algorithm for analysis of experimental designs i n a regression framework. Communications i n S t a t i s t i c s 3: 179-184. Mulloy, G.A. 1944. Empirical density y i e l d tables. Dominion Forest Serv. S i l v i c u l t u r a l Res. Note 73. 22p. - 193 -1947. Empirical stand density y i e l d . Dominion Forest Serv. S i l v i c u l t u r a l Res. Note 82. 41p. Nair, K.R. 1941. A note on the method of " f i t t i n g of constants" for analysis of non-orthogonal data arranged i n a double c l a s s i f i c a t i o n . Sankhya 5: 317-328. Outhwaite, A.D. and A. Rutherford. 1955. Covariance analysis as an a l t e r n a t i v e to s t r a t i f i c a t i o n i n the control of gradients. Biometrics 11: 431-440. Paterson, S.S. 1956. The forest area of the world and i t s p o t e n t i a l p r o d u c t i v i t y . Goteborg Royal Univ. N.V. ( o r i g i n a l not seen, c i t e d from 1957 Forestry Abstracts. 18: 1176). Quenouille, M.H. 1948. The analysis of covariance and non-orthogonal comparison. Biometrics. 4: 240-246. Rao, CR. 1946. On the l i n e a r combinations of observations and the general theory of least squares. Sankhya 7: 237-256. 1965. Linear S t a t i s t i c a l Inference and Its Applications. John Wiley and Sons, New York. 522p. Richmond, A.E. 1969. An analysis of some physical and economic c r i t e r i a f o r the determination of rotations for B r i t i s h Columbia f o r e s t s . Univ. B.C., Faculty of Forestry, M.F. Thesis. 135p. Rowe, J.S. 1959. Forest Regions of Canada. Can. Dept. of Northern A f f a i r s and Natural Resources. For. Br. B u l l . 123p. . "- 1962. S o i l , s i t e and land c l a s s i f i c a t i o n . For. Chron. 38: 420-432. Scheffe, H. 1959. The Analysis of Variance. John Wiley and Sons. New York. 477p. Searle, S.R. 1971. Linear Models. John Wiley & Sons, Inc. New York. 532p. Smith, D.M. 1962. The Practice of S i l v i c u l t u r e . 7th Ed. John Wiley and Sons, New York. 578p. Smith, J.H.G. 1973. F e a s i b i l i t y of preparing v a r i a b l e density y i e l d tables. Report to the Productivity Committee of B. C. For. Serv. on P.C. 006. Faculty of Forestry, UBC, Vancouver, B.C. 74p. - 194 -. 1976. Methods for use of timber inventory data to estimate average and upper l i m i t s to growth and y i e l d of biomass. Paper prepared for a meeting of the Working Party on Mensuration of Forest Biomass at the XVI IUFRO Congress i n Oslo, Norway, June 21-27, 1976. lOp. . 1977. Y i e l d estimates and preliminary guides f o r spacing and thinning B r i t i s h Columbia f o r e s t s . Faculty of Forestry, UBC, Vancouver, B.C. 26p. Snedecor, G.W. 1934. The method of expected numbers tables of multiple c l a s s i f i c a t i o n with disproportionate subclass numbers. Jour. Amer. Stat. Assn. 29: 389-393. . 1956. S t a t i s t i c a l Methods. 5th ed. Iowa State Univ. Press. Ames, Iowa. 534p. and G.M. Cox. 1935. Disproportionate subclass numbers i n tables of multiple c l a s s i f i c a t i o n . Iowa Agric. Expt. St. Res. B u l l . 180: 233-272. Stanek, W. 1966. Occurrence, growth, and r e l a t i v e value of Lodgepole pine and Engelmann spruce i n the I n t e r i o r of B r i t i s h Columbia, Ph.D. Thesis, U n i v e r s i t y of B.C., Vancouver, B.C. 252p. Steel , R.G.D. and J.H. T o r r i e . 1960. P r i n c i p l e s and Procedures of S t a t i s t i c s with Special Reference to the B i o l o g i c a l Sciences. McGraw-Hill Co., New York. 481p. Stevens, W.L. 1948. S t a t i s t i c a l analysis of a non-orthogonal t r i -f a c t o r i a l experiment. Biometrika 35: 346-367. Tarrant, R.E. 1961. Stand development and s o i l f e r t i l i t y i n a Douglas-fir - red alder plantation. Forest S c i . 7: 238-245. Tourney, J.W. and C F . Korstian. 1947. Foundations of S i l v i c u l t u r e . John Wiley and Sons. New York. 414p. Turnbull, K.J. 1963. Population dynamics i n mixed forest stands. A system of mathematical models of mixed stand growth and structure. Ph.D. Thesis. Univ. of Washington, Seattle, Wash. 186p. Urquhart, N.S., D.L. Week and CR. Henderson. 1973. Estimation associationed with l i n e a r models: a r e v i s i t a t i o n . Communications i n Stat. 1. 303-330. Vacovski, H. 1967. Growth structure and pr o d u c t i v i t y of pure and mixed stands of Ouercus s e s s i l i f l o r a and Fagus o r i e n t a l i s i n the Strandzda Planina. Govskostop nauka s o f i a 4: 29-50. ( o r i g i n a l not seen, c i t e d from Forestry Abstract. 29: 400). - 195 -Vuokila, V. 1966. Functions for v a r i a b l e density y i e l d tables of pine based on temporary sample p l o t s . For. Fenn. 60: 1-86. Walters, J. and J.H.G. Smith. 1973. Review of methods used i n establishment and summary of early r e s u l t s from spacing t r i a l s on the U.B.C. Research Forest. Faculty of Forestry, U.B.C., Vancouver, B.C. 40p. Week, J. 1955. F o r s t l i c h e Zuwochs-und Ertragskunde. Neumann Verlag, Raaebenl und B e r l i n . 92p. . 1957. Neuer Versuch zum Problem der K o r r e l a t i o n Klima und F o r s t l i c h e s Produktions p o t e n t i a l . Forstarchiv. 28: 223-227. Whitford, H.N. and R.D. Craig. 1918. Forests of B r i t i s h Columbia. Commission of Conservation Canada. Ottawa. 409p. Whyte, A.G.D. 1973. Pr o d u c t i v i t y of f i r s t and second crops of Pinus ra d i a t a of the Moutere Gravel S o i l of Nelson. New Zealand Jour. Forestry. 18: 84-103. Wilks, S.S. 1938. Analysis of variance and covariance i n non-orthogonal data. Metron. 13: 141-154. Yang, R.C., A. Kozak and J.H.G. Smith. 1978. The p o t e n t i a l of Weibull-type functions as f l e x i b l e growth curves. Accepted. Can. Jour. For. Res. Yates, F. 1934. The analysis of multiple c l a s s i f i c a t i o n s with unequal numbers i n the d i f f e r e n t classes. Jour. Amer. Stat. Assn. 29: 51-66. Young, W.E.L. 1966. Modern forest inventory, i t s purpose, goal and message. WFCA. Portland. Proceedings 1965 meeting 16-20. " . 1974. The resource and forest management p o l i c y — B r i t i s h Columbia. Proceedings Poplar U t i l i z a t i o n Symposium. P17-23. Western Forest Products Lab., Canadian For. Serv. VPX-127. Zakopal, V. and V. Mares. 1968. The importance of Norway spruce admixtures for volume and value production of oak stands on the r i c h e r types of the beech/oak c l a s s . Lesn Cas 14: 923-942. ( o r i g i n a l not seen, c i t e d from Forestry Abstract 32: 6181). - 196 -APPENDICES - 197 -APPENDIX 1 . B. C. Forest Inventory Zones L i s t i n g of a l l P.S.Y.U. 's and T.F.L. *s to show areas i n c l u d e d i n each o f the 12 Broad Groups used as a b a s i s f o r planning Growth and Y i e l d and Loss F a c t o r S t u d i e s . 1. Northern and C e n t r a l Coast Reqion - Cedar, Hemlock Types P.S.Y.U.«s: Dean Hecate Queen C h a r l o t t R i v e r s I n l e t Lower P a r t Of Skeena (Terrace Block of Skeena) T.F.L «s: No. 1, 2, 12, 17, 24, 25, (block 5 ) , 39, 41 K i t i m a t T.F.L r e s e r v e 2. Southern Coast Region- F i r , P. S. Y. U. 's: T.F.L 's: No. 2, 6, 7, 10, (block 1, 2, 3, 4 ) , 27, 36, Hemlock Types : Nootka Quadra Kingcome 12, 17, 19, 20, 21, 22, 25, 38, 39, (block 1, 2, 3, 4, 5) 3. South Coast T r a n s i t i o n B e l t - Hemlock, F i r , Balsam Types P.S.Y.U.'s: - Dewdney Soo Vancouver T.F.L «s: No. 26 4. South Western I n t e r i o r Dry B e l t - F i r , Lodgepole P i n e , Spruce Types P.S.Y.U.' s: - Ashnola Barton H i l l B i g Bar Botanie Kamloops fieqion Lac La Hache N i c o l a Okanagan Similkameen Williams Lake Yalakom - 198 -South part of Special Sale Area T.F.L «s: No. 5, 9, 15, 16, 32, 35 West Kootenay Reqion - Spruce, Balsam Cedar, F i r Types P.S.Y.U.'s: - Creston Edqewood Granby Kettle Salmo T.F.L 's: No. 8 East Kootenay Reqion - Spruce, Lodqepole Pine, F i r Types P.S.Y.U.'s: Cranbrook Fernie Upper Kootenay Windermere Crows Nest Pass Coal Co. T.F.L «s: No. 13 Central Columbia Reqion - I n t e r i o r Wet Belt - Spruce, Cedar, Hemlock, F i r Types P.S.Y.U.'s: Adams Arrowhead Barriere Canoe Eaqle Kinbasket Nakusp Nehalliston North Thompson Quesnel Lake Raft Robson Salmon Arm Shuswap Spallumcheen T.F.L «s: No. 3,14,18,23 (blok 1, 2, 3, 4), 32, 33 Nechako - Fraser Plateau Reqion - Lodqepole Pine, F i r , Spruce Types P.S.Y.U.'s: Burns Lake Nareosli Necha ko - 199 -Stum Westlake C h i l k o 9. C e n t r a l I n t e r i o r Reqion - Spruce, Balsam, Lodqepole Pine Types P.S.Y.U.•s: Babine Bowron Carp Cottonwood Crooked River Lonqworth Honkman Morice Naver Pa r s n i p Pur den Smithers S p e c i a l S a l e Area (N) S t u a r t Lake T a k l a Willow R i v e r P o r t St. James S. S. A, T.F.L »s: No. 30 10. North-Western P l a t e a u Reqion - S t i k i n e and Skeena Drainaqes Hemlock, Balsam, Spruce Types P.S. Y. U. «s: B e l l I r v i n q Skeena Kitwanqa; Hazelton B l k s . of Skeena Proposed: S t i k i n e , A l s e k , Boundary, Taku T.F.L »s: No. 1 11. N o r t h - C e n t r a l P l a t e a u Reqion - C a s s i a r and Omineca P l a t e a u s - Spruce, Balsam Deciduous Types P.S. Y. D. »s; F i n l a y Proposed; Dease, Kechika, Klappan 12. North-Eastern P l a i n s Reqion - Spruce, Decidous Types P.S.Y.U. «s: Blueberr y Moberly Peace Wapita Proposed: Kotcho Fort Nelson Fontas Sikanni Liard - 201 -Appendix 2. Computer Program. MICHIGAN TFRMINAL SYSTEM FORTRAN G141336) MAI N 02-10-78 13:39:47 0001 DOUBLE PRECISION PR10,AA,3,AV,SSC,COEF 0002 DIMENSION PROO(120,120) ,AA(120,120) ,R(80,00) ,AV(30,120) ,SSC(120) . 1 COEFf 80),NAME!120),NIT(80),FMT{20),NFL(10) 0003 COMMON NF.NI ,NI NT,NX,NCOV,NY,NAM,ILS, IRL 0004 EXTERNAL MATINV 0005 CALL T ITLEIFMT.NFL.NAME) 0006 CALL OATAIPRnO.AV,SSC, NFL.FMTI 0007 CALL INF0P.M(PP.0D,AV.SSC,NAME ,NFL) 0308 IF(ILS.E0.1)CALL LSMAT(PROD,NAME ) 0009 6 F0RMAT(26I3) 0010 CALL REOUCEtPROO.NX) 0011 00 I 1=1,NX 0012 00 1 J=l,NX 0013 1 AA( J , I ) = PRODU,J) 0014 CALL REDUCE!AA,NX 1 0015 99 READ(5,6)NEX,(NIT(I),1=1,NEX) 0016 DO 7 1=1,MEX 0017 COEF(II=0.0 0018 DO 7 J=l,MEX 0019 7 R(I,J)=AAfN IT( I ) ,NIT(J )) 0020 I F U R L . E Q . l t CALL RLSMATIR, NAME, NEX, NCOV. NY. NIT) 0021 N0C=NEX-1 00?? CALL GSPACE(C,N0C*N0C*8) 0023 CALL CALLERfMATINV,C.IPTR(R).IPTR(NDC)) 0024 CALL FSPACE(C) 0025 DO 25 I=1,NDC 0026 00 25 J=l,NDC 0027' 25 COEF! I>=C0EF! I ) + R( I, J )*R(NJEX, J ) 0028 WRITEI6.90) 0029 90 FORM AT! ' 1 ' , 20X, 'MATRIX INVERSE TO THE COMPLETE V ARI ANCE-COVAR I ANCF 1 MATRIX' / / ) 0030 DO 2 6 I=1,N0C 0031 K=MIT(1> 0032 WRITE16,8)K,NAME(K) 0033 8 FORMAT ( ' ' . ' R O W , 2X.I4.1X.A4) 0034 26 W R I T E ( 6 , 2 7 M R ( I , J ) , J = 1,I) 0035 27 FORMAT(* ' ,9014.61 0036 28 F O R M A T ! ' 1 « , / / / , 3 X , ' E S T I M A T E D C O E F F I C I E N T S ' , / / / ) 0037 WRITE(6,28) 0038 DO 31 I=1,NEX 0039 NA=N IT( I ) 004 0 WRIT E(6,32INAME(NA),CO EFIII 0041 31 CONTINUE 0042 32 FORMAT! '0 ' ,A4 ,2X ,G13 .6 ) 0043 N0FT=PR0D(1,1) 0044 SSOR=0.0 0045 DO 33 1=1,NEX 0046 33 SSOP. = SSDR*COEF(I l*R( NEX, I 1 0047 ERR=R!NEX,NEX)-SSDR 0048 CALL ANOVA(R,COEF,ERR,NOFT) 0049 IRL=IRL-l 0050 REA0(5,6)NREP,NC0V 0051 IF(NREP.GE.l )G0T0 99 0052 STOP 0053 END "OPTIONS TN EFFECT* ID,EBC DIC,SOURCE«NOL1ST, NODECK,LOAD, NOMAP •OPTIONS IN EFFrCT* NAME = MAIN , LI NECNT = 60 •STATISTICS* SOURCE STATEMENTS = 53,PROGRAM S I Z E = 315032 1.000 1.500 2.00 J 3. 000 4.000 5.000 6.000 7.000 8.000 10.000 11.000 12.000 13.000 14.000 15.000 15.500 16.000 1 7.000 18.000 19.000 20.000 21.000 22.000 23.000 24.000 25.000 26.000 27.000 28.000 29.000 30.000 31.000 32.000 33. 000 34.000 35.000 36.000 37.000 38.000 39.000 40.000 41.000 42.000 43.000 44.000 45.000 46.000 47.000 48.000 49.000 49.200 49.500 49.700 50.000 51.000 - 202 -MICHIGAN TERMINAL SYSTEM FORTRAN GI41336) TITLE 02-10-78 10:08:48 0001 SUBROUTINE T IT LE f FMT, N FL , N A*l E ) 52.000 0002 DIMENSION NAME(120). T I T L E ( 2 0 ) , F M T ( 2 0 1 . N F L ( 1 0 » 53.000 0003 COMMON NF.NI ,N INT,NX,NCOV,NY ,N AM , I LS, I Rl. 54.000 0004 NI=0 55.000 0005 NINT=0 56.000 0006 READ(5 » 1 )T ITL E 57.000 0007 1 FORMAT(20A4) 58.000 0008 READ(5,l)FMT 59.000 0009 READ!5,2 IMF,(NFL (I ) . I=1,NF) ,NCOV,NY,NAM,ILS,IRL 60.000 0010 2 FORMAT(20141 61. 000 0011 NK=NF-1 62.000 0012 DO 3 1=1,NK 63.000 0013 IJ=I+1 64.000 0014 DO 3 J=IJ,NF 65.000 0015 3 NINT=MINT+NFL(I) *N FL(IJ) 66.000 0016 DO 4 I = 1 , NF 67. 000 0017 4 NI=NI+NFL(I) 68.000 001R NX=NI+NINT+NCOV*NY-l 69. 000 0019 READ!5, I)( N4MEU ) ,1 = 1, NX) 70.000 0020 WRITE(6,5)TITLE 71.000 0021 5 FORMAT (• 1' , / / ,2 0*4 / / ,1 OOC * • M 72.000 0022 WRITE(6,6 INF 73.000 0023 6 FORMyr(i • ,5X. ' * ' , IX, 'NUMBERS OF FACTOR'»12( ' . ' ) , I 4) 74.000 0024 WRI TE(6 ,7 ) (NFL( I ) , 1 = 1,NFI 75.000 0025 7 FORM AT(• ' , 5 X , ' * ' , l X , ' L E V E L S OF F A C T O R S ' , 1 2 ( ' . ' ) ,1 014) 76.000 0026 WRITE<5 ,81NC0V 77. 000 0027 8 F ORM AT(• < ,5X, , IX , ' NUMBERS OF COVARIATES ' , 3 ( ' . ' ), 14) 78.000 0028 WRITE<6. 10)NY 79.000 0029 10 FORMAT ( ' ' ,5X , * * ' , IX ,'NUMBERS OF YIELD VARIABLES ' , 3( ' . « ),I 4) 80.000 0030 WRITE(6,9)FMT 31.000 0031 9 FORMAT(• • , 5 X , ' * ' , I X , ' FORMAT', 1 0 ( ' . ' ) , 20A4) 82.000 0032 RETURN 83.000 0033 END 84.000 *PPTIONS IN EFFECT* ID,EBCDIC,SOURCE,NOLIST,NODEC<,LOAD,NOMAP "OPTIONS IN EFFECT* NAME = TITLE , LINECNT = 60 *STATISTICS* SOURCE STATEMENTS = 33,PROGRAM SIZE = 1428 ^STATISTICS* NO DIAGNOSTICS GENERATED NO ERRORS IN TITLE - 203 -MICHIGAN TERMINAL SYSTEM FORTRAN G(41336) DATA 02-10-78 10:08:48 0001 SURROUT INE OAT A (PROD, A V, SSCNFL.FMT) 0002 DOUBLE PRECISION PROD,AV,SSC,X 000? DIMENSION PRODI 120, 120),AVI 30, 120) ,SSC(120) ,X( 120) 0004 DIME MS I ON NFL(10 ).NS (10 I,FMT(20 ),A( 30 1 0005 COMMON NF ,NI ,NI NT,NX, NCOV,NY,N AM , ILS , I RL 0006 00 I 1=1, NX 0007 SSC( I 1=0.0 OOOR DO 1 J=1,NX 0009 1 PPOD(I,J)=0.0 0010 NRFA 0 = NCOV+NY 0011 10 READ(4,FMT,EN0 = 99)(NS( I ) ,1=1 ,NF) , (A(I ) , 1=1 .NREAD) 0012 DO 15 1=1,NX 001 3 15 X(II=0.0 0014 K=l 001 5 KK=1 0016 X(l)=1.0 0017 00 2 T=1,NF 0018 K=K+NS(I) 0019 X(KI=1.0 00 20 KK=KK+NFL(I) 0021 K = KK 002? 2 CONTINUE 0023 NK=NF-1 0024 DO 8 1=1,NK 0025 I J=T +1 0026 00 8 J=IJ,NF 0027 K = K-(NS(II-11*NFL(J)-NS(J ) 0028 X(K) = 1.0 0029 KK=KK+NFL(J) 0030 K=KK 0031 8 CONTINUE 0032 K=1+NI+NINT 0033 DO 6 1=1,NREAD 0034 K=K+1 0035 6 X(K)=A(I) 0036 DO 4 1=1,NX 0037 SSC<I) = SSC(I)+X( I )*X(NX)*X(MX) 0038 00 4 .1 = 1 ,NX 0039 PROD(I,J)=PR00(I,J ) + X( I ) *X( J ) 0040 4 PRODI.), II=PROD(I,J ) 0041 GOTO 10 0042 99 RETURN 0043 ENO 85.000 85.500 86.000 87.000 88.000 89.000 90.000 91.000 9 2 .000 93.000 94.000 95.000 96.000 97.000 98.000 99.000 100.000 101.000 102.000 103.000 104.000 105.000 106.000 107.000 108.000 109.000 110.000 111.000 112.000 113.000 114.000 115.000 116.000 117.000 118.000 119.000 120.000 121.000 122.000 123.000 124.000 125.000 126. 000 •OPTIONS TN EFFECT* 10,ESC01C,SOURCE,NOLIST,NODE CK.LOAD,NOMAP • OPTIONS TN EFFECT* NAME = 0 AT A , LINECNT = 60 •STATISTICS* SOURCE STATEMENTS = 43,PROGRAM SIZE = •STATISTICS* NO DIAGNOSTICS GENERATED NO ERRORS IN DATA 2582 - 204 -MICHIGAN TERMINAL SYSTEM FORTRAM G!41336) LSMAT 02-10-78 1008 :48 0001 SUBROUTINE LSMAT(PROD,NAMF) 127.000 0002 DOUBLE PRECISION PROD 127.500 0003 01 MENS ION PRODI 1 20, 120 >, NA ME ( 1 20) 128.000 0004 COMMON NF.NI , NI NT , NX , N COv , Nr1, N AM, ILS. IPL 129.000 0005 CALL FTNCMDt 'SET ZEROSUPPRESS=ON' ,19) 130.000 0006 1 FORMAT ( •l ' , / / / 1 0 X , ' L E A S T SQUARES EQUATIONS' , / / / ) 131.000 0007 NTB=1 132.000 0008 NLAST=NX-NY-NCOV 133.000 0009 10 NTE=NLAST 134.000 0010 IF((NTE-NTBI.GT.251NTE=NTB*25 135.000 0011 WRITE!6, I ) 136.000 0012 WRITE(6.2 1 (NAME( I ), I = NTB,NTE > 137.000 0013 2 FORMAT(•0',26{A4,1X1> 138.000 0014 DO 3 I=1,NLAST 139.000 0015 3 WRIT E(6 ,6 ){ PROD ( I, J 1 , J =\lT B , NT E ) 14 0. 000 0016 1F(NLAST-NTE)4,4,5 141.000 0017 5 NTE=NLAST 142.000 0018 NTB = NT B-26 143.000 0019 GOTn 10 144. 000 0020 6 FORMAT(• « , 2 6 F 5 . 0 ) 145.000 0021 4 WRITE(6,11I 146.000 0022 11 FORMAT( • I' , / / / , 1 0 X , ' L E A S T SQUARES EQUATIONS - - COVARIATF.S AND RHS« 147.000 1/// ) 148.000 0023 NC=NLAST*1 149.000 0024 WRITE!6,12) I NAME ( I ),I = NC,NX) 150.000 0025 12 FORMAT('0' ,10(5X,A4,4X )) 151.000 0026- 00 7 1=1,NX 152.000 0027 7 WRITE(6,8)IPROO(I,J),J=NC,NX) 153.000 002R CALL FTNCMDI'SET ZEROS U P P P ES S= OFF • , 20 I 154.000 0029 8 FORMAT! > « , 10G 13.6) 155.000 003 0 RETURN 156.000 0031 END 157.000 *DPTIONS IN EFFECT* ID,EBCDIC,SOURCE,NOLIST,NODECK,LOAD, NOMAP *OPTIONS IN EFFECT* NAME = LSMAT , LINECNT = 60 • STATISTICS* SOURCE STATEMENTS = 31.PROGRAM SIZE = 1182 •STATISTICS* NO DIAGNOSTICS GENERATED v|0 ERRORS IN LSMAT v - 205 -MICHIGAN TERMINAL SYSTEM FORTRAN GI41336) REDUCE 02-10-78 0001 SUBROUTINE REDUCE(D,NX) 0002 DOUBLE PRECISION D 0003 DIMENSION DU20 , 120 1 , I RV( 15) , I AV (1 5t, NTRD( 101 0004 REWIND 1 0005 READ (1 , U N , (NTRDI I), 1= l . N ) 0006 1 F0RMATI20I4) 0007 NAA=NTR D(1 I 0008 00 2 1 = 1 , NA A 0009 REAO{ 1, I ) 1A , IF , IL 0010 00 3 IJ=IF,IL 0011 IF(IJ .EO.IAIGOTO 3 0012 DO 4 J=1,NX 0013 4 OtI J , J ) = -D( IA,J)+0(IJ , J) 0014 3 CONTINUE 0015 2 CONTINUE 0016 I F l N . E Q . l JRFT'JRN 1 0017 00 5 IJK=2,N 0018 REA0(1, 1 )NVA,( IAV( I ) , I = 1 » NV4) 0019 NAA=NTRDIIJK) 0020 00 6 11=1,NAA 0021 READ!1,1 )NVR,(IRV(I) , I = 1.NVR) 002? DO 7 1=1 ,MVR 0023 DO 8 J=l,NX 0024 8 o(iRvm,j) = - o ( i A v m , j ) + D U R v m . j i 0025 7 CONTINUE 0026 6 CONTINUE 0027 5 CONTINUE 0028 RETURN 0029 END •OPTIONS IN EFFECT* ID,EBCDIC,SOURCE,NOLIST,NOOECK,LOA 0, NOMAP "OPTIONS IN FFFECT* NAME = REDUCE , LINFCNT = 60 MO •STATISTICS* SOURCE STATEMENTS = •STATISTICS* NO DIAGNOSTICS GENERATED ERRORS IN REDUCE 29,PROGRAM SIZE = 1278 10:08:49 158. 158. 159. 160. 161. 162. 163. 164. 165. 166 . 167. 168. 169. 170. 171. 172. 173. 174. 175. 176. 177 178. 179. 180. 181. 182. 183. 184. 135. 000 500 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 - 206 -MICHIGAN TERMINAL SYSTEM FORTRAN G!41336) RLSMAT 02-10-78 10:08:49 0001 SU8R0UT INE RLSMAT (P., NAME, N EX , NCO V, NY.NIT) 186.000 0002 DOUBLE PRECISION R 186.500 0003 DIMENSION R ( 80,80 ),NAM E( 12 0) , N I K 8 0) 137.000 0004 NL=NEX-MCOV -NY 188.000 0005 4 WRITE(6,51 189.000 0006 5 FORMAT!<1',20X,'REDUCED SET OF LEAST SQUARES EQUATIONS' , / / / ) 190.000 0007 DO 1 1=1,NL 191.000 OOOR 1 WRITE!6 ,3 ) tR( I , J ) , J= I,I) 192.000 0009 3 FORMAT(' ' . 2 0 F 6 . 0 ) 193.000 0010 7 WRITE(b,8) 194.000 0011 8 FORMAT! ' 1' ,20X,'REDUCED SET OF LEAST SQUARES E OUAT IONS — COVARIAT 195.000 1 ES AND YIELD VARIATES' / / / 1 196.000 0012 NLl=NL*l 197.000 0013 DO 12 1=1,NEX 198.000 0014 12 WRITE!6,10)(R(I , J ) , J=NL1,NEX) 199.000 0015 10 FORMAT! ' • , 10G12.6) 200.000 0016 RETURN 201.000 0017 END 2 02.000 •OPTIONS IN EFFECT* ID,EBCDIC,SOURCE,NOLIST,NODECK ,LOAD,NOMAP •OPTIONS IN E FFECT* NAME = RLSMAT , LINECNT = 60 • STATISTIC S* SOURCE STATEMENTS = 17,PROGRAM SIZE = 868 • S T A T I S T I C S * N O D I A G N O S T I C S G E N E R A T E D MO ERRORS IN RLSMAT - 207 -MICHIGAN TERMINAL SYSTEM FORTRAN GC41336) MATINV 02- 10-78 10:08:49 0001 0002 0003 0004 0005 0006 0007 0008 0009 00 10 0011 SUBROUTINE MATINV<C,R,NDCI DOUBLE PRECISION R,C,DDET,DCOND DIMENSION R(80,801,C(NOC,NDC) 00 1 1=1,NOC DO 1 J=l,NOC 1 C (I , .11 =R (T . JI CALL D1NVRT(C,NDC,NDC,DDET,DCOND) 00 2 1=1,NDC DO 2 J=1,NDC 2 R(I , J)=C(1,J) RETURN END EFFECT* 0012 •OPTIONS IN * ID,EBCDIC,SOURCE,NOL1ST,NODECK,LOAD,NOMAP •OPTIONS IN EFFECT* NA ME = MATINV , LI NECNT = 60 *STATISTICS* SOURCE STATEMENTS = 12,PROGRAM SIZE = •STATI ST ICS* 203. 2 03. 204. 205. 2 06. 207. 208 . 2 09. 210. 211. 212. 213. 000 500 000 000 000 000 000 000 000 000 000 000   MO DIAGNOSTICS GENERATED FRPOPS IN MATINV 666 O. - 208 -MICHIGAN TERMINAL SYSTEM FORTRAN GI41336) ANOVA 02-10-78 10:08:49 SUBRDUTINE A NOVA IC,COEF,ERR, NDFT) DOUBLE PRECISION C.COEF EXTERNAL SUMSO DIMENSION C180,80),COEF(80 ), IOF( 30 I, SS( 30) , SMI 30) DOUBLE PRECISION NC AT ( 30 I READ(5, 1 IN,(IDF( I ) , 1 = 1,N) FORMAT (2014) RE A 0(5 ,9 I (NC A TU ) , I= l ,N) FORMAT(10A8 ) Nl = l N2 = l NOF=0 ST=0.0 DO 2 1=1,N Nl=N2+l N2 = N2*-I DF (I I N3=IDF(I) CALL GSPACEIZ,N3*N3*8) CALL CALLER! SUMSO,Z,IPTR(C),IPTR(COEF) , IPTRfNl ), IPTR(N3), IPTR(SO) ) CALL FSPACE(Z) SSI I )=S0 SM|I)=SO/IDF(I) NDF=N DF + 1DFI I) CF=COEF(1 ) * (1 .0 /C(1 .1 I l *COEF( l I NERR-NDFT-NDF-1 RES= ERR/NERR WRITE(6,10) FORM A T ( « I S / / , 1 5 X , • A N A L Y S I S OF VARIANCE TABLE ' / / ) WRITE(6,11) F O R M A T ! ' 0 ' , 6 X , « SOURCES',3X,' SUM SOUARES' , 5 X . ' D . F . • , 5 X , 'MEAN SQUARE l ' , 4 X , ' F - V A L U E ' , / / ) DO 12 1=1,N F=SM(I)/RES WRITE!6,131NCAT!I ) , SS ( I ) , IDF! I ) , SMI I ) ,F FORMAT( '0 ' , 6X ,A8 ,3X ,G13 .6 ,3X , 14, 3X , GI2.6,F12.4> FORMAT! • 0 « , 6 X , ' E R R O R • , 5 X , G 1 3 . 6 , 3 X , I 4 ,3X,G12.6J WRITE!6,14)ERR,NERR,RES WRITEI6.15) FORMATJ * 0 ' » / / / » 1 2 0 ! * * * ) , / ' • , • * • • END OF ANALYSIS • * * • ) RETURN END • OPTIONS IN EFFECT* I'D. EBCDIC . SOURCE, NOL I ST, NODECK .LOAD, NO MAP •OPTIONS IN EFFECTS NAME = ANOVA , LINECNT = 60 •STATISTICS* SOURCE STATEMENTS = 4 0, PROGRAM SIZE = 2120 •STATISTICS* NO DIAGNOSTICS GENERATED MO ERRORS IN ANOVA 0001 0002 0003 0004 0005 0006 0007 1 0008 0009 9 0010 001 1 0012 0013 0014 0015 0016 0017 0018 0019 0020 0021 0022 0023 2 0024 0025 0026 0027 0028 10 0029 0030 11 0031 0032 0033 12 0034 13 0035 14 0036 0037 0038 15 0039 0040 214 214 215 216. 217. 213. 219. 220. 221. 222. 223. 224. 225. 226. 227. 228. 229. 23 0. 231. 232. 233. 234. 235. 236. 237. 238. 239. 240. 241. 242. 243. 244. 245. 246, 247. 248, 249, 250. 251, 252, 253 000 500 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 ,000 ,000 ,000 ,000 000 000 000 000 000 000 000 ,000 000 000 ,000 000 ,000 - 209 -MICHIGAN TERMINAL SYSTEM FORTRAN 0(41336) SUM SO 02-10-78 10:08:50 0001 SUBROUTINE SUMSOIZ.C,COEF,NB,NE.SO I 0002 DOUBLE PRECISION C Z . C O E F 0003 DIMENSION C (80 ,8 01 , COE F( 80 t, SM (30 ) 0004 01 MENS I ON Z(NE ,NE) 0005 S0=0.0 0006 00 3 1=1,NE 0007 3 SM( 1 )=0.0 0008 00 5 1=1,NE 0009 K1=I-1*NB 0010 DO 5 .1=1, NE 0011 K2=J-1+NB 0012 5 Z ( I » J I = C ( K 1 , K 2 ) 0013 CALL DINVRT(Z,NE,NE,DDET, DCDND) 0014 DO 1 1=1,NE 0015 DO I J=l,NF 0016 K=J-1+NB 0017 1 SMII l=SM(I)*COEF(K(*Z( I, J) 0018 00 2 1=1,NE 0019 K=!-1+NB 0020 2 SO=SO*SM(II*COEF(K) 0021 RETURN 002? END •OPTIONS IN EFFECT* IO.EBCDIC,SOURCF,N0L1ST,NO DECK,LOAD, NC M A P •OPTIONS IN EFFECT* N&ME = SUMSO , L1NE"NT = 60 •STATISTICS* SOURCE STATEMENTS = •STATISTICS* NO DIAGNOSTICS GENERATED ERRORS IN SUMSO 22,PROGRAM SIZE = 1098 254. 254. 255. 256. 257. 258. 259. 260. 261. 262. 263. 264. 265. 266. 267. 268. 269. 270. 271 . 272. 273. 274. 000 5 00 000 000 000 000 000 000 000 000 00 0 000 00 0 000 000 000 000 000 000 000 000 000 - 210 -TERMINAL SYSTEM FORTRAN G(41336) INFORM 02-10-78 10:08 :50 0001 0002 0003 0004 0005 0006 0007 0008 0009 0010 001 1 0012 0013 21 0014 20 0015 0016 001 7 0018 0019 26 0020 0021 25 0022 0023 8 0024 0025 002 6 0027 0028 0029 0030 1 0031 0032 0033 0034 0035 0036 0037 0038 3 0039 2 0040 0041 SUBROUTINE INFORM I PROD,AV,SSC, NAME,NFL > DOUBLE PRECISION PROD,AV,S D,SS C COMMON NF,NI,NINT,NX,NCOV.NY,NAM,ILS,IRL DI MENS ION PROD(120, 120 I,A V(30,120),SD(1201•NAME(120) 1 ,NFL(10) NDUM=1+NI+MINT NREAD=NCOV+NY DO 20 1=1,NREAD IK=I+NDUM DO 20 J=l,NX I F ( P R O D ( I , J » . L E . O . O ) G O TO 21 AV( I ,JI = PROD(IK,Jl/PROD(1,JI GOTO 20 A V U . J ) =0.0 CONTINUE DO 25 I=1,NDUM IFCPRODd ,1 ) . L E . l .01G0TO 26 SD( I )=SSC( I) —PROD(NX,I )**2/PR0D( 1,1) SD(I ) = DSORT(SD( I )/(PRODI 1 , I l - l . D O ) ) GOTO 25 SD(I)=0.0 CONT INJF-WRITE(6,8) FORM AT{ • ' , / / / , 1 0 X , • M E A N , HONS AND COVARIATE MEANS ,SSC(120) NUMBERS OF OBSERVATIONS, STANDARD DEVI AT IN CLASSES AND SUBCLASSES' / / / ) IF = 1 NL= 1 CALL DO 1 STAT(PR0D ,4V ,SSC,SD,NAME, IF,ML I I = 1 , NF IF=NL+1 NL=NL+NFL(II CALL STAT(PROD,AV,SSC,SO,NAME,IF , NL) 'NE=NF-1 DO 2 1=1,NE K=NFL(I ) K1 = NFL(1+1 I DO 3 IJ=1,K IF=NL + 1 NL=NL+K1 CALL STAT(PR00,4V,SSC,SD,NAME,IF,NL) CONTINUE RETURN END •OPTICNS 1M EFFECT* 10,EBCD!C,SOURCE,NOLI ST, NO DECK,L0AD,NO MAP •OPTIONS TN EFFECT* NAME = INFORM , LINFCNT = 60 •STATISTICS* SOURCE STATEMENTS = 41,PROGRAM SIZE = • S T A T I S T i r s " NO DIAGNOSTICS GENERATED ERRORS IN INFORM 275. 000 275.500 2 76.000 277.000 278.000 279.000 280.000 281 .000 282.000 283.000 284.000 235.000 286.000 287.000 288.000 289.000 290.000 291.000 292.000 293.000 294.000 295.000 296.000 297.000 298.000 299.000 300 .000 301.000 302.000 303.000 304.000 305.000 3 06.000 307.000 308.000 309.000 310.000 311.000 312.000 313.000 314.000 315.000 316.000 2382 - 211 -M I C H I G A N T E R M I N A L S Y S T E M F O R T R A N 0 ( 4 1 3 3 6 ) S T A T 0 2 - 1 0 - 7 8 1 0 : 0 8 : 5 0 0 0 0 1 SIJBROUT I N E S T A T ( P R O D , A V , S S C , S D , N A M E , I F , N L ) • 3 1 7 . 0 0 0 0 0 0 2 D O U B L E P R E C I S I O N P R O D , A V , A A X , S D 3 1 7 . 5 0 0 0 0 0 3 D I M E N S I O N P R O D ( 1 2 0 , 120 I , AV ( 3 0 , 1 2 0 1, SS C ( 1 2 0 1, SD ( 120 1, NA ME ( 1 2 0 > 3 1 8 . 0 0 0 0 0 0 4 COMMON N F , N I , N I N T , N X , N C O V , N Y . N A M , I L S , I R L 3 1 9 . 0 0 0 0 0 0 5 N 0 U M = 1 + N I + N I N T 3 2 0 . 0 0 3 0 0 0 6 N R E A D = N C O V ~ N Y 3 2 1 . 0 0 0 0 0 0 7 W R I T E ! 6 , 1 ) ( N A M E ( I ) , I = I F , N L ) 32 2 . 0 0 0 0 0 0 8 1 F O R M A T ( ' 0 ' , / / , 6 X , 1 0 ( 5 X , A 4 , 3 X ) ) 3 2 3 . O O J 0 0 0 9 W R I T E ( 6 , 2 I ( A V ( N R E A 0 , I I , I = I F , N L ) 3 2 4 . 0 0 0 0 0 1 0 W R I T E I 6 , 7 ) ( P R O D I 1 , I ) , ! = I F , N L ) 3 2 5 . 0 0 0 001 1 W R I T E 1 6 , 3 1 ( S D ( I 1 , I = I F , N L 1 3 2 6 . 0 0 0 0 0 1 2 I F ( N C O V . E O . O ) G O T O 10 3 2 7 . 0 0 0 001 3 0 0 5 1 = 1 , N C O V 3 2 8 . 0 0 3 0 0 1 4 K = N D U M - 1 32 9 . 0 0 0 0 0 1 5 5 W R I T E ! 6 , 4 ) N A M E ( K ) , ( A V ( I , J ) , J = 1 F , N L ) 3 3 0 . 0 0 3 001 6 2 F O R M A T ( ' 0 ' . ' U E A N ' » 2 X , 1 0 I F 1 0 . 0 , 2 X 1 1 3 3 1 . 0 0 0 0 0 1 7 7 F O R M A T ( ' 0 ' ii OB • , 2 X , 1 0( F l 0 . 0 . 2X ) 1 3 3 2 . 0 0 0 0 0 1 8 3 F O R M A L ! < 0 ' , ' S . D . ' , 2 X , 1 0 ( F 1 2 . 4 ) ) 3 3 3 . 0 0 3 0 0 1 9 4 F O R M A T ! ' ' , A 4 , 2 X , 1 0 ( 2 X , F 8 . 2 , 2 X ) ) 3 3 4 . 0 0 3 0 0 2 0 10 R E T U R N 3 3 5 . 0 0 0 0 0 2 1 END 3 3 6 . 0 0 3 • O P T I O N S T N E F F E C T * I D , E B C D I C , S O U R C E . N O L 1ST , NO.DEC'< , L 0 A D , N DM AP • O P T I O N S TN E F F E C T * NAME = S T A T , L I N E C N T = 6 0 • S T A T I S T I C S * ' - S O U R C E S T A T E M E N T S = 2 1 , P R O G R A M S I Z E = 1 0 R 3 • S T A T I S T I C S * NO D I A G N O S T I C S G E N E R A T E D NO E R R O R S I N S T A T NO S T A T E M E N T S C L A G G E 0 I N T H E A B O V E C O M P I L A T I O N S . N AM F N U M B E R O F E R R O R S / W A R N I N G S S E V E R I T Y M A I M 0 0 T I T L E 0 0 O A T A 0 0 L S MA T 0 0 9 E D U C F 0 0 R L S M A T 0 0 MA T I N V 0 0 A N O V A 0 0 S U M S O 0 0 T NF ORM 0 0 S T A T 0 0 E X E C U T I O N T E R M I N A T E D $ S I G APPENDIX 3.1 Least squares equations for Coast Douglas-fir inventory data-V \ P<C y\ SS / v . SS y\ /* Mf VI P P*H 1 2 Z 3 PZ2 PZ3 PC22 PC 23 PH22 PH23 SITE AGE L AGE HT 635 4C4 172 59 501 134 347 57 107 65 47 12 76.J60.0 34655.0 1076.38 5540.00 404 404 34 7 57 347 57 49J10.0 20440.0 674.488 3469.00 172 172 1C7 65 107 65 19450.0 11460.0 305.422 1564.00 59 59 47 12 47 12 76JO.O0 . 2755.00 96.9688 507.030 5C1 347 107 47 501 347 107 47 62 .20 .0 23545.0 824.084 / 4178.00 134 57 ts 12 134 57 65 12 14J60.0 11110.0 252.794 1362.00 347 347 347 347 4 3-, 40.0 15855.0 567.740 29 06.00 57 57 57 57 5870.00 45E5.0C 1C6.748 563.000 1C7 107 IC7 107 12460.0 5575.00 179.719 379.J00 65 65 65 65 6970.00 5835.00 125.702 685.000 47 47 47 47 6C30.00 2115.00 76.6253 393.000 12 12 • 12 12 1520.00 640.000 20.3435 114.000 0 .9o l48CF-07 0.402540E-07 0 . 4 0 2 5 4 0 E « C 7 0.228623E*07 12o236. 68". 170. 0.144496E*08 8824?7. 0. 1 1C651E*08 81624. 1 ' 0 .115507E-C7 0.3725426*09 61633.7 328510. 0.681219E*07 422841. C.565701E*07 34654.6 610394. 0.192616E*09 128236. 61633.7 1848.59 9599.36 205333. 12456.1 156664. 1076.87 16075.5 639170. 328510. 9599.36 53C56.0 O . 109727E*07 66 364 .8 894136. • 59 33.30 97164.1 0.519648E*07 0.307847E*08 NT 1 19107. 79032.0 30565.0 9510.00 93169.0 25938.0 68462.0 I 0570.0 17017.0 13548.0 7690.00 1820.00 0.144496E*C8 0 . 6 8 1 2 1 9 E » C 7 205383. 0. 109727E*07 0.271085E*03 0 .136112E-C7 0.195618E*C8 134201. 0.196880E-07 0. 646765E*09 OBH 7225.88 4455.53 2095.98 674.366 5403 .58 1*22.30 3702.64 752.B91 1183.84 912.142 517.097 157.269 882427. 422341. 12456.1 66864.8 0.1361125*07 87191.9 O . U 1 6 1 6 E * 0 7 7518.67 119368. 0.380133E*08 B 4 88591.0 54776.0 26807.0 7008.00 62769.0 25822.0 44818.0 9958.00 12629.0 14178.0 5 32 2.00 1666.00 0. U0651E*08 0.565701E*07 156664. 894136. 0 . 1 9 5 6 1 B E » 0 8 0. 11 1616E-07 0 . 1 6 7 0 3 0 E » 0 8 104135. 0.184973E*07 0.5844S4E*O9 SO 634.999 416.565 160.461 57.9730 501.200 133.799 362.653 53.9070 93.3360 67.0750 45.1560 12.8170 61624.1 34654.6 1076.37 5933.SO 134231. 7513.67 104136. 771.074 10854.7 0.353960; HT Ba 8941.36 -5363.56 2393.20 679.600 5933.74 2957.62 4243.23 1125.23 1249.12 1644.03 491.340 1S8.260 C . l l S 5 0 7 E * O 7 610394. 16075.5 9 7164.1 0.196330E*37 119363. 0. 13<-973E*07 10854.7 213792. 1*07 0.672933E*03 0.291393E*D7 0.176400E*07 901337. 225535. 0 .2 - . 2344EO7 ' 0 . i " ? 9 = E-37 , 34 1017. ; 412521. 433316. 167729. 57655.3 0.372543=*09 0. 1 = 261SE* 09 C - . 5 V 9 r » 3 = - :7 0 .307547; .03 0.646765E-39 C . 3 e : i 5 3 = - 0 ? I 0 . 5 2 4 4 5 4 c » 3 9 0 .35 :963E'07 0.672933E*~3 |_i 0 - 2 1 2 1 5 1 E . i li^j Remarks: P - Pure; P+C- Douglas-fir conifer mixed; P+fl— Douglas-fir hardwood mixed; A — F o r e s t Inventory Zone 2; zV-Forest Inventory Zone 3; PZ&, PZ3, PCZ2, PCZ3, PHZ2, and PHZ3 are interactions for types and zones, LAGE—logarithmic age; HT—height; NT—number of trees per acre; BA—basal areaj S D — r e l a t i v e stand density; HTBA-—HTxBA; VOL—volume. APPENDIX 3.2 Least squares equations for Interior Douglas-fir inventory data—qualitative ^ #*\ / V - A A> A >V / V / \ PZ5 PZ6 PZ7 PZ8 PCZ4 PCZ5 PCZ6 PCZ7 PCZ8 PHZ4 PHZ5 PHZ6 PHZ7 PHZ8 164 44 320- 231 232 62 83 400 21 8 4 3 69 164 44 320 231 232 62 83 400 21 8 4 . 3 69 232 8 v a r i a b l e s / \ /S >V /* /V A MFAN P P + C P+H Z4 Z5 Z6 Z7 Z8 PZ4 2332 1450 793 64 931 230 130 719 252 6 91 145C 1450 793 691 164 44 320 231 691 793 232 62 83 4 CO 21 34 84 8 4 3 69 931 691 232 8 931 691 230 164 62 4 230 130 44 ' 83- , 3 130 759 320 400 69 789 252 231 21 252 691 691 691 691 164 164 164 44 44 44 320 320 320 231 231 231 232 232 232 62 62 62 S3 83 83 * 430 400 400 21 21 21 8 8 8 4 4 4 3 3 3 69 69 69 164 6 2 44 83 32 0 400 231 21 164 44 320 231 232 69 62 83 400 21 ro 69 Remarks: P, P-i-C, p+H, y\ s\ y\ ^ Z4, Z5, EZ4, PCZ4, PIIZ4 etc., are defined i n Appendix 3.1. APPENDIX 3.2 Least squares equations f or I n t e r i o r Douglas-fir inventory data — quantitative v a r i a b l e s , S I T E 13^110. i 10593. 63553.0 7170.00 53*10.0 18j70 . 0 9 4 s 3 . 00 6 6 P 3 0.G I S J S O . O • S G J O O . O 1 3 4 5 0 . 0 3105.03 26/00.0 16*55.0 I7 0 5 . 0 50=0.00 61 j 5 . 00 33055.0 1 4 j 5 . 00 70j.or,o 3 2./ • 3 0 0 210.300 55J5.G0 0. 145573F*C3 0. lol313E+33 3 A,21 A. 0.1j62e7f*07 0.22S483E+C3 0 . 2 i l l 6 6 E - : 7 0.1 uS6 34S*C3 192393. 0.14176SE+C7 0.4-.4357E + C9 4GE 2C6770. 132030. 69150.0 5590.00 . , 86345.0 21690.0 11900.0 612E5.0 25050.0 64455.0 155O.0 4C6C.00 24763.0 23215.0 21350.0 5840.00 7605.00 32020.0 1835. 00 540.000 310.000 235.000 4505.00 0.161318E+08 0.206559E+08 438524. C.157359E+07 0.253076E+0S C.245165E-07 0. 156617E+08 2C6438. 0.U5517E+07 0.513444E+09 LAGS 4472.00 2794. 10 1526.55 151.342 1806.58 446.643 251.232 1470.93 496.516 1338.96 218. 356 6 5. 32 53 594.987 456.474 453. 093 120.956 160.342 752.120 40.0'<21 14.5238 7.33130 5.614C0 123.8 73 349214. 40S524. 664 1.05 32364.6 543302. 52015.0 4CC719. 4466.18 33019.1 0. IO3825E+08 HT 16900.0 10260.0 6012.00 623.000 6370.00 1842. 00 072.000 6119.00 1697.00 4671.00 1304.00 233.000 2433.00 1569.00 1640.00 508.000 569. 000 3167.00 123. COO 59.0000 30 .0000 20.0000 51 9. 000 ,0. 1362875 + 07 0.157899E+07 32 864.6 132004. 0.220146E+07 200576. 0. 167125E + 07 18319.1 143900. 0.447164E+08 NT 281783. 151174. 119055. 11554.0 91542.0 28066.0 15883.0 123784. 22503.0 62928.0 10932.0 4757. 00 43991 .0 20566.0 2 7544.0 3711.00 10827.0 69631.0 I 942 .00 670. 000 4 23.000 299.000 10162.0 0.229483E+08 0.253076E+C8 543802. 0. 22C146E+G7 0.471926E+08 0.319500E+07 0.326559E+C8 368638. 0. 275397E + C7 0.864974E+C9 26829.7 17067.2 8892.33 869.653 10873.2 2701.96 1423.81 8770.63 3055 .1 1 8153.60 • 1949 .85 507.755 3626 .7 1 2829.26 2643.22 711.047 • 884.476 4428.24 225.844 81.3400 41 .0600 31 .5780 715.675 0.21U66E+07 0.249 165E+07 52015.0 2006 76. 0.319500E+07 323261 . 0.249127E+07 27615.9 208263. 0.651238E+08 84 204380. 115236. 82170.0 6974.00 63337.0 21550.0 11061.0 84750.0 18182.0 49170.0 14788.0 3551.00 30380.0 16047.0 19300.0 6486.00 7320.30 47729.0 1335.00. 367.000 276.000 190.000 614 1. 00 0. 168634E + 08 0. 196617E + 08 400719. 0.1A7125E+07 0.326559E + 08 0.249127E+07 0.254053E+08 274790. 0.2224566+07 0.689218E+09 SO 2333 .35 1309.05 92*..321 96.9330 739. 205 242.026 117.377 1039.60 191.646 531.423 172.119 33.7930 389.148 177.562 202.215 66.5720 76.9400 564.502 14.0340 5.56700 3.33500 • 2.13600 85.9500 192390. 206433. 4463.13 13319 .1 363638. 27615.9 274790. 3227.84 23465.9 0.736113E+07 H T H A 16712.5 9231.71 6 503.43 572.360 5371.95 1910.71 863.300 7 199.51 1306.53 3815. e9 1299.70 2 54.400 2587.39 1274.33 1523.46 586.770 595.280 4.105.72 92.2000 27.6000 24.2400 14.1200 5 36.4 CO 0.141763E+07 0.165517E+07 33319.1 143900. 0.275397E-07 203263. 0.222456E+07 23465.9 202912. 0.621015E+03 V O L 0.527S53=+07 0 . 2 ° 0 1 2 2 r - 0 7 0.219713E + 07 1S0277. 0.173624E+07 5771 40. 2S6!)5S. 0.276735S + 07 441223. 0 . 1 19638E-07 3554 75. E6706.0 811962 . 4032 05. 501377. 171245. 1961 33. 0.125539E-07 33Ci S.O 8434 .03 7420.03 4377.30 159996. , 4 4 4 8 5 7 H « 0 9 ,5l3444=+09 . 1 0 3 ? . 2 5 F » " 3 , 4 4 7 1 6 4 = » 0 3 .364974F-05 • 651 23S3 + 0S ,6&5?16F+35 .736S13E-07 .621015= + 03 .194312E + U Remarks: LAGE, HT, NT, 3A, SD, HTBA, and VOL are defined i n Appendix 3.1. J Appendix 3.3 Least squares equations for the Interior spruce inventory data —qualitative variables X X SS ^ S*. S\ ^ f> •»> ^ f\ / \ f \ C211 CZ12 H z " \ H p S HZ^6 HZ^7 HZ^e HZ*9 K i l l HZ 12 « i N PURE P»C P*H 87 4 8Z_5 EZ 6 B7 7 81 8 B Z 9 BZ11 BZIZ PZ 4 P Z 5 PZ 6 PZ 7 PZ 8 P 9 PZ1 P » . « * CZ * CI | « 7 CZ . CZ * ^ ^ ^ „ . L S . L U . 2 . 4 4 . 90. % B 1 . B 0 0 / l 2 ; / V , / z 7 ; . " 3 3 . . 3 ^ . ' ' 80. 7 « . » . 8 4 . " l 6 . 4 5 . 9 6 . 292. 3 7 . 522. ! 4 4 . 44. 5 9 . 2 2 0 . , 1 * . 439 1 U < . . ! 1 « . 52. 3 4 . . 1 6 . 4 5 . 9 6 . 292. 37. 522. 52. 8 4 . 16. 4 5 . 9 6 . 292. 3 7 . 522. 144. 4 4 . 59. 2 2 0 . 114. 4 3 9 . 2 4 . 146. * * • " ° " * 3 ' - * ' + I, 200 1150. 1 190. 231. 281. 2C C . . 52. 144. 4 . 12C. 8 4 . 4 4 . 1. 79. 16. 5 9 . 4 . 274. 4 5 . 22C. 9 . 2 3 3 . 9 6 . 114. 23. B34 . 2 9 2 . 439. 1C3. a c . 37. 24. 19. 7c t . 5 2 2 . 146. l i e . 52. 52. 8 4 . £4. 16. 16. 4 5 . 4 5 . 9 6 . 9 6 . 2 9 2 . 2 9 2 . 37. 37. 52 2. 522. 144. 144. 4 4 . 44. 5 9 . 5 9 . 2 2 C . 2 2 0 . 114. 114. 4 3 9 . 4 3 9 . 2 4 . 2 4 . 146. 146. 4 . 4 . 1. 1. 4 . 4. 9 . 9 . 2 3 . 2 3 . 103. 1C3. 19. 19. H E . 11$. 1. 4 . 9 . 2 3 . 103. 19. 113. . 144. 129. 79. 2 7 4 . 5 9 . 2 2 0 . 9 6 . 292. Remarks: 'pDRE-pure spruce type; P^-spruce-conifer mixed type; Prt-.pruce-h.rdw.od mixed type; other symbols are similarly defined in Appendix 3.1. \ \ t Appendix 3.3 Least squares equations for the I n t e r i o r spruce inventory data — quantitative v a r i a b l e s S I T E 2 1 J 3 4 ? . 9 U 5 7 . 0 9 5 , 4 2 . 0 2 2 s 0 3 . C 1 6 C 3 5 . 0 1 J i 1 7 . 0 6 2 / 0 . C O 2 3 * 2 6 . 0 1 9 J 2 0 . 0 7 C C 5 3 . 0 4 3 3 4 . C O 5 5 - . 3 6 . G 4 2 0 7 . C O 71 J 5 . C C 1 2 = 3 . C O 3 5 . 5 . C O 3 1 v . C . C C 2 4 ; 5 8 . C 2 1 o l . 0 0 4 C i 9 3 . C 1 1 - 3 7 . 0 3 3 - 6 . C O 4 6 < ; 4 . 0 C 1 3 3 9 5 . C 9 C •» 2 . C 0 3 7 . ^ 1 . 0 1 4 ; ; . cc 5 7 a t . C C 3 4 > . C 0 C «*.:oco 36j. :cc 7 3 u . 0 O O 1 3 . i 7 . C 0 £334.C C 1 l o 3 . C O 9 * ^ 2 . C C C . 1 7 5 2 & 1 E + C S 0 . 2 i 6 5 3 2 S + C S 4 2 * 3 6 2 . 0 . 1 ; 2 0 4 7 E + C 7 C . 3 c C 5 < : 6 f + C 8 C . 2 5 2 3 2 6 E + C 7 2 l v C 4 4 . 2 6 * 2 4 6 . C . 2 T 3 5 4 2 E + C S C . 2 / 1 5 2 7 E + C 9 A G E C A G E H T . N T SO D B H 8 A H T B A V O L 2 S 2 C 5 5 . 5 2 6 4 . 1 3 2 1 8 3 4 . 0 4 7 C 6 9 3 . 3 0 8 4 3 . 4 2 6 1 5 . 0 3 3 1 5 2 . 6 2 3 5 2 7 3 6 . 0. 1 0 2 7 3 5 E - 0 3 1 2 7 5 3 0 - 2 3 2 1 . 8 3 9 6 5 9 . 0 0 ' 2 0 2 1 2 6 . 1 3 6 0 3 . 5 1 1 4 4 . 0 1 1 3 9 0 . 9 9 1 5 4 2 3 2 . 0 . 4 5 5 7 S 4 E + 0 7 1 2 7 6 3 3 . 2 3 9 1 . 7 6 9 9 6 3 . 0 0 2 1 4 7 5 3 . 1 4 1 5 4 . 2 1 1 9 C . 0 2 1 4 4 7 - 4 4 1 6 2 7 8 4 . 0 . 4 7 4 9 9 7 E + 0 7 2 6 E 7 5 . 0 ' 5 5 3 . 5 3 3 2 2 6 2 . 0 0 5 3 8 1 9 . 0 3 0 8 5 . 6 9 2 8 0 . 9 9 6 3 1 ' - . 1 3 4 3 5 7 2 0.0 9 6 6 1 2 5 -2 1 9 4 0 . 0 4 C 5 . 0 6 5 1 6 8 6 . 0 0 3 7 3 5 2 . C 2 3 4 2 . 9 8 2 0 0 . 0 3 2 2 4 6 . 9 2 1 2 7 9 3 6 . 0 8 2 3 7 4 1 . 1 4 3 4 5 - 0 2 6 1 . 5 3 2 1 1 8 0 . 0 0 2 3 1 2 4 . 0 1 6 1 6 . 2 2 1 2 9 . 0 0 2 1 3 6 . 8 4 6 1 9 6 7 8 . 0 53 7 6 6 9 . 9 2 1 5 . C C 1 6 1 . 3 9 3 6 7 7 . 0 0 0 1 2 6 3 6.0 9 5 5 . 1 4 8 7 8 . 9 9 8 0 9 4 . I 9 6 0 1 0 2 9 9.0 3 0 7 1 1 4 . 2 9 9 3 0 . 0 5 5 1 . 0 4 5 2 4 6 9 . 0 0 4 2 5 3 3 . 0 3 5 8 2 . 0 0 2 7 4 . 0 4 4 3 6 0 . 5 8 7 3 7 9 3 5.0 0 . 1 14 3 S 3 F + 0 7 2 4 3 5 5 . C ' 6 5 . 9 1 7 1 9 3 6 . 0 0 4 6 3 3 3.0 2 5 1 2 . 2 5 2 3 3 . 0 3 1 2 7 5 . 3 1 3 3 1 6 6 3.0 9 3 6 4 4 7 -£ - 4 6 0 . 0 1 6 7 2 . 5 9 72 3 3 . 0 0 1 4 2 0 3 8 . 1 0 2 1 6 . 8 8 3 3 . 9 3 6 1 0 6 6 . 7 3 1 1 5 2 3 9 . 0 . 3 4 3 3 5 5 E + 0 7 9 C 3 0 . 0 0 16 3 . 2 9 0 53 5 . C O O 1 5 C S 3 . 0 C 5 7 . 1 5 7 8 0 . 0 0 2 0 6 8 . 3 1 2 0 9 5 0 1.00 2 3 7 7 7 0 . 6 3 7 3 0 . 0 1 5 3 3 . 2 5 61 6 4 . 0 0 1 5 1 0 4 4 . 8 6 5 6 . 7 9 . . 7 3 5 . 9 9 2 3 5 3 . U 9 1 C 0 5 7 0 . 0 - 2 7 9 3 6 2 E + 0 7 6 0 1 0 . 0 0 • 1 C 6 . 5 9 0 4 5 5 . 0 0 C 1 C 6 9 6.0 6 2 6 . 4 3 2 5 2 . 0 0 0 0 7 2 . 4 1 4 0 8 1 5 9 . 0 0 2 5 2 5 5 1 . 9 5 4 3 . C O 1 7 1 - 2 2 2 7 7 4 . 0 0 0 . 1 4 4 3 5 .0 1 0 6 1 . 2 0 8 4 . C 0 2 0 1 1 9 . 0 4 5 1 2 4 3 5 . 0 3 7 1 3 1 7 . 2 2 C C . C 0 3 4 . I 9 6 0 1 5 7 . 0 0 0 2 4 5 4 . 0 0 2 1 1 . 2 9 4 1 6 . 0 0 1 0 2 2 . 4 2 6 0 2 2 3 1.00 7 9 1 3 S . 0 5 2 9 5 • C O 5 1 . 5 9 C 0 4 2 C . 0 0 0 6 7 4 9 . 0 0 5 9 3 . 9 9 2 4 5 . 0 0 2 0 5 9 . 0 2 2 0 6 0 3 3 . 3 0 2 0 1 0 6 6 . 1 C 7 9 0 . 0 " 1 9 4 . 9 9 0 8 4 5 . 0 0 0 1 8 2 0 9.0 1 1 2 7 . 1 5 9 6 . 0 0 0 0 1 2 3 . S 9 8 1 3 4 7 1.0 4 1 3 0 3 2 . 3 2 7 C 0 . 0 5 5 C . 7 7 C 2 5 3 6 . 0 0 4 4 8 3 0 . 0 3 6 8 0 . 3 8 2 9 2 . 0 0 2 3 6 3 . 0 3 5 3 3 8 9 0.0 0 . 1 1 6 1 2 2 E + 0 7 4 3 9 5 . 0 0 7 4 . 9 4 6 0 2 3 5 . 0 0 0 6 5 3 3 . 0 0 3 3 4 . 1 2 9 3 7 . C 0 1 3 2 3 . 3 1 1 0 3 9 3 5 . 0 0 9 9 1 1 0 . 0 5 6 9 5 0 . C 1 C 5 7 . 0 3 4 2 3 7 . 0 0 9 3 1 7 0 . 0 5 9 1 5 . 9 2 5 2 2 . 0 0 0 6 0 3 . 0 3 9 6e973.0 C . I 9 7 9 9 1 E + 0 7 1 5 5 9 0 . 0 2 5 C . 3 5 1 1 1 9 9 . 0 0 2 5 5 5 5 . 0 1 6 7 5 . 3 2 1 4 4 . 0 0 2 ' 1 7 0 . 4 3 5 1 9 3 0 9 . 0 5 5 3 7 C 3 . 4 7 3 0 . 0 0 8 3 . 3 3 5 0 3 9 6 . 0 0 0 8 3 5 1 . C O 5 4 5 . 7 3 0 4 4 . C O O O 6 5 . 8 0 1 0 7 0 4 3 . 0 0 2 0 5 9 7 2 . 6 7 C 5 . 0 0 1 1 9 . 6 8 8 4 S 3 . 0 0 0 9 5 7 6 . C O 7 0 0 . 5 9 4 5 8 . 9 9 7 0 6 7 . 9 9 2 0 7 5 3 0 . 0 0 2 1 6 6 6 0 . 2 4 0 2 0 . 0 4 4 3 . 1 4 2 1 9 3 9 . 0 0 3 4 9 9 9 . 0 2 8 8 8 . 5 9 2 2 0 . 0 4 2 2 9 4 - 3 1 3 3 0 3 9 6 . 0 9 2 1 6 2 0 . 1 1 5 4 0 . C 2 2 6 . 7 2 7 9 2 1 . 0 0 0 2 3 9 3 4 . 0 1 2 5 5 . 8 7 1 1 4 . 0 0 0 1 3 2 - 4 2 7 1 5 7 9 6 - C 4 6 0 7 C 7 . 4 7 2 2 5 . C 3 8 1 . 9 7 9 . 3 3 1 3 . 0 0 7 7 5 3 6 . 0 5 3 5 1 . 6 7 4 3 8 . 9 3 6 5 6 9 . 7 7 8 6 1 3 3 6-0 0 . 1 S 6 5 7 0 E + 0 7 2 7 5 C . C 3 4 9 . 2 3 7 0 1 6 3 . 0 0 C 5 1 5 3 - 0 0 2 6 4 . 3 5 7 2 4 . 0 3 0 0 2 4 . 5 7 4 0 3 3 9 7 . 3 3 3 7 3 C 0 . 0 14 94 0 . 0 2 5 1 . 7 5 3 9 8 4 . 0 0 0 2 9 7 4 5 . 0 1 4 7 1 . 0 3 1 4 5 . 9 5 6 1 2 2 . 0 7 5 1 6 3 3 7 . 0 4 2 5 2 9 3 . 3 - - 0 . 0 C 0 7 . 6 2 4 C 0 3 2 . 0 0 0 C 6 9 7 . 3 0 0 4 0 . 7 3 0 0 4 . 0 0 0 3 0 4 . 0 2 2 C 0 4 3 3 . 0 0 0 1 2 4 . 5 2 . 0 7 5 . C O C O • 1 . 3 7 5 0 3 1 0 . 0 0 0 0 2 3 8 . 0 0 0 1 1 . 2 8 4 0 1 . 0 0 0 3 3 2 . 0 3 0 0 0 2 0 0 . C O O 5 3 3 3 . 0 0 3 1 0 . 0 0 0 7 . 5 C 5 0 0 3 2 . 0 0 0 C 6 5 6 . 3 C 0 4 3 . 2 6 0 0 4 . 0 C 0 0 0 3 . 7 7 8 0 0 4 3 3 . 0 0 0 1 1 2 1 6 . 0 5 5 5 . 0 0 0 1 5 . 9 1 3 0 6 0 . 0 0 0 0 1 1 3 5 . 0 0 9 9 . 4 1 7 0 9 - 0 0 3 0 0 7 . 3 5 2 0 0 . 8 7 6 . 0 0 0 2 1 1 3 4 . 0 2 3 2 5 . 0 0 4 4 . 2 3 0 0 1 7 0 . 0 0 0 4 2 4 0 . 0 0 2 2 9 . 2 3 7 . 2 3 . 0 0 1 0 . 1 9 . 4 3 3 0 2 4 0 1 . 0 0 6 7 7 0 3 . 0 5 5 3 5 . C C 1 5 9 . 3 3 3 8 7 9 . 0 0 0 1 9 7 2 2 . 0 1 1 8 4 . 8 0 1 0 2 . 9 9 3 1 3 3 . 8 6 2 1 4 4 6 3 - 0 41 1 7 7 3 . 2 1 9 5 . 0 0 3 9 . 1 C 7 0 1 3 6 . 0 0 0 3 4 C 2 . 0 0 2 0 6 . 1 7 1 1 9 . 0 9 1 0 ' 1 5 . 72 7 0 2 1 1 9 . 3 0 5 1 3 5 0 . 0 1 1 S 4 0 . 0 2 3 4 . 4 7 2 9 4 3 . O O C 2 3 6 2 9 . 0 1 2 6 3 . 8 0 1 1 7 . 9 9 6 1 2 7 . 5 5 5 1 4 7 3 5 . 0 .3 . - .54 1 7 . 0 . 2 2 6 5 2 2 E + C 8 4 2 2 8 6 2 . 0. 1 3 2 C 4 7 E + 0 7 . C . 3 S C 5 5 6 E + C 8 0 - 2 5 2 3 7 6 E + 0 7 2 1 0 0 4 4 . 2 6 9 2 4 3 . 0 - 2 9 3 5 4 2 E + 0 8 0 . 3 7 1 5 7 7 F + C 9 C - 3 2 6 1 3 6 E « - 0 3 5 7 0 1 7 9 . 0 . 2 ' . 4 3 0 8 E * 0 7 0 . 5 1 2 5 3 3 E * C 8 0 - 3 4 2 2 O 4 E + O 7 2 8 2 0 5 9 . 3 6 4 0 6 3 . 0.3994 54 E - 0 3 •3 . 1 17 4 0 3 F - I Q 5 7 6 1 7 5 . 1 C 6 4 3 . 3 4 4 4 5 2 . 6 9 5 1 7 4 2 . 6 2 5 3 3 . 1 5 2 6 4 . 1 9 6 4 6 6 . 9 4 7 1 9 9 5 5 . C - 2 1 0 2 7 2 E + C S C . 2 4 4 3 C 3 E + 0 7 4 4 4 9 2 . 6 1 9 4 3 4 4 . 0 . 3 5 3 9 9 4 E+C7 2 6 7 5 7 5 . 2 2 0 0 2 . 9 2 9 3 3 9 . 3 0 . 3 1 5 2 6 2 E • 0 7 0 . 544 3 ! 2 F + 0 ? O . 5 1 7 5 8 3 E + 0 8 9 5 1 7 4 2 . 0.393994E+07 0 . 101 3 7 9 E - C 9 0 . 5 4 2 6 9 7 E - 0 7 5 0 4 9 3 3 . 6 3 0 9 1 9 . 0 . 7 1 5 7 U E - O S 0 . 2 C 5 - 5 5 E - 1 C C - 3 4 2 2 0 4 E + 0 7 6 2 5 3 3 . 1 2 6 7 5 7 5 . 0- 542 3 5 7 E + C 7 3 7 8 9 9 8 . 3 U 4 J . 1 3 9 8 3 2 . 3 0 . 4 3 4 8 0 7 E + C 7 0 . 1 2 3 5 3 P E + C 9 2 3 2 0 5 5 . 5 2 6 4 . 19 2 2 C 0 2 . 9 5 C 4 9 3 8 . 3 1 1 4 3 . 1 2 3 9 6 . 3 8 3 3 9 5 . 1 6 3 8 1 J 6 3 . C. 1 1 0 3 9 3 = 0 3 3 6 4 0 6 3 . 6 4 6 6 . 9 4 29 3 3 9 . 3 6 3 0 9 1 9 . 3 9 8 8 2 . 3 3 3 9 5 . 1 6 4 9 7 7 . 73 5 2 1 9 0 6 . C . l 5 3 S 7 1 E + 0 6 0 . 3 5 9 4 5 4 E + 0 8 7 1 5 9 5 5 . 0 . 3 1 5 2 6 2 " + 07 C. 7 1 5 7 1 1 E + C 8 0 . 4 3 4 6 0 7 E + C 7 3 S 1 3 6 3 . 5 2 1 9 3 6 . 0 . 5 6 6 2 2 7 H + 0 3 0 . 1 6 3 3 ? ? E + 1 C C I 1 7 6 9 3 S + 10 C 2 1 0 3 2 3 S + 0 8 0 - 9 4 4 3 1 2 5 + 0 6 0. 2 0 5 8 5 5 E + 1 0 0. 1 2 8 9 3 8 E + 0 9 0 . 1 1 0 3 9 3 E + 0 8 0 . 1 5 3 5 7 1 S - 0 8 0 . 1 5 d J ? : S c - 1 0 O . S l i ^ i t - i l A p p e n d i x 3.4 L e a s t s q u a r e s e q u a t i o n s f o r t h e I n t e r i o r l o d g e p o l e p i n e i n v e n t o r y d a t a — q u a l i t a t i v e v a r i a b l e s * = AN PCSE 4199 2356 2356 2356 1010 833 1130 273 274 306 1434 533 47 197 642 153 142 137 964 213 27 73 250 49 132 154 133 217 1 69 23 8 76 15 332 103 19 5C 1C10 1C1C 642 153 142 137 964 213 27 78 642 153 142 137 964 213 27 78 250 49 132 154 133 217 1 69 /*> / X HMIX P.Z 4 ez 5 ei 6 833 1130 278 274 642 152 142 250 49 132 333 233 76 238 1130 76 278 6Z 7 BZ 8 306 1434 137 964 154 138 15 332 / s />» / v BZ 9 BZ10 BZ12 533 47 197 213 27 73 217 1 69 103 19 50 / X /"•-PZ 4 PZ 5 PZ 6 6 4 2 1 5 3 1 4 2 6 4 2 1 5 3 1 4 2 6 4 2 1 5 3 / X PZ 7 137 137 P Z 8 P Z 9 P Z 1 0 P Z 1 2 C Z 4 C Z 5 9 6 4 2 1 3 2 7 7 8 2 5 0 4 9 , 6 4 2 1 3 2 7 7 8 2 5 ( ) ^ 2 5 0 C Z 6 C Z 7 1 3 2 1 5 4 C Z 8 C Z 1 1 3 8 2 1 7 S*\ / S s*\ C Z 1 0 C Z 1 2 H Z 4 6 9 2 3 8 HZ 5 HZ 6 1 3 2 1 5 4 1 3 8 2 1 7 2 3 8 2 3 8 s** S*< S\ s** K Z 7 H Z 8 H Z 9 H Z 1 0 H Z 1 2 1 5 3 3 2 1 0 3 1 9 5 3 1 5 3 3 2 1 0 3 1 5 3 3 2 1 C 3 1 9 5 C 3 0 6 1 3 7 1 5 4 1 5 1434 9 6 4 1 3 8 197 7 8 6 9 250 49 132 154 133 217 1 69 6 4 2 2 5 0 238 238 76 15 33 2 103 19 50 R e m a r k s 6 4 2 1 3 7 1 5 4 1 3 7 9 6 4 2 5 0 1 5 4 1 3 8 1 3 8 3 3 2 I r-1 I 6 9 1 5 5 0 P u K - p u r e l o d g e p o l e p i n e t y p e ; C M K - l o d g e p o l e p i n e - c o n i f e r m i x e d t y p e ; H M I X - l o d g e p o l e p i n e ^ h a r d w o o d m i x e d ; B z V - F o r e s t I n v e n t o r y Zone 4; o t h e r s y m b o l s a r a d e f i n e d i n A p p e n d i x 3 , 1 . Appendix 3.4. Least squares equations for the I n t e r i o r lodgepole pine inventory data — quantitative v a r i a b l e s S I T E 3 U 8 1 4 . 1 6 u 4 6 6 . 7 7 o 3 3 . 0 6 4 7 1 0 . 0 76<:3S . 0 2 2 2 4 0 . 0 • 2 0 ; 3 0 . 0 2 5 8 4 0 . 0 1 0 3 3 0 6 . 4 5 * 1 0 . 0 4 1 1 0 . 0 0 1 2 ^ 9 0 4 0 s SO 1 2 u 9 0 1 0 * 0 0 1 l i 3 0 6 S 2 S 6 1 S . , 50 2 3 ^ 0 . 0 0 4 7 o C . 0 0 1 6 * 5 3 . 0 4 1 ^ O . C C 9 6 o 0 . 0 0 I 3 . . 6 0 . 0 1 0 1 2 0 . 0 1 3 * 3 0 . 0 7 0 . 0 0 0 0 4 4 s 0 . OC 1 6 / 6 0 . 0 6 0 . 0 0 1 2 5 3 . 0 0 2 4*30.0 S o j O . 00 1 7 : 0 . 03 34-.0. 00 0. 2J 9 9 3 5E * 0 3 0. 2 715 7 S E * C 3 5 So 0 4 3 . 0 . 2 . i 5 7 4 S E + C3 0 . 5-50E4E+ C 8 3 3 J C 9 4 . 0 . 3 J 7 0 2 9 E + 0 7 0 . 2 J 1 1 3 3 E - C 7 0 . 3 : 3 7 6 1 E + 0 3 0 . 9 J 4 7 6 4 E + C 9 AGE 3 7 0 0 3 5 . 2 0 9 4 0 0 . 9 3 1 8 0 . 0 6 7 4 5 5 . 0 5 9 3 3 0 . 0 2 3 4 1 0 . 0 2 1 7 4 0 . 0 2 3 0 3 0 . 0 1 2 9 6 2 3 . 5 0 1 C 5 . 0 4 1 1 5 . 0 0 1 8 6 3 5 . 0 5 7 9 4 0 . 0 1 3 3 4 5 . C 1 0 3 2 0 . 0 . 1 0 4 5 5 . C 5 8 2 1 0 . 0 1 9 2 6 5 . 0 2 6 9 5 . 0 0 7 1 7 0 . 3 0 2 2 7 9 0 . 0 4 1 6 5 . 0 0 1 1 4 2 0 . 0 1 1 6 3 0 . 0 1 4 0 2 0 . 0 2 2 4 7 5 . 0 1 4 5 . 0 0 0 6 5 3 5 . O C 1 S 6 5 0 . 0 5 9 0 0 . 0 0 9 4 5 . 0 0 0 2 7 3 9 3 . 0 3 3 6 5 . 0 0 1 2 7 5 . 0 0 4 5 3 0 . C O 0 . 27 1 5 7 8 E + 0 8 0 . 3 5 7 2 1 2 E + 0 3 7 2 7 6 1 8 . 0 . 2 6 7 7 6 0 E + 0 8 0 . 7 C 7 5 7 4 E - 0 8 3 6 9 9 2 3 . 0 . 3 6 7 3 9 5 E + 0 7 0 . 3 0 5 2 1 7 E - 0 7 0 . 3 8 9 3 0 6 E + 0 8 0 . U 3 6 0 3 E + 1 0 L G G A 8070.78 4538. 90 1961 .69 1570.19 2175. 43 527.380 512.361 -5 6 5 .C67 2774.78 • 1034.58 90.1251 386 . 6 6 5 1243.70 2 9 2 . 5 4 9 2 6 0 . 5 4 6 2 5 6 . 3 3 4 1 3 7 2 . 2 0 4 0 7 . 5 2 6 53 . 5 9 5 9 L 5 2 . 0 5 6 485. 518 93.0037 251 . 8 1 5 2e6.152 2 7 4 . 4 5 3 4 3 2 . 0 6 8 2 . 1 6 1 4 0 135 . 4 7 6 4 4 6 . 2 1 4 1 4 1 . 8 2 3 2 6 . 5 4 1 5 6 2 8 . 1 2 3 1 9 3 . 5 8 1 3 4 . 3 6 7 3 9 9 . 1 3 3 5 5 9 6 0 4 3 . 7 2 7 6 1 8 . 1 5 6 0 3 . 8 57 1 2 7 4 . . 0 . 148309E*07 S C 6 S . 1 5 7 9 1 1 3 . 2 6 1 3 9 1 . 7 8C1567. 0.231700E+03 H T 2 9 4 8 5 0 . 1 6 0 3 4 0 . 7 5 5 5 0 . C 5 8 9 6 0 . 0 7 2 4 1 0 . 0 2 1 0 2 0 . 0 1 3 4 0 0 . 0 2 3 0 6 0 . 0 98 93 0 . 0 . 4 4 1 8 0 . 0 3 9 3 0 . 0 0 1 2 8 2 0 . 0 3 8 3 6 0 . 0 1 1 6 e o. o 9 3 5 C . O 0 1 0 2 8 0 . 0 66 C O O . 0 1 7 0 0 0 . 0 2 4 3 0 . 0 0 4 7 4 0 . 0 0 1 6 6 5 0 . 0 3 8 9 0 . 0 0 9 0 5 0 . 0 0 117", 0 . 0 1 0 3 2 0 . 0 1 9 2 9 0 . 0 8 0 . 0 0 0 0 4 5 0 0 . 0 0 1 6 9 C 0 . 0 5 4 5 0 . 0 0 1 0 1 0 . 0 0 2 2 6 6 0 . 0 7 3 5 0 . 0 0 . 1 4 7 0 . 0 3 3 5 3 0 . 0 0 0 . 2 2 5 7 4 3 E + 0 8 0 . 2 6 7 7 6 0 E + 0 8 5 7 1 2 / 4 . 0 . 2 1 7 9 0 1 E + 0 8 0 . 5 6 8 9 1 8 E - 0 8 31 4 C i 7 . 0 . 2 5 3 6 9 9 E + 0 7 0 . 2 5 0 1 S 4 E - 0 7 0 . 3 1 3 9 2 9 ; + 0 8 0 . 9 2 7 2 5 2 E + 0 9 NT 7 6 3 3 1 2 . 4 1 6 4 2 5 . 1 9 7 2 8 5 . 1 4 9 6 0 2 . 1 8 1 7 3 6 . 4 7 1 9 6 . 3 4 7 6 3 5 . 0 6 3 4 3 6 . 0 2 6 3 2 7 2 . 1 1 2 2 3 4 . 8 7 9 0 . 0 0 3 8 9 1 3 . 0 9 5 2 4 7 . 0 2 5 3 3 3 . 0 2 4 2 1 2 . 0 3 1 3 2 8 . 0 1 8 0 5 6 0 . 4 2 4 4 0 . 0 5 6 C 9 . 0 0 1 1 6 9 6 . 0 4 3 9 6 0 . 0 8 7 9 6 . 0 0 2 3 4 7 3 . 0 2 9 8 2 4 . 0 2 3 4 8 1 . 0 4 7 3 4 0 . 0 2 2 6 . 0 0 0 1 5 1 8 5 . 0 4 2 5 2 9 . 0 1 3 0 6 7 . 0 22 84 . CO 5 4 2 3 1.0 2 2 5 C 4 . 0 2 9 5 5 . 0 0 1 2 0 3 2 . 0 0 . 5 S 5 0 8 4 E + C 8 0 . 7 0 7 9 7 4 E - C 8 0 . 1 4 8 8 C 9 E + C 7 0 . 5 6 3 5 1 8 E + C S 0 . 1 7 6 6 1 7 E * 0 9 9 2 1 7 7 8 . 0 . 7 4 S 9 2 6 E + C 7 0 . 7 2 3 2 6 3 E + 0 7 0 . 9 2 9 3 9 8 E + C 3 0 .271543E+10 SO 4 1 9 7 . 6 0 2 2 0 6 . 9 5 1 1 6 9 . 0 4 8 2 1 . 6 0 8 9 5 9 . 0 6 9 2 9 4 . 9 0 4 2 7 8 . 8 3 5 3 6 6 . 2 0 1 1 3 6 3 . 7 2 7 0 0 . 6 9 5 5 9 . 0 2 5 0 1 7 4 . 1 5 3 4 7 1 . 9 3 0 1 6 2 . 7 1 5 1 3 9 . 9 0 3 1 5 7 . 2 4 6 9 1 7 . 4 9 0 2 7 3 . 6 2 0 3 7 . 1 3 3 0 4 6 . 8 6 5 0 • 2 4 9 . 8 5 4 6 0 . 1 4 3 0 1 3 8 . 9 3 2 1 9 1 . 4 7 0 1 5 7 . 1 6 9 2 9 4 . 4 3 6 0 . 7 5 3 0 0 0 7 6 . 2 8 6 0 2 3 8 . 0 8 5 7 2 . 0 4 6 0 1 7 . 4 8 5 0 2 8 9 . 0 6 2 1 3 2 . 8 3 9 2 1 . 0 8 9 0 5 1 . 0 0 2 0 3 3 0 0 9 4 . 3 6 9 9 2 3 . 8 0 6 8 . 1 5 3 1 4 0 1 7 . 9 2 1 7 7 8 . 5 3 3 2 . 9 6 4 2 3 0 3 . 1 4 0 2 4 6 . 0 • 5 0 9 2 4 6 . 0 . 1 4 9 6 0 9 E + 0 S 0 3 H ' 4 1 0 0 2.8 2 2 5 8 2.8 1 0 5 2 1 . 9 7898.05 ' 1 0 7 8 5 . 1 • 2797.69 2 5 7 7 . 2 3 . 2 9 1 8 . 3 2 13358.8 5 7 1 4 . 1 3 5 1 7 . 9 4 6 1 8 3 3 . 6 2 5 9 8 1 . 6 6 1 5 8 4 . 9 7 . 1 2 9 0 . 3 4 1 2 3 3 . 6 3 9 2 6 0 . 6 9 2 2 4 4 . 5 7 3 1 3 . 7 6 4 6 6 8 . 0 1 0 2 5 4 8 . 8 7 5 1 3 . 1 5 8 1 2 8 6 . 89 1 5 3 8 . 4 1 1 4 5 4 . 4 1 2 4 7 2 . 8 2 i 9 . 0 0 3 0 0 6 9 8 . 3 5 0 2 2 5 4 . 57 6 9 9 . 5 5 8 1 4 6 . 0 7 2 3 1 4 3 . 6 3 9 9 6 . 7 4 2 1 9 0 . 1 7 4 4 6 7 . 2 5 6 0 . 3 0 7 0 2 9 E + 0 7 0 . 3 6 7 3 9 5 E - 0 7 7 9 1 1 8 . 2 0 . 2 5 3 6 9 9 E + 0 7 0. 7 4 8 9 2 6 E-07 4 2 3 0 3 . 1 4 1 0 9 7 6 . 3 2 5 9 2 7 . 0 .417546E+07 0 .121526E+09 B A H T 31392.9 1 6 3 2 6 . 3 9 4 0 3 . 0 8 5 6 5 7 . 9 9 6 5 0 7 . 7 9 2 2 2 4 . 2 1 1 3 7 4 . 7 6 2 4 5 5 . 0 7 1 0 2 9 3 . 9 6 1 6 4 . 0 3 5 2 1 . 7 7 0 1 3 5 1 . 3 8 3 1 4 4 . 9 4 1 2 7 6 . 7 8 8 8 2 . 8 1 0 1 0 7 0 . 8 1 6 9 7 6 . 3 7 2 2 2 9 . 1 3 38 1 . 7 3 0 3 6 4 . 2 6 0 1 7 9 8 . 4 1 4 4 2 . 8 0 0 9 9 1 . 9 5 0 1 3 0 3 . 9 9 1 3 2 1 . 3 0 2 9 3 0 . 9 6 8 . 0 0 0 0 0 5 5 5 . 1 7 0 1 5 6 4 . 4 4 5 0 4 . 6 3 0 7 5 . 2 7 0 0 1 9 9 5 . 7 2 9 5 3 . 9 4 0 1 3 2.040 4 3 1 . 9 5 0 0 . 2 5 1 1 3 3 E + 0 7 0. 3 0 5 2 1 7 E * 0 7 6 1 8 9 1 . 7 0 .250184E+07 0 . 7 2 8 2 6 3 E*07 4 0 2 4 6 . 0 3 2 5 9 2 7 . 3 5 0 3 7 5 . 0 .42277IE+07 0 . 1 2 8 2 9 4 E*09 RA 4 0 3 6 5 0 . 2 1 6 5 4 8 . 1 1 7 6 3 3 . 7 4 4 6 9 . 0 9 2 9 1 5 . 0 2 7 1 2 0 . 0 2 4 3 4 0 . 0 3 1 1 3 6 . 0 1 3 7 2 4 5 . 7 0 3 3 1 . 0 5 8 6 7 . 0 0 1 9 0 9 6 . 0 4 6 3 1 0 . 0 1 5 3 5 3 . 0 1 1 8 2 2 . 0 1 3 7 3 0 . 0 9 3 3 0 1 . 0 2 6 2 0 6 . 0 4 1 2 6 . 0 0 5 2 0 0 . 0 0 2 5 3 2 0 . 0 5 3 3 4 . 0 0 1 3 0 1 8 . 0 1 6 3 7 4 . 0 1 6 9 4 2 . 0 3 2 4 2 5 . 0 1 0 0 . C O O 8 1 2 0 . 0 0 2 0 7 3 5 . 0 6 4 3 3 . 0 0 1 0 3 2 . 0 0 2 7 0 0 2 . 0 1 1 7 5 0 . 0 1 6 4 1 . 0 0 5 7 7 6 . C O 0 . 3 1 S 7 6 1 E + 0 8 0 . 3 3 9 3 0 6 E » 0 8 8 0 1 5 6 7 . 0 . 3 1 3 9 2 9 E + 0 3 0 . 9 2 9 3 9 8 E + C 3 5 0 9 2 4 6 . 0 . 4 1 7 5 4 6 E - 0 7 0 . 4 2 2 7 7 1 E + C 7 0 . 5 2 5 1 2 8 E + 0 3 0 . 1 5 6 0 7 6 E - 1 0 VOL 0 . 1 1 7 7 4 7 E - 0 8 0 . 6 2 0 8 3 3 E + 0 7 0. 3 3 3 4 9 0 E + 0 7 0 . 2 1 3 0 4 4 E - 0 7 0. 2 5 3 5 4 4 E + C 7 79 52 6 4 . 6 9 4 5 5 3 . 91 1 2 5 9 . 0.397 5 4 1 E - 0 7 0 . 2 1 3 4 3 3 E + 0 7 1 7 5 0 4 4 . 4 9 4 1 4 1 . 0. 1 2 4 5 5 R E + 0 7 4 5 4 4 9 2 . 3 3 3 0 9 6 . 4 0 5 5 3 9 . 0 . 2 6 9 5 2 1 £ ^ 0 7 8 0 0 2 3 4 . 13 0 5 1 5 . 1 3 5 1 6 2 . 6 7 5 1 5 4 . 1 4 9 7 3 6 . 3 5 6 3 9 2 . 4 7 3 7 8 4 . 4 9 7 2 4 3 . 0. I 0 3 2 2 1 E + 0 7 2 7 3 7 . 0 0 2 0 2 1 4 5 . 6 1 4 7 0 S . 1 9 5 0 3 6 . 2 7 9 7 6 . 0 7 3 7 9 5 5 . 3 5 2 1 3 = . 4 5 79 2 . 0 15 6 3 3 4 . 0 . 9 3 4 7 6 4 E + 0 9 0 . 1 1 3 6 0 3 E - 1 0 0 . 2 31 7 C O E * Co 0 . 9 27 7 5 2 5 * 0 9 0 . 2 7 1 5 4 3 ? * 1 C 0 . 1 4 9 5 3 5 : * 0 3 0 . 1 2 1 5 7 6 E - 0 9 0 . 1 2 3 2 9 4 E + 0 9 0 . 1 5 6 0 7 6 5 + 10 0 . 4 7 7 6 4 3 E - 1 1 I 00 I Remarks: LAGE, HT, NT. SD, BA, HTBA,' and VOL are defined i n Appendix 3.1. 

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