A STUDY OP FORM AND TAPER OF STEMS, OF • DOUGLAS FIR, WESTERN HEMLOCK, AND WESTERN RED CEDAR ON THE UNIVERSITY RESEARCH FOREST, HANEY, BRITISH COLUMBIA by R. M. NEWNHAM, B. Sc., U n i v e r s i t y o f Wales, 1956. A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF FORESTRY i n the-department o f FORESTRY We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1958. i i ABSTRACT. A knowledge o f the f o r m of t r e e stems, and. o f the manner i n w h i c h such stems v a r y i n t a p e r , are o f importance t o the f o r e s t e r i n the d e t e r m i n a t i o n o f volume and i n the c o n s t r u c t i o n o f volume t a b l e s . A t p r e s e n t t h e r e are two t h e o r i e s r e l a t i n g t o stem-form, namely t h o s e o f Metzger and Gray, and s e v e r a l f o r m u l a e f o r c o n s t r u c t i n g s t e m - p r o f i l e s . The t h e o r y " o f M e t z g e r i s the o l d e s t and most w e l l known. He c l a i m e d t h a t the f o r m o f the f o r e s t t r e e stem i s n o t f o r t u i t o u s b u t depends on c e r t a i n f o r c e s a c t i n g on i t , o f which wind i s the most i m p o r t a n t . M e t z g e r d e s c r i b e d the t r e e stem as a "beam o f u n i f o r m r e s i s t a n c e " w h i c h , a c c o r d i n g t o the laws o f s t a t i c s , i s a c u b i c p a r a b o l o i d . More r e c e n t l y Gray has d i s a g r e e d w i t h t h i s t h e o r y , a r g u i n g t h a t because the base o f the beam i s not f i x e d i n a s o l i d s t r a t u m , as Metzger supposed, the c u b i c p a r a b o l o i d would i n f a c t be t o o s t r o n g and t h e r e f o r e u n economical. He c l a i m e d t h a t the most e c o n o m i c a l stem f o r m i s t h a t o f a q u a d r a t i c p a r a b o l o i d . The q u a d r a t i c p a r a b o l o i d was t e s t e d f o r a number o f t r e e s f r o m th e U n i v e r s i t y R e s e a r c h P.orest and i n each case was f o u n d t o be c l o s e l y c o r r e l a t e d w i t h the a c t u a l stem p r o f i l e , e x c e p t a t the b u t t due t o b u t t s w e l l , and a t the top where the stem r e s e m b l e d a cone. The c u b i c p a r a b o l o i d was f o u n d to g i v e a good f i t i n the l o w e r p a r t o f the stem b u t u n d e r - e s t i m a t e d the d i a m e t e r i n the upper p a r t o f the stem. An A m e r i c a n i n v e s t i g a t o r , C E . Behre, d e s c r i b e d the stem p r o f i l e as a h y p e r b o l a . A l t h o u g h no t h e o r y has been p u b l i s h e d as t o why t h i s s h o u l d be the c a s e , the f o r m u l a d e r i v e d by Behre has g a i n e d wide acceptance i n N o r t h A m e r i c a and many o t h e r i i i parts of the world. The formula, i n theory at le a s t , gives a perfect f i t to a cylinder or a cone and also good f i t s to the major parts of quadratic and cubic paraboloids and i n t e r -mediate forms of stem p r o f i l e . When used i n t h i s study, Behre's formula was found to give a s l i g h t l y better f i t than the quadratic paraboloid. The formula describing the stem p r o f i l e of a quadratic paraboloid, c a l l e d the "taper-line" by Gray, i s of the form 2 D = a - bH, where D i s the diameter of the stem at a height H, "a" i s the regression constant, and "b" i s the regression c o e f f i c i e n t . The regression c o e f f i c i e n t i s an index of the slope of the taper-line, and therefore of taper, and can be used to trace the pattern of taper v a r i a t i o n with various f a c t o r s which are thought to be re l a t e d to taper. By using multiple regression techniques i t i s possible to reduce these factors to one or two. In one such t e s t on ten Douglas f i r trees from a mixed f i r , hemlock, and cedar stand, average age about 65 years, i t was found that age and s i t e index were not important but that diameter at breast height, D^ ,-, and t o t a l height, Ht, were. The f i n a l regression obtained was b = 0.021+2 + 0.99881+0 D^ g 2 / ^ . A si m i l a r regression was obtained f o r "a", the regression constant, and tables of values of "a" and of "b:" were constructed. From these tables i t i s possible to derive the taper-line f o r any tree of known d.b.h. and height. By thi s means i t i s possible to calculate the volume of standing trees. The main conclusions reached i n t h i s thesis are that the form of fo r e s t tree stems i s complex and cannot be ascribed read i l y to e i t h e r of the known theories; the quadratic i v paraboloid gives a s u f f i c i e n t l y good f i t over the main part of the stem (between about 15 and 80 per cent, of the t o t a l height) that i t can be used as a basis f o r studying taper v a r i a t i o n ; and f i n a l l y , that the amount of taper present i n a stem varies d i r e c t l y as the square of diameter at breast height and inversely as the t o t a l height. The amount of taper appears to increase throughout the l i f e of the tree as long as the tree remains i n the dominant crown class. As soon as the tree passes into the codominant class the rate of increase i n taper f a l l s o f f and when the tree passes into the intermediate and suppressed classes the amount of taper decreases with increasing age. The form of a tree grown i n the open, that i s apparently free from competition from surrounding trees, resembles a cone, or a n e i l o i d i n some cases, and not the quadratic paraboloid of a close-grown or fo r e s t tree. I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h C olumbia, I agree t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by t h e Head o f my Department o r by h i s r e p r e s e n t a t i v e . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f F o r e s t r y The U n i v e r s i t y o f B r i t i s h Columbia, Vancouver S, Canada. Date 11th March, 1958. V Acknowledgement s. The writer wishes to acknowledge the help and useful c r i t i c i s m given him by Dr. J. W. Ker, Associate Professor, and other members of the Faculty of Forestry at the University of B r i t i s h Columbia i n the writing of t h i s thesis. Thanks are due also to Dr. J. H. G. Smith and J. Walters, respectively Assistant Professor and Research Forester, Faculty of Forestry, who, with Dr. Ker, supplied the hulk of the data. F i n a l l y the writer wishes to thank the Faculty for the award of two Assistantships which provided the necessary money f o r t h i s work to be ca r r i e d out. V X TABLE OF CONTENTS. Page ABSTRACT i i ACKNOWLEDGEMENTS . . . . . . . . . v TABLE OF CONTENTS v i INTRODUCTION . 1 Source o f Dat a 1 D e f i n i t i o n s 2 Cone k Q u a d r a t i c P a r a b o l o i d k C u b i c P a r a b o l o i d . k N e i l o i d k H y p e r b o l a k Form .5 Taper 5 PART I . P a s t work 7 Form F a c t o r 7 Form Q u o t i e n t ' 8 Form-point 8 M e t z g e r ' s G i r d e r t h e o r y 9 H o j e r ' s and Jonson's f o r m u l a e 11 Behre's f o r m u l a 12 Gray's t a p e r - l i n e t h e o r y . . . . . . 13 PART I I . The form o f f o r e s t t r e e stems on the U n i v e r s i t y R e s e a r c h F o r e s t , Haney . . . . 17 Q u a d r a t i c P a r a b o l o i d . . . . . . . . 17 C u b i c P a r a b o l o i d • • 19 v a. a. Page Behre's f o r m u l a 19 The f o r m o f c l o s e - g r o w n t r e e s . . . . 23 Subnormal d i a m e t e r s . . . . . . 2k The f o r m o f open-grown t r e e s 27 PART I I I . The v a r i a t i o n i n t a p e r o f t r e e stems on the U n i v e r s i t y R e s e a r c h F o r e s t , Haney ., . 3k Close-grown t r e e s . 3k Methods o f s t u d y i n g v a r i a t i o n i n t a p e r . . 3k D i f f e r e n c e i n t a p e r "between s p e c i e s . . 3k D i f f e r e n c e i n t a p e r due t o d i f f e r e n c e i n d e n s i t y o f s t o c k i n g 3& The v a r i a t i o n o f ""b" w i t h d i a m e t e r a t b r e a s t h e i g h t , t o t a l h e i g h t , age and s i t e i n d e x f o r D ouglas f i r 37 The m u l t i p l e r e g r e s s i o n o f "b" on d i a m e t e r a t b r e a s t h e i g h t , t o t a l h e i g h t , age and s i t e i n d e x f o r Douglas f i r 37 The m u l t i p l e r e g r e s s i o n o f "b" on d i a m e t e r a t b r e a s t h e i g h t (D^ ^ ) , t o t a l h e i g h t ( H t ) and D^ ^ f o r Douglas f i r k3 p The r e g r e s s i o n o f "b" on D^ ^ /H^ f o r Western hemlock k5 2 The common r e g r e s s i o n o f "b" on D^ ^ /H^ f o r Douglas f i r and Western hemlock . . . . k6 2 The r e g r e s s i o n o f " a " on D^ ^ /H^ . . . k7 Use o f t a b l e s V I I and V I I I . . . . 53 Example V X 1 1 Page The v a r i a t i o n of "a" and "t>" i n the young cedar-hemlock stands 55 The v a r i a t i o n of taper with age . . . 55 The v a r i a t i o n i n taper among open-grown trees . 6h The v a r i a t i o n of "a" and "t>" with diameter at "breast height . . . CONCLUSIONS 67 BIBLIOGRAPHY 70 ILLUSTRATIONS FIGURE Page 1 Geometrical shapes related to stem-form: (1) n e i l o i d ; (2) cone; (3) hyperbola; (k) quadratic paraboloid; and (5) cubic paraboloid 6 2 Graphs of D^ p^/d^ over L f o r (1) cylinder; (2) quadratic paraboloid; (3) hyperbola of the form d^/D^ ^ = L/0.1+ + 0 . 6L; (1+) cubic paraboloid; and (5) cone . . 11+ 3 a Diameter-height curve f o r dominant Douglas f i r . Age 27 years 18 3b Diameter squared-height curve f o r the. same tree 18 3c Graph of diameter cubed over height . . 20 3d Graph of D^ ^L/d^ over L . . . . 20 U The occurrence of subnormal diameters i n P. S.P. 101 . Age 78 years . . . . 25 5a Stem p r o f i l e s of open-grown and close-grown Douglas f i r 28 5b Stem p r o f i l e s of open-grown and close-grown Western hemlock 29 5e Stem p r o f i l e s of open-grown and close-grown Western red cedar , 3 0 6 Graph of D^ ^l/d-^ over L f o r t y p i c a l open-grown tree. Hemlock, height 3 2 . 9 feet; D.b.h. o.b. 6.1+0 inches 33 FIGURE Page 7a Graph of "b" over diameter at breast height f o r Douglas f i r . . . . 38 7b Graph of "b" over t o t a l height f o r Douglas f i r 39 7c Graph of "b" over age f o r Douglas f i r . UO 7d Graph of "b" over s i t e index f o r Douglas f i r . . . . . . . . k1 8 Graph of "b" over D^ ^2/^t for Douglas f i r and Western hemlock kk 9 Graph of "a" over D^ ^ /H t f o r Douglas f i r and Western hemlock hS> 10 Alinement chart f o r a = 131 . 70lfD #^ ^/Et - 3 k . 8 5 6 . . 51 11 Alinement chart f o r b = 0.0758 + 1.056985D^ ^ / ^ t ' 5 2 2 12 Graph of "a" over D^ ^ /H t f o r Western hemlock and Western red cedar (Age 20-30 years) 56 13 Graph of "b" over D^ 5 2/H" t for Western hemlock and Western red cedar (Age 20-30 Years) 57 12+ Alinement chart f o r a = 69.6033D^ 5 2/H t - 5 .553 . . 60 15 Alinement chart f o r b = 0.01+066 + 1.778311)^ 5 2 / H t . . 61 16 Va r i a t i o n of taper with age; close-grown hemlock. P.S.P. 115 . • • • 63 FIGURE Page 17a Graph o f " a " o v e r d i a m e t e r a t b r e a s t h e i g h t f o r open-grown t r e e s . . . 65 . 17b Graph o f "b" over d i a m e t e r a t b r e a s t h e i g h t f o r open-grown t r e e s . . . 65 x i i TABLES TABLE Page I D i s t r i b u t i o n of sample trees i n P. S. Ps. 101-109, 115 and 118 3 II Observed and expected values of diameter at various heights on the stem obtained by the three methods investigated . . . 22 III Analysis of variance f o r difference i n taper between Douglas f i r and Western hemlock on P.S.P. 108 35 IV Analysis of variance f o r the regression of "b" on diameter at breast height (D^ t o t a l height (H t), age (A), and s i t e index (S) 1+2 V Analysis of variance f o r the regression of "b" on diameter at breast height (D^ ^ ) , t o t a l height (H t) and D^ 5 2 / H t . k5 VI Analysis of variance for difference i n taper between Douglas f i r and Western hemlock 1+6 VTI Values of regression constant "a", f o r 50-80 year Douglas f i r and Western hemlock . 1+9 VIII Values of regression c o e f f i c i e n t "b", f o r 50-80 year Douglas f i r and Western hemlock 50 IX Values of regression constant "a", f o r 20-30 year Y/estern hemlock and Western.red cedar 58 X Values of regression c o e f f i c i e n t "b", for 20-30 year. Western hemlock and Western red cedar 59 0 1 INTRODUCTION, The t i t l e o f t h i s t h e s i s was s u g g e s t e d f o l l o w i n g a s t u d y o f the r e c e n t work o f H. R. Gray (1956) on the f o r m and t a p e r o f f o r e s t t r e e stems. I n t h i s paper Gray d e a l t w i t h the v a r i o u s t h e o r i e s t h a t have "been put f o r w a r d t o d e s c r i b e the f o r m o f f o r e s t t r e e stems, such as M e t z g e r ' s g i r d e r t h e o r y , and a l s o the v a r i o u s terms, such as f o r m q u o t i e n t and f o r m -p o i n t , t h a t have been u s e d t o g i v e a n u m e r i c a l v a l u e t o form. He expounded h i s own t h e o r y w h i c h d e s c r i b e d the main p a r t o f the stem as r e s e m b l i n g a q u a d r a t i c p a r a b o l o i d and gave many examples t o s u p p o r t h i s t h e o r y . F u r t h e r examples were g i v e n t o show the v a r i a t i o n o f t a p e r w i t h age, d i a m e t e r a t b r e a s t h e i g h t , and crown c l a s s . I f , as Gray supposed, the m i d d l e p a r t o f the stem does resemble a q u a d r a t i c p a r a b o l o i d , t h e n the e q u a t i o n d e s c r i b i n g p the t a p e r l i n e I s D = a - bH, where D i s the d i a m e t e r a t any p o i n t on the main p a r t o f the stem, H i s the h e i g h t o f the p o i n t o f measurement o f D, " a " i s the r e g r e s s i o n c o n s t a n t , and "b" the r e g r e s s i o n c o e f f i c i e n t . The f i r s t o b j e c t o f t h i s t h e s i s has been t o d e s c r i b e the f o r m o f t r e e s g rowing on the U n i v e r s i t y R e s e a r c h F o r e s t a t Haney and t o t e s t the goodness o f f i t o f the above f o r m u l a t o a s e l e c t i o n o f these t r e e s t a k e n f r o m as wide a range o f c o n d i t i o n s as p o s s i b l e . Source o f d a t a . The U n i v e r s i t y R e s e a r c h F o r e s t i s a b l o c k o f n e a r l y 10,000 a c r e s o f f o r e s t l o c a t e d a t the s o u t h e r n end o f P i t t Lake a t a d i s t a n c e o f a p p r o x i m a t e l y 28 m i l e s f r o m the mouth o f the F r a s e r R i v e r . C l i m a t i c a l l y , the f o r e s t i s i n the i n n e r c o a s t a l r e g i o n 2 ( B r i t i s h Columbia Natural Resources Conference, 1956) which i s characterised hy high l e v e l s of p r e c i p i t a t i o n and a small range of annual temperature. The predominant tree species are Douglas f i r (Pseudotsuga menziesii (Mirh.) Franco), western hemlock. (Tsuga heterophylla (Raf.) Sarg.), and western red cedar (Thu.ja p l i c a t a Donn). These three species have been considered i n t h i s study. The bulk of the data were obtained from two se r i e s of permanent sample plots (P. S.Ps). P. S.Ps 101-109 are located i n mixed stands of f i r , hemlock,, and cedar, where the f i r usually forms the main part of the upper crown canopy. P.S.Ps 115 and 118 are located i n young stands predominantly of hemlock and cedar, with occasional dominant f i r trees present. Both these sets of data have the disadvantage that they were obtained from trees removed during thinning operations and are therefore not necessarily representative of the plots. The data are summarised i n table I. The remaining data were obtained from 26 trees taken from the cut-over land on the eastern side of the Forest. The cut-over i s an extensive area which was logged i n the ' 2 0 s and e a r l y ' 3 0 s . Most of the reproduction which came i n following logging was destroyed by a severe f i r e i n July, 1931 so that a l l the trees taken from t h i s area were under 30 years of age. Over much of the area regeneration i s scattered, the trees open-grown, and com-pe t i t i o n , apart from ground vegetation, i s at a minimum. Consequently a l l the trees have vigorous crowns which extend down to the bases of the trees. D e f i n i t i o n s . In order that the reader may understand some of the TABLE I: D i s t r i b u t i o n of sample trees i n P.S.Ps. 101-109, 115 and 118 P.S.P 'No. Average ' Age i n Site index (feet at Stocking Normality (corrected) F i r Hemlock Crown Class^ Cedar Total years 100 years) D C I S D C I S D C I S 101 78 145 72.8% 7 1 1 - - 2 k 1 - - - - 16 102 73 100 131.1+% - 10 3 13 103 Control 10k 61 111 105.0% 3 7 3 - - 3 - — — 16 105 60 110 111.6% 1 10 k 15 106 62 9k 11+2.5% 1 11 1 - 2 2 17 107 Co] ntrol -108 55 160 53.7% k 1 - - k 6 15 109 59 112 72.5% k 7 1 - - 2 3 17 115 26 1 170 1 _2 1 - - — 2 1 2 1 - - 2 1 10 118 26 1 1801 _2 - - - - - 52 21 73 Total 21 1+7 81 13 - 8 16 8 7 9 2 - - 2 21+ 1 192 1 Figures only approximate 2 Not known 3 D = dominant; C = codominant; I = intermediate; S = suppressed. 1+ technical terms used i n thi s thesis, they are defined and i l l u s t r a t e d below. Cone: a geometric s o l i d tapering to a point, i n which the diameter at any point (D) i s d i r e c t l y proportional to i t s distance above the base (H). The equation describing the p r o f i l e of a cone i s : D = a - bH Quadratic paraboloid: a geometric s o l i d i n which the square of the diameter at any point i s d i r e c t l y proportional to i t s distance above the base. The equation describing the p r o f i l e , c a l l e d the "taper-line" by Gray (1956), i s : D 2 = a - bH Cubic paraboloid: a geometric s o l i d i n which the cube of the diameter at any point i s d i r e c t l y proportional to i t s distance above the base. The equation describing the p r o f i l e i s : D 3 = a - bH Ne i l o i d : a geometric s o l i d i n which the square root (or cube root) of the diameter at any point i s d i r e c t l y propor-t i o n a l to i t s distance above the base. The equation describing the p r o f i l e i s : D 2 = a - bH Hyperbola: a curve produced when a cone i s cut by a plane making a larger angle with the base than the side makes with the base. A formula commonly used to describe hyperbolic taper curves i s that of C. E. Behre (Spurr, 1952): D, c c + CL 5 where cL^  i s the diameter at a r e l a t i v e distance L from the t i p to the diameter at "breast height, ^, the distance L i s expressed as a percentage of t o t a l height above breast height, and "c" and "C" are constants. Fig. 1. i l l u s t r a t e s the f i v e "shapes" which may be associated with stem-form of forest trees. Form: the shape of a s o l i d , the p r o f i l e of which i s determined by the power index of D. Taper: the rate of narrowing i n diameter i n r e l a t i o n to increase i n height. Except for the hyperbolic curve, taper i s expressed by the regression c o e f f i c i e n t , "b" i n each of the above formulae. A low value of "b" indicates a slender stem with gentle taper; a high value of "b" indicates, a stout stem with rapid taper. 7 PART I. Past work. A knowledge of the form and taper of tree stems i s of great importance i n the construction of taper and volume tables. Diameter at breast height and t o t a l height can be measured from the ground with reasonable accuracy on the standing tree but i t i s impossible, without e i t h e r climbing or f e l l i n g the tree, to obtain an estimate of the form and taper of the tree which are required i f the volume i s to be computed accurately. I f a knowledge of the v a r i a t i o n of form and taper under d i f f e r e n t conditions could be obtained i t would be possible to estimate the form of any standing tree, providing that the variables, e,.;g. , height and d.b.h. which a f f e c t form and taper are known for that p a r t i c u l a r tree. The relationship between taper and the variables aff e c t i n g i t can be obtained from sample trees. Form factor. The e a r l i e s t method of expressing the form of a tree was by the use of form factors. The form factor of a tree i s the amount by which the volume of a cylinder of i d e n t i c a l basal cross-sectional area and height, has to be m u l t i p l i e d to give the actual volume of the tree. I t may be calculated from the formula: f = v b.h. where " f " i s the form factor, "v" i s the actual volume of the tree, "b" i s the basal cross-sectional area (taken at breast height), and "h" i s the t o t a l height of the tree. This method of expressing form has the disadvantage that 8 i t cannot be measured d i r e c t l y on the standing tree and has been larg e l y superseded by the form quotient method i n North America, the B r i t i s h Commonwealth, and i n Scandinavia (Spurr, 1952). Form quotient. The form quotient was evolved i n Germany i n 1891. I t was defined as the r a t i o between the diameter at h a l f the height of the tree and the diameter at breast height. A major d i f f i c u l t y with a form quotient measured i n t h i s way was that the upper diameter was measured at a variable height and the lower diameter at a f i x e d height. This caused the form quotient to vary with the s i z e of the tree. This d i f f i c u l t y was moderated by Jonson i n 1910 when he redefined form quotient as the r a t i o between the diameter at h a l f the height above breast height and diameter at breast height. I t i s i n t e r e s t i n g to note that, from t h i s d e f i n i t i o n , the form quotient f o r a quadratic paraboloid i s 0.711. Therefore, i f i t i s assumed that the stem form of a tree closely resembles such a paraboloid, the form quotient should not vary much from 0.71.1. That t h i s i s often not the case i s due mainly to butt-swell a f f e c t i n g the breast height measurement. Form-point. In 1912, Jonson evolved the form-point method of describing form. The form-point i s that point i n the crown at which the wind pressure can be considered to be centred. I t corresponds generally with the centre of gravity of the crown as seen from the side (Jerram, 1939). I t s p o s i t i o n i s determined by v i s u a l estimate and i t s height measured with a hypsometer. Basing h i s argument upon a theory o r i g i n a l l y ' proposed by Metzger, Jonson claimed that the r a t i o of the height of the form-point to the 9 t o t a l h e i g h t p r o v i d e d an i n d e x o f stem-form. T h i s method has the o b v i o u s d i s a d v a n t a g e t h a t i t i s i m p o s s i b l e t o f i x the p o s i t i o n o f the f o r m p o i n t a c c u r a t e l y . M e t z g e r ' s g i r d e r t h e o r y . M e t z g e r , i n 1893> s u g g e s t e d t h a t the f o r m o f t r e e stems was not f o r t u i t o u s b u t r e s u l t e d f r o m c e r t a i n f a c t o r s i n the environment w h i c h c o m p e l l e d the t r e e t o c o n s t r u c t a b o l e i n w h i c h t h e r e were d e f i n i t e d i a m e t e r r e l a t i o n s h i p s i n d i f f e r e n t p a r t s o f i t s l e n g t h . A c c o r d i n g t o M e t z g e r the p r i n c i p l e o f wood f o r m a t i o n i n the stem was governed by the r e q u i r e m e n t o f . the t r e e f o r s t r e n g t h (Busgen and Munch, 1929). The main f o r c e s a c t i n g on the stem are the l a t e r a l p r e s s u r e o f the wind on the crown and the w e i g h t o f the crown., and upper p a r t o f the stem. A c c o r d i n g t o Hohenadl ( S p u r r , 1952) i t was the l a t t e r f o r c e which e x e r t e d most i n f l u e n c e on the f o r m and t a p e r o f the stem whereas Metzger b e l i e v e d t h a t wind p r e s s u r e was the prime f a c t o r . The p r e s s u r e o f the w i n d a c t i n g on the crown o f the t r e e i s conveyed t o the l o w e r p a r t o f the stem and i s i n c r e a s e d by l e v e r a g e . The h i g h e r the t r e e , and c o n s e q u e n t l y the l o n g e r the arm o f the l e v e r , the g r e a t e r i s t h i s p r e s s u r e on the stem. I t r e q u i r e s a r e i n f o r c e m e n t o f the stem i n c r e a s i n g downwards i n p r o p o r t i o n t o the h e i g h t o f the t r e e and the exposure o f the crown t o the wind. T r e e s growing i n s h e l t e r e d l o c a l i t i e s or. under crowded c o n d i t i o n s w i t h s m a l l crowns, are more s l e n d e r t h a n i s o l a t e d o r exposed t r e e s as t h e y are more weakly con-s t r u c t e d i n the lower p a r t o f the stem. The w e i g h t o f t h e crown r e n d e r s n e c e s s a r y r e s i s t a n c e i n the stem t o b r e a k a g e which t h r e a t e n s whenever the crown i s bent out o f the e q u i l i b r i u m . 10 The f o r e s t t r e e a l s o r e q u i r e s a c e r t a i n r i g i d i t y t o p r e v e n t s h a k i n g and b e n d i n g o f the s h o o t s , which has been shown by c e r t a i n German and S c a n d i n a v i a n i n v e s t i g a t o r s t o be p r e j u d i c i a l t o good growth. A c c o r d i n g t o M e t z g e r , however, the f u l f i l l i n g o f these demands i s opposed t o the t r e e ' s endeavour t o expand i t s crown and r o o t system as much as p o s s i b l e and t o produce seed, because s t r e n g t h e n i n g o f the stem r e q u i r e s b u i l d i n g m a t e r i a l s which must be t a k e n f r o m the crown, r o o t s , and f r u i t s . To s a t i s f y t hese r e q u i r e m e n t s , the stem must have the h i g h e s t p o s s i b l e b e n d i n g s t r e n g t h w i t h the most e c o n o m i c a l e x p e n d i t u r e o f m a t e r i a l . To do t h i s , the stem must be a "beam o f u n i f o r m r e s i s t a n c e " . I f a beam i s f a s t e n e d a t one end and a f o r c e a p p l i e d a t the o t h e r i t b r e a k s most r e a d i l y a t the f i x e d end. To economise i n m a t e r i a l , the beam s h o u l d not be t h e same t h i c k n e s s t h r o u g h -out i t s l e n g t h but s h o u l d decrease i n t h i c k n e s s ( t a p e r ) f r o m the f i x e d end t o the o t h e r . Near the p o i n t o f a p p l i c a t i o n o f the f o r c e much t h i n n e r p a r t s o f the beam w i l l s u s t a i n i t t h a n towards the f i x e d end, where the f o r c e s a c t w i t h g r e a t e r l e v e r -age. A beam, whose t a p e r i s so a d j u s t e d t h a t i t has a u n i f o r m r e s i s t a n c e t h r o u g h o u t i t s e n t i r e l e n g t h t o a f o r c e a p p l i e d a t i t s end, i s a beam o f u n i f o r m r e s i s t a n c e h a v i n g the f o r m o f a c u b i c p a r a b o l o i d . The p a r t o f the stem w i t h i n a crown r e s e m b l e s a cone as the a r e a o f a t t a c k o f the wind f a l l s o f f w i t h the d i m i n u t i o n o f the crown. The s w e l l i n g a t the base o f the t r e e stem ( b u t t s w e l l ) i s a l s o due t o m e c h a n i c a l r e a s o n s c o n n e c t e d w i t h the t r a n s m i s s i o n o f s t r e s s e s f r o m the v e r t i c a l stem to the h o r i -z o n t a l l a t e r a l r o o t s . B u t t s w e l l may e x t e n d q u i t e f a r up t h e 11 stem, e s p e c i a l l y i n i s o l a t e d t r e e s , and. i n extreme c a s e s may cause the stem to he n e i l o i d a l i n p r o f i l e . M e t z g e r ' s h y p o t h e s i s t h a t the stem o f a f o r e s t t r e e resembles a c u b i c p a r a b o l o i d has had c o n s i d e r a b l e s u p p o r t i n Germany. Any d e v i a t i o n s f r o m the h y p o t h e s i s were a t t r i b u t e d t o v a r i a t i o n s i n the s t r e n g t h o f the stem t i m b e r . A l t h o u g h no f u r t h e r t h e o r i e s r e g a r d i n g stem f o r m were p u t f o r w a r d u n t i l q u i t e r e c e n t l y ( G r a y , 195&) , s e v e r a l a t t e m p t s were made to f i n d s t e m - p r o f i l e e q u a t i o n s t h a t w o u l d g i v e b e t t e r f i t s t h a n t h e s i m p l e c u b i c p a r a b o l o i d . Most o f the work i n t h i s f i e l d was c a r r i e d o u t i n the S c a n d i n a v i a n c o u n t r i e s a t the b e g i n n i n g o f the p r e s e n t c e n t u r y . Ho.jer's and Jonson's f o r m u l a e . tt I n 1903 A. J . H o j e r , w o r k i n g w i t h Norway s p r u c e , d e v e l o p e d the f o r m u l a : d r c + L h . = c l o g — ; — D 4 . 5 where d^ i s the d i a m e t e r a t a d i s t a n c e L f r o m the t i p , J D ^ ^  i s the d i a m e t e r a t b r e a s t h e i g h t , and C and c are c o n s t a n t s v a r y i n g w i t h the f o r m q u o t i e n t s o f the t r e e . T h i s f o r m u l a was used i n the c a l c u l a t i o n o f t a p e r and volume t a b l e s . J o n s o n , tt however, i n comparing the v a l u e s g i v e n by H o j e r ' s f o r m u l a w i t h the a c t u a l t a p e r o f S c o t s p i n e , noted a c o n s i d e r a b l e f a l l i n g o f f i n the t o p s . T h i s he at t e m p t e d t o c o r r e c t by i n t r o d u c i n g a " b i o l o g i c a l c o n s t a n t " i n t o H o j e r ' s f o r m u l a so t h a t i t became: dr c + L - 2 . 5 X j — = C l o g — 4 . 5 where 2 . 5 i s the b i o l o g i c a l c o n s t a n t . "I d Many attemp t s were made i n Europe t o e x p r e s s the f o r m o f t r e e s "by the e q u a t i o n o f some known s o l i d o f r e v o l u t i o n such as the p a r a b o l o i d , cone, o r n e i l o i d . These attempts f a i l e d , a p p a r e n t l y because the i n v e s t i g a t o r s d i d not r e a l i z e t h a t the stem c o n s i s t e d not o f one g e o m e t r i c a l f o r m b u t o f p a r t s o f two o r t h r e e such forms. O t h e r workers e v o l v e d complex l o g a r i t h m i c e q u a t i o n s which a l t h o u g h t h e y gave a good f i t t o the stem p r o -f i l e , were too c o m p l i c a t e d t o be o f much p r a c t i c a l use. Behre* s f o r m u l a . Behre (1923) w o r k i n g w i t h w e s t e r n y e l l o w p i n e , f i r s t e l i m i n a t e d b u t t s w e l l f rom the r e a d i n g s by g r a p h i c a l means. it He t h e n f o u n d t h a t f o r the l o w e r c l a s s e s the H o j e r f o r m u l a gave a good f i t b u t i n the h i g h e r f o r m c l a s s e s t h e r e was a c o n s i d e r a b l e f a l l i n g o f f i n the upper d i a m e t e r s (as n o t e d by J o n s o n ) . Behre's v a l u e f o r the b i o l o g i c a l c o n s t a n t was about 7 . 5 . Behre t h e n p r o c e e d e d t o e v o l v e h i s own f o r m u l a w h i c h d e s c r i b e d a h y p e r b o l a : _ - L / = L \ . 5 " C + C L tt where the symbols are the same as those u s e d f o r H o j e r ' s f o r m u l a e x c e p t t h a t L i s e x p r e s s e d as a perc e n t a g e o f the t o t a l h e i g h t above b r e a s t h e i g h t and the c o n s t a n t s c and C t a k e d i f f e r e n t v a l u e s . T h i s f o r m u l a has g a i n e d wide acceptance i n N o r t h A m e r i c a , G r e a t B r i t a i n and many o t h e r p a r t s o f the w o r l d . T h i s may be due i n p a r t t o the f a c t t h a t the f o r m u l a , b e s i d e s g i v i n g a p e r f e c t f i t t o e i t h e r a cone o r a c y l i n d e r , a l s o g i v e s a good f i t t o the major p a r t o f q u a d r a t i c and c u b i c p a r a b o l o i d s and i n t e r m e d i a t e forms when i s p l o t t e d o v er L, as c a n be seen 13 f r o m f i g . 2. P r o v i d i n g b u t t s w e l l i s e l i m i n a t e d f r o m the d i a m e t e r measurements, the sum of the two c o n s t a n t s s h o u l d e q u a l a p p r o x i m a t e l y one. Tourney and K o r s t i a n (1947) s t a t e t h a t " t h e t y p i c a l f o r m o f b o l e i s e x h i b i t e d when t r e e s grow i n w e l l - s t o c k e d s t a n d s . The b o l e i s more o r l e s s c i r c u l a r i n c r o s s - s e c t i o n and i n f o r m approaches t h a t o f a n e i l o i d " . . T h i s i s not i n a c c o r d w i t h M e t z g e r ' s t h e o r y and one would not e x p e c t t o f i n d t r e e s o f such f o r m u n l e s s t h e y were i s o l a t e d . Gray's t a p e r - l i n e t h e o r y . The most r e c e n t work on the f o r m and t a p e r o f f o r e s t t r e e stems was t h a t o f Gray (1956). I t was b ased m a i n l y on h i s e x p e r i e n c e w i t h e x o t i c c o n i f e r s and e u c a l y p t s i n A u s t r a l i a b u t a l s o i n c l u d e d a number o f the P a c i f i c Northwest s p e c i e s grown i n p l a n t a t i o n s i n G r e a t B r i t a i n . Gray agreed g e n e r a l l y w i t h M e t z g e r ' s t h e o r y t h a t the r e s i s t a n c e t o wind was the d o m i n a t i n g f a c t o r i n f l u e n c i n g the shape o f a f o r e s t t r e e stem. However, he thought t h a t M e t z g e r ' s d e d u c t i o n t h a t the main stem s h o u l d t h e r e f o r e conform t o the d i m e n s i o n s o f a t r u n c a t e d c u b i c p a r a b o l o i d was r a t h e r an a r t i -f i c i a l one, because o n l y i f the t r e e were embedded i n a m a t e r i a l s u f f i c i e n t l y s t r o n g t o e nsure t h a t the attachment a t the base would have h e l d a g a i n s t f o r c e s g r e a t e r t h a n those n e c e s s a r y t o br e a k the stem, would i t have r e q u i r e d s u c h d i m e n s i o n s t o o f f e r u n i f o r m r e s i s t a n c e to l a t e r a l p r e s s u r e s c e n t r e d on the crown. As a t r e e had i t s r o o t s embedded i n a r e l a t i v e l y weak m a t e r i a l i t appeared t h a t Metzger's stem was u n n e c e s s a r i l y s t r o n g and i n e f f i c i e n t . Gray c l a i m e d t h a t the q u a d r a t i c p a r a b o l o i d , w i t h 15 about 20 per cent, l e s s volume than the cubic p a r a b o l o i d , would s t i l l have been, c o n s i s t a n t w i t h the mechanical requirements of the t r e e . He submitted the f o l l o w i n g hypothesis, d e r i v e d mathematically w i t h the a i d of G. Odgers: "The mechanical s t r e s s averaged over the whole s e c t i o n underbark i s constant along the l e n g t h of the main stem, and i f t h i s i s c i r c u l a r , the area of the s e c t i o n i s p r o p o r t i o n a l to the s t r e s s on i t " . Gray showed with numerous examples from A u s t r a l i a and Great B r i t a i n t h a t although graphs of D^ over H gave good f i t s t o the p stem p r o f i l e , graphs of D over H gave even b e t t e r f i t s and over a greater p o r t i o n of the stem. According to Gray, r a p i d taper was a s s o c i a t e d with t r e e s which had exposed crowns. Thus dominant tr e e s tapered r a p i d l y and dominated t r e e s slowly. Dominant trees tapered more r a p i d l y as the tree grew older but i f the tree was changing from a dominant to a dominated crown c l a s s , the taper remained appro-ximately constant or even decreased w i t h i n c r e a s i n g age. Although l i t t l e work other than that of Gray has been c a r r i e d out w i t h the ;aim of showing th a t the form of f o r e s t tree stems i s that of a quadratic p a r a b o l o i d , many f o r e s t men-s u r a t i o n i s t s assume that t h i s i s so. For instance Bruce and Schumacher (191+2) r e f e r to "the c h a r a c t e r i s t i c (quadratic) p a r a b o l o i d a l shape of the centre p o r t i o n of the b o l e . " Chapman and Demeritt ( 1 9 3 6 ) , and Chapman and Meyer ( 1 9 U 9 ) , support t h i s theory and also c l a i m that the very t i p of the stem above the conic p o r t i o n i s p a r a b o l o i d a l as w e l l . I t . should a l s o be remembered t h a t the Huber and Smalian formulae, two of the formulae most commonly used f o r c a l c u l a t i n g the 16 volume of logs, are "based on the assumption that the stem i s the frustrum of a quadratic paraboloid, although a t h i r d , Newton1 s formula, gives the volume of a cone, n e i l o i d or cubic paraboloid as well as that of a quadratic paraboloid. The work of Duff and Nolan (1953) i s also of i n t e r e s t although they were more concerned with the pattern of growth i n the tree stem than with the actual form of the stem. Sections of the stem were taken at the middle of each internode and the width of each annual r i n g recorded to the nearest 1 / 1 0 0 t h . of an inch. By taking such fi n e measurements the writers were able to obtain d e f i n i t e patterns of r a d i a l growth, both h o r i z o n t a l l y and v e r t i c a l l y . I t can be seen from t h i s summary of the past work that the v a r i a t i o n of form and taper i s a complex subject and that -no hypothesis has been suggested which s a t i s f i e s a l l the conditions encountered. There i s obviously s u f f i c i e n t v a r i a t i o n i n stem form i n d i f f e r e n t parts of the world to lend support to both of the theories and the formulae outlined above. The object of the present thesis i s to study the more important of these theories, p a r t i c u l a r l y that of Gray, and t h e i r a p p l i c a t i o n to data from the University Research Forest at Haney. 17 PART I I The f o r m o f f o r e s t t r e e stems on the U n i v e r s i t y R e s e a r c h F o r e s t , Haney. The stem-form o f a t y p i c a l c l o s e - g r o w n f o r e s t t r e e i s shown i n f i g . 3a. I n t h i s t h e s i s , d i a m e t e r , o r d i a m e t e r s q u a r e d , w i l l always he p l o t t e d o v er h e i g h t i n s t e a d o f v i c e v e r s a as h e i g h t w i l l always he the independent, o r x - v a r i a b l e , and d i a m e t e r the dependent o r y - v a r i a b l e . To o b t a i n a m e n t a l p i c t u r e o f the f o r m o f the t r e e , the r e a d e r must v i s u a l i z e the t r e e as l y i n g on i t s s i d e w i t h i t s b u t t t o the l e f t o f the f i g u r e . I t c a n be se e n f r o m t h i s f i g u r e t h a t t h e b u t t s e c t i o n t o a h e i g h t of about f i v e f e e t r e s e m b l e s the f r u s t r u m o f a n e i l o i d and the top, f r o m about k5 f e e t , resembles a cone. The main p a r t o f the stem i s a c u r v e , the f o r m o f whi c h cannot be d e t e r m i n e d by eye. Q u a d r a t i c p a r a b o l o i d . F i g . 3b. shows how t h i s c u r v e i s t r a n s f o r m e d t o a s t r a i g h t l i n e by p l o t t i n g d i a m e t e r s q u a r e d , i n s t e a d o f d i a m e t e r , o v e r h e i g h t . The e q u a t i o n d e s c r i b i n g t h i s l i n e , t he t a p e r - l i n e , i s p D = a - bH and the c o n s t a n t s " a " and "b" c a n be d e t e r m i n e d by the method o f l e a s t s q u a r e s . I n t h i s case the v a l u e s o f " a " and "b" are 29.16 and 0.610 r e s p e c t i v e l y . The r e g r e s s i o n c o -e f f i c i e n t , "b", i s a u s e f u l i n d e x o f t a p e r as i t f i x e s t h e s l o p e o f the t a p e r - l i n e . The r e g r e s s i o n c o n s t a n t , " a " , f i x e s the p o s i t i o n o f the t a p e r l i n e i n r e l a t i o n t o the h e i g h t a x i s . H a v i n g o b t a i n e d a t a p e r - l i n e i t i s n e c e s s a r y t o o b t a i n some measure o f the goodness o f f i t o f the l i n e t o the b a s i c d a t a . The method commonly u s e d i s t o c a l c u l a t e the c o r r e l a t i o n c o -e f f i c i e n t , " r " . The v a l u e o f " r " i n t h i s example i s -0.98k 19 which f o r eight degrees of freedom i s s i g n i f i c a n t at the 0.1 per cent, l e v e l . The minus sign indicates negative c o r r e l a t i o n , that i s , the value of the y-variahle decreases with increase i n the value of the x-variable. I t can he seen that f o r t h i s p a r t i c u l a r tree the.quadratic paraboloid gives a good f i t to the main part of the stem. Cubic paraboloid. I f Dr i s p l o t t e d over height as i n f i g . 3 c , i t can be seen that the points tend to l i e i n a curve but i n the opposite d i r e c t i o n to the curve i n f i g . 3 a . However, f o r a portion of t h i s curve, from a height of 5 feet to a height of about 30 feet, the points do appear to l i e close to a s t r a i g h t l i n e and i t i s possible to calculate the "cubic" taper l i n e f o r t h i s section. The equation f o r the l i n e i s D = 153.31 - 3.999H (r = - 0 . 9 9 8 with h degrees of freedom). It can be seen that the "cubic" and "quadratic" taper l i n e s l i e close together to a height of 30 feet. Behre* s formula. The f i n a l curve f i t t e d to t h i s tree i s that of Behre: D, L 5 c+CL which may also be written: Di+.5.L - = c+CL P X O 1 i n which form ^U. 5 . L p l o t t e d over L should give a s t r a i g h t l i n e as i n f i g . 3 d . I t i s i n t h i s form also that the constants, "c" and "C", can be calculated by the method of least squares. For the tree 21 u s e d i n the p r e v i o u s examples, th e Behre f o r m u l a i s : L D 'k.5 0.1+78+0.533L The c o r r e l a t i o n c o e f f i c i e n t i s +0.999 w h i c h i n d i c a t e s t h a t f o r t h i s t r e e , the Behre c u r v e g i v e s a s l i g h t l y b e t t e r f i t t h a n the q u a d r a t i c p a r a b o l o i d . The d a t a on which the above t a p e r c u r v e s a r e b a s e d are shown i n t a b l e I I . Average e r r o r s and s t a n d a r d e r r o r s o f the e s t i m a t e have been c a l c u l a t e d f o r p o i n t s on the stem between 5 and 1+5 f e e t f o r the q u a d r a t i c p a r a b o l o i d and Behre's c u r v e , and between 5 and 30 f e e t f o r t h e c u b i c p a r a b o l o i d . The s t a n d a r d e r r o r of the e s t i m a t e , S , i s c a l c u l a t e d f r o m the f o l l o w i n g f o r m u l a : Prom f i g s . 3a-d and t a b l e I I i t c a n be se e n t h a t f o r t h i s example: (a) the base o f the stem resembles a n e i l o i d and t h a t none o f the c a l c u l a t e d c u r v e s g i v e a good f i t i n t h a t r e g i o n ; (b) the main p a r t o f the stem i s b e s t f i t t e d by a h y p e r -b o l i c curve c a l c u l a t e d by Behre's f o r m u l a a l t h o u g h a q u a d r a t i c p a r a b o l o i d g i v e s an almost e q u a l l y good f i t . The c u b i c p a r a b o l o i d o n l y f i t s the p o r t i o n between 5 and 30 f e e t and e v e n t h e r e not as w e l l as the o t h e r c u r v e s ; ( c ) the top o f the stem approximates a cone. The q u a d r a t i c and c u b i c p a r a b o l o i d s c a l c u l a t e d f o r the main p a r t o f the stem g i v e u n d e r e s t i m a t e s o f the d i a m e t e r i n t h i s r e g i o n b u t Behre's cur v e g i v e s a r e a s o n a b l y good f i t . TABLE I I : Observed and expected values of diameter at various heights on the stem obtained hy the three methods investigated. Height (feet) Observed Diameter ins. Quadratic Parabola Cubic Parabola Behre' s Hyperbola Expected Diameter Difference(2 - 3 ) Expected Diameter 'Difference (2-6) Expected Diameter Difference (2-9) ( D (2) ins. (3) ins. (U) per cent (5) ins. (6) ins. (7) per cent (8) ins. (9) ins. (10) per cent (11) 0.5 6 . 9 7 5.37 +1.60 +29.80 5.33 +1.64 +30.77 5.33 +I.64 +30.77 1.25 6.30 5.33 +0.97 +18.20 5.29 +1.01 +19.09 5.30 +1.00 +18 .87 . 2.5 5.59 5.27 +0.32 +6.O7 5.23 +0^6 + 6.88 5.24 +0.35 +6.68 5.0 5.15 5.11 +0.04 +0.78 5.11 +0.04 + O.78 5.11 +0.04 +0. 78 10.0 4.85 4.80 +0.05 +1.04 4.84 +0.01 + 0.21 4.84 +0.01 +0.24 15.0 4.44 4.47 -0.03 -O.67 4.54 -0.10 -2.20 4.52 -0.08 -1.77 20.0 4.14 4.12 +0.02 +0.49 4.18 -0.04 - O.96 4.16 -0.02 -0.48 25.0 3.78 3.73 +0.05 +1.34 3.76 +0.02 + 0.53 3.76 +0.02 +0.53 30.0 3.30 3.30 0 0 3.22 +0.08 + 2.48 3.28 +0.02 +0. 61 35.0 2.78 2.80 -0.02 -0.71 2.37 +0.41 +.27.4-3 2.73 +0.05 +1.83 40.0 2.02 2.18 -0.16 -7-34 -1.97 +3-99 2.08 -0.06 -2.88 1+2.5 1.65 1.80 -0.15 -8.33 -2. 60 +if. 25 1.70 -0.05 -2.94 45.0 1.34 1.30 +0.04 +3.08 -3.03 +4.37 1.29 +0.05 +3.88 U7.5 0.86 . 0.43 + 0 ^ +100.00 -3-35 +4..11 0.83 +0.03 +3. 61 50.0 0. 24 -1.15 -3.63 0.31 -0.07 -22.58 Average Error 0.056 2.378 0.048 1.193 0.032 0.735 Standard Error of the Estimate ±0.085 ±4.145 ±0.112 ±1.792 ±0.050 ±2.237 Differences underlined were not used i n the ca l c u l a t i o n of the average and standard errors. 23 Having outlined the main aspects of form and described the method of determining form f o r the main part of the stem f o r one tree, i t i s possible to proceed to a more detailed des-c r i p t i o n of the v a r i a t i o n i n form among the sample trees from the two close-grown stands and from the open, cut-over land on the University Research Forest. The trees from the cut-over are considered separately owing to the completely d i f f e r e n t conditions of growth encountered. The form of close-grown trees. The method used was to obtain the quadratic paraboloidal 2 taper-line f o r each tree by c a l c u l a t i n g the regression of D on 2 H. Where the plotted points of D on H showed a marked devia-t i o n from t h i s taper-line, or the c o r r e l a t i o n c o e f f i c i e n t , " r " , was not s i g n i f i c a n t , attempts were made to f i t one of the other curves to the data to obtain a better correlation. I t should be noted that, as f o r the example used above, taper-lines were obtained only f o r the part of the stem above butt-swell and below the conic top. Taper-lines were determined i n t h i s way f o r 73 trees. For 60 trees the c o r r e l a t i o n c o e f f i c i e n t was s i g n i f i c a n t at the 0.1 per cent, l e v e l , and f o r the remaining 1 3 , " r " was s i g n i f i -cant at the 1 per cent, l e v e l . For most of the trees that were s i g n i f i c a n t only at the 1 per cent, l e v e l there were only two degrees of freedom ( i . e . , four measurements) and i t seems probable that had more measurements been taken, most of these also would have been s i g n i f i c a n t at the 0 .1 per cent, l e v e l . Behre's curves were calculated f o r two trees which showed apparently marked deviations from the parabolic curve over the main part of the stem. The standard error of the Behre curve was, i n both cases, about one-quarter of an inch and that f o r the parabolic curve about one-twentieth of an inch greater. Subnormal diameters. Although the quadratic paraboloid appeared to s a t i s f y the form of the tree stem very well there were departures from t h i s taper-line which, although not s i g n i f i c a n t , were too regular to be ignored. Towards the base of the tree there quite often occurred what Gray (1956) described as "subnormal" diameters, that i s , the observed diameters appeared to be le s s than those expected from the calculated taper-line. These subnormal diameters occurred commonly i n the suppressed and intermediate crown classes, occasionally i n the codominant, but r a r e l y i n the dominant crown classes. This i s shown f o r a s e l e c t i o n of trees from P.S.P.101 i n f i g . 4. I t i s d i f f i c u l t to show c l e a r l y a l l the trees i n the plot i n t h i s figure and i t should be r e a l i s e d that there are departures from t h i s pattern. However, so regular was the occurrence of these "subnormal" diameters i n the lower crown classes that i t i s probable they are i n f a c t "normal". There are two possible explanations f o r these deviations from the normal taper-line. Gray has shown fo r Alpine Ash (Eucalyptus gigantea) growing i n A u s t r a l i a that there i s a close rel a t i o n s h i p between subnormality and e c c e n t r i c i t y of section and that symmetrically grown specimens do not have subnormal diameters. Unfortunately i t i s impossible to conform or refute t h i s with the data available as the majority of the diameters were obtained using a diameter tape and not from the mean of two diameters at r i g h t angles to each other. A l t e r n a t i v e l y , another hypothesis i s submitted. According to Gray's theory the mechanical stress i s constant 26 a l o n g the l e n g t h o f the main stem and, i f t h i s i s c i r c u l a r i n c r o s s - s e c t i o n , the a r e a o f the s e c t i o n i s p r o p o r t i o n a l t o the s t r e s s on i t . T h i s assumes t h a t the d e n s i t y o f the wood, i s c o n s t a n t t h r o u g h o u t the stem. I t i s known, however, t h a t g e n e r a l l y the h e a v i e s t wood, i s found, a t the "base o f the t r e e and t h a t t h e r e i s a g r a d u a l d e c r e a s e i n d e n s i t y w i t h i n c r e a s e i n h e i g h t (Desch, 1953). I t i s a l s o known t h a t , f o r s o f t w o o d s , wood d e n s i t y i n c r e a s e s f r o m p i t h t o cambium e x c e p t i n v e r y o l d t r e e s when t h e r e i s a g r a d u a l d e c l i n e near the cambium ( S p u r r and Wen-Yeu H e s i u n g , 195k). D e n s i t y i s u s u a l l y a c c e p t e d as the b e s t , s i n g l e i n d i c a t i o n o f s t r e n g t h so t h a t i t c a n he see n t h a t the s t r o n g e s t wood i s found towards the cambium a t the base o f the t r e e . Thus the s e c t i o n s f r o m the base o f the t r e e c a n have a s m a l l e r c r o s s - s e c t i o n a l a r e a t h a n e x p e c t e d and y e t s t i l l be i n agreement w i t h the t a p e r - l i n e t h e o r y . T h i s may a l s o account f o r the f a c t t h a t the h y p e r b o l i c c u r v e c a l c u l a t e d f r o m Behre's formu-l a u s u a l l y g i v e s a s l i g h t l y b e t t e r f i t t h a n the q u a d r a t i c p a r a b o l o i d o r t a p e r - l i n e . The f a c t t h a t subnormal d i a m e t e r s are not found v e r y f r e q u e n t l y i n dominant t r e e s i s p r o b a b l y due t o the e x t e n s i o n o f b u t t - s w e l l up the stem w h i c h masks any p o s s i b l e s u b n o r m a l i t y . B u t t - s w e l l i s more pronounced i n dominant t r e e s t h a n i n t r e e s n o t h a v i n g t h e i r crowns i n the canopy due t o the g r e a t e r wind p r e s s u r e e x e r t e d on the crowns. I n c o n c l u s i o n i t can be s e e n f r o m the stem p r o f i l e s o f the 73 t r e e s s t u d i e d t h a t the f o r m o f c l o s e - g r o w n t r e e s remains f a i r l y c o n s t a n t . The b u t t s e c t i o n i s n e i l o i d a l i n p r o f i l e and the top o f the t r e e r e s e m b l e s a cone.- The main s e c t i o n o f the stem resembles a q u a d r a t i c p a r a b o l a . The e x t e n t o f these s e c t i o n s 27 depends m a i n l y on the crown c l a s s . I n dominant t r e e s the cone and n e i l o i d s e c t i o n s u s u a l l y e x t e n d f a r t h e r toward the m i d d l e o f the stem t h a n i n dominated t r e e s . I t i s d i f f i c u l t t o g i v e a c c u r a t e e s t i m a t e s o f the h e i g h t s a t which the n e i l o i d a l and the cone-shaped s e c t i o n s change t o the c h a r a c t e r i s t i c p a r a h o l o i d a l p r o f i l e o f the main p a r t o f the stem. The n e i l o i d a l p o r t i o n u s u a l l y e x t e n d s t o about 15 p e r c e n t , o f the t o t a l h e i g h t i n dominant t r e e s , d r o p p i n g t o about 10 per c e n t , i n s u p p r e s s e d and i n t e r m e d i a t e t r e e s . The cone-shaped top u s u a l l y s t a r t s a t about 80 p e r c e n t , o f the t o t a l h e i g h t . The c e n t r a l s e c t i o n o f the stem c a n g e n e r a l l y be b e s t f i t t e d by a h y p e r b o l i c curve c a l c u l a t e d by Behre's f o r m u l a . The q u a d r a t i c p a r a b o l o i d g i v e s almost an e q u a l l y good f i t and, as i t i s e a s i e r t o det e r m i n e and t o u n d e r s t a n d , i s p o s s i b l y o f most p r a c t i c a l use. I t a l s o has the advantage t h a t the r e g r e s s i o n c o e f f i c i e n t , b, i s an e x c e l l e n t i n d i c a t i o n o f t a p e r . The f o r m o f open-grown t r e e s . The f o r m o f the open-grown t r e e s f r o m the c u t - o v e r l a n d on the U n i v e r s i t y R esearch F o r e s t i s i l l u s t r a t e d i n f i g s . 5 a-c. The p r o f i l e s o f the open-grown stems are shown by the s o l i d l i n e s w hich can be compared w i t h the p r o f i l e s o f some of the t r e e s i n P.S.Ps. 115 and 118, w h i c h are shown by b r o k e n l i n e s . I t c a n be seen f r o m t h e s e f i g u r e s t h a t the main p a r t o f the stem r e s e m b l e s a cone and t h a t the top and b u t t s e c t i o n s are n e i l o i d a l i n shape. I n some o f the s m a l l e r t r e e s , p a r t i c u l a r l y f i r , the c o n i c a l p a r t i s n o t p r e s e n t , g i v i n g the stem the c h a r a c t e r i s t i c n e i l o i d a l shape d e s c r i b e d by Tourney and K o r s t i a n (1947). B u t t - s w e l l does n ot appear as- marked as i t i s i n c l o s e - g r o w n ^ f^trtt4rH:q;•ife ;^tC•^ u^^^^ dF+GE i x r t : ' 31 t r e e s , p r o b a b l y because of the r a p i d t a p e r o f the r e s t of the stem. There i s no e v i d e n c e o f any p a r a b o l o i d a l r e g i o n i n the stem, d o u b t l e s s due"to the f a c t t h a t the l i v e crown ext e n d s p r a c t i c a l l y t o the base o f the t r e e and annual r a d i a l growth r e m a i n s f a i r l y c o n s t a n t a l o n g the l e n g t h o f the stem. A l t h o u g h i t would appear t h a t the c o n i c a l stem f o r m o f these t r e e s does not s u p p o r t Gray's h y p o t h e s i s , i t c a n be shown t h a t t h i s i s not so. Gray's h y p o t h e s i s s t a t e s t h a t "the m e c h a n i c a l s t r e s s i s c o n s t a n t a l o n g the l e n g t h o f the main stem," and t h a t "the a r e a o f the s e c t i o n i s p r o p o r t i o n a l t o the s t r e s s on i t . " T h i s e x p l a i n s the p a r a b o l o i d a l f o r m o f f o r e s t t r e e stems demonstrated e a r l i e r . However, t h i s a p p l i e s o n l y t o c l o s e - g r o w n t r e e s where the l a t e r a l p r e s s u r e can be c o n s i d e r e d t o be concen-t r a t e d a t one p a r t i c u l a r p o i n t i n the crown and o n l y t o the main p a r t o f the stem which g e n e r a l l y l i e s b elow t h i s p o i n t . When a t r e e grows i n the open and i s not p r o t e c t e d f r o m the w i n d by s u r r o u n d i n g t r e e s , the l a t e r a l p r e s s u r e i s s p r e a d out o v e r the e n t i r e h e i g h t o f the t r e e . I f i t i s assumed, as Gray c l a i m s , t h a t the s e c t i o n a l a r e a i s p r o p o r t i o n a l to t h e s t r e s s on the s e c t i o n i t can be seen t h a t the f o r m o f the stem would resemble a cone due t o the e x t r a s t r e s s e s a p p l i e d t o the l o w e r p a r t o f the stem. The stem o f an open-grown t r e e t h e r e f o r e c l o s e l y r e s e m b l e s the p a r t of the stem w i t h i n the crown o f a c l o s e - g r o w n t r e e . To t e s t the h y p o t h e s i s t h a t the stem f o r m o f open-grown t r e e s resembles a cone, 13 t r e e s were s e l e c t e d a t random f r o m the 26 sample t r e e s a v a i l a b l e and t a p e r - l i n e s o f the f o r m D = a - bH, c a l c u l a t e d f o r each tree." The method o f measurement o f these t r e e s 32 d i f f e r e d from that normally adopted i n stem analyses. Instead of diameters "being taken at f i x e d heights as i s normal, diameters were taken at mid-nodal points, thus eliminating the p o s s i b i l i t y of nodal swelling a f f e c t i n g the measurement. I t was found that f o r each of the 13 trees used, the c o r r e l a t i o n c o e f f i c i e n t was s i g n i f i c a n t at the 0.1 per cent, l e v e l , showing that the stem-form of open-grown trees does resemble a cone. It can he seen from f i g s . 5 a-c. that the stem p r o f i l e s of open-grown trees do not resemble e i t h e r the quadratic or cubic forms of paraboloid. I t i s s t i l l possible however, that a good f i t to the stem p r o f i l e can be obtained using Behre's formula (see f i g . 2 . ) . Behre's curves were calculated for three trees chosen at random and i n no case was " r " s i g n i f i c a n t at the 5 per cent, l e v e l . This was undoubtedly due to butt-swell extending to above breast height and thus a f f e c t i n g the d.b.h. measurement. When ^k. 5** i s p l o t t e d over L the points tend to l i e -on a dr curve ( f i g . 6) instead of a straight l i n e as shown i n f i g . 2. Prom t h i s study of the form of open-grown trees i t can be seen that f o r p r a c t i c a l purposes, the stem closely resembles a cone above butt-swell and below the top. Behre's formula would only give a good f i t where the diameter at breast height i s not affected by butt-swell. 34 PART I I I The v a r i a t i o n i n t a p e r o f t r e e stems on the U n i v e r s i t y R e s e a r c h F o r e s t , Haney. C l o s e - g r o w n t r e e s . I t has "been e s t a b l i s h e d i n P a r t I I t h a t the f o r m o f the c e n t r a l s e c t i o n o f the stems o f f o r e s t , o r c l o s e - g r o w n , t r e e s i s e s s e n t i a l l y t h a t o f a q u a d r a t i c p a r a b o l o i d and t h a t the e q u a t i o n d e s c r i b i n g the stem p r o f i l e o r t a p e r - l i n e o f t h i s p a r t 2 o f the t r e e i s o f the fo r m D = a - bH, where D i s the d i a m e t e r a t a h e i g h t , H, and " a " and w b " are the r e g r e s s i o n c o n s t a n t and c o e f f i c i e n t r e s p e c t i v e l y . The r e g r e s s i o n c o e f f i c i e n t , "b", i s an i n d i c a t i o n o f the s l o p e o f the t a p e r - l i n e , and t h e r e f o r e o f t a p e r , and c a n be u s e d t o s t u d y the v a r i a t i o n o f t a p e r under d i f f e r e n t c o n d i t i o n s . A h i g h v a l u e o f "b" i n d i c a t e s a stem h a v i n g r a p i d t a p e r and a low v a l u e a stem w i t h g e n t l e t a p e r . Methods o f s t u d y i n g v a r i a t i o n i n t a p e r . There are two methods o f s t u d y i n g v a r i a t i o n o f t a p e r . The f i r s t method i s t o c a l c u l a t e a common t a p e r - l i n e f o r a p a r t i c u l a r group o f t r e e s growing under one s e t o f c o n d i t i o n s and t h e n t o t e s t f o r a s i g n i f i c a n t d i f f e r e n c e between t h i s t a p e r - l i n e and the common t a p e r - l i n e f o r a n o t h e r group o f t r e e s g rowing under d i f -f e r e n t c o n d i t i o n s . F o r t h i s t e s t t o y i e l d any r e l e v a n t i n f o r m a -t i o n i t i s e s s e n t i a l t h a t the two groups d i f f e r i n o n l y one r e s p e c t , f o r example s i t e i n d e x , and t h a t a l l o t h e r f a c t o r s s uch as age, s p e c i e s , and d e n s i t y o f s t o c k i n g , r e m a i n a p p r o x i m a t e l y the same f o r b o t h groups. A s t u d y o f t a b l e I , showing the d i s -t r i b u t i o n o f sample t r e e s , r e v e a l s t h a t s uch t e s t s a re o n l y p o s s i b l e i n one o r two i n s t a n c e s . D i f f e r e n c e i n t a p e r between s p e c i e s . The f i r s t t e s t c a r r i e d o u t 35 was to see i f there was a s i g n i f i c a n t difference between the two species, Douglas f i r and Western hemlock. The two groups chosen were the four dominant f i r trees and the four dominant hemlock trees from P.S.P. 108. The common regression of diameter on height, or taper-line, was calculated f o r each group from the equation: y = a + bx where y = ^ d < 0 , 1 3 ^ 9 x 100% (d.b.h.o.b.)^ and x - ( t o t a l height - height to point of measurement) x t o t a l height For f i r the values of "a" and "b" were - 13.932 and 1.01+9891 respectively and f o r hemlock, -12.706 and 1.162919. An analysis of variance to t e s t f o r a s i g n i f i c a n t difference between "b" values was carried out by the method described by Fisher (192+2+) and by Wishart (1950). The following table shows the r e s u l t of t h i s analysis: TABLE I I I : Analysis of variance f o r difference i n taper between Douglas f i r and Western hemlock on P.S.P. 108. Source of v a r i a t i o n D.f. S.s. M. s. V.r. Removed by regression 1 21,836.130 21,836.130 310.172 Due to difference between values of "b' 1 55.816 55.816 0.793 N S Residual 37 22,601+. 801 70.2+00 Total 39 21+, 1+96. 72+7 Sig n i f i c a n t at the 1 per cent, l e v e l . NS Not s i g n i f i c a n t . 36 I t can "be seen from the analysis of variance that, f o r the populations sampled, the difference i n taper between species was not s i g n i f i c a n t . A close examination of the data f o r the i n d i v i d u a l trees, however, indicated that there was considerable v a r i a t i o n i n taper within each species which may have masked any possible difference between species. Difference i n taper due to difference i n density of stocking. A s i m i l a r test was carried out to determine whether there i s a difference i n taper between trees growing i n d i f f e r e n t densi-t i e s of stocking but under s i m i l a r conditions of s i t e q u a l i t y , age, species, and crown class. Por this t e s t the four codominant f i r trees from each of P. S.Ps. 10k and 109 were used. Density of stocking i n each case was expressed by the stocking normality fo r f i r which was based on basal area and corrected f o r the presence of hemlock i n mixture. The analysis of variance revealed that there was no s i g n i f i c a n t difference i n taper between the two stocking normalities. Prom the experience gained above i t was decided that such tests were of very l i m i t e d value i n studying the v a r i a t i o n of taper i n forest trees. A second method, si m i l a r to that used by Hummel (1955) > was devised. This method, instead of t e s t i n g f o r s i g n i f i c a n t differences between groups of trees, traced the pattern of the v a r i a t i o n i n taper of i n d i v i d u a l trees over the range of conditions being studied. By p l o t t i n g values of "b" f o r i n d i v i d u a l trees over the corresponding values of diameter at breast height, t o t a l height, age, ;or s i t e index, i t i s possible to obtain a pattern of the v a r i a t i o n of taper 37 with these variables either i n d i v i d u a l l y or c o l l e c t i v e l y by multiple regression techniques. The v a r i a t i o n of "b:" with diameter at breast height, t o t a l height, age, and s i t e index f o r Douglas f i r . Figs. 7 a-d show the pattern of the va r i a t i o n of taper with d.b.h., height, age, and s i t e index f o r 10 Douglas-fir trees chosen at random from P. S.Ps. 101-109. I t can be seen that the regression c o e f f i c i e n t , "b", increased with increase of each of the four variables although only the regression of "b" on d.b.h. and on height were s i g n i f i c a n t . The multiple regression of "b" on diameter at breast height, t o t a l  height, age, and s i t e index f o r Douglas f i r . The multiple regression of "b" on diameter at breast (D^ ^ ) , t o t a l height (H^), age (A), and s i t e index (S), was then determined. The r e s u l t i n g regression equation was: b = 1.1591 + 0.279521D^ 5 - O.Ol67i4-7Ht - 0 . 01 902+OA - 0.003888S The multiple c o r r e l a t i o n c o e f f i c i e n t , "R", fo r t h i s regression was - 0.9&825 which fo r 5 degrees of freedom and 5 variables i s s i g n i f i c a n t at the 1 per cent, l e v e l . I t was seen that the i n c l u s i o n of age and s i t e index i n the equation removed, only a small amount of the v a r i a t i o n present so that although the regression was s i g n i f i c a n t , the regression was recalculated omitting the age and s i t e index variables. The equation then became: b = 0.262+1 + 0.25950713^ 5 - 0.018558Ht "R" was - 0.97972 which f o r 7 degrees of freedom and 3 variables was s i g n i f i c a n t at the 1 per cent, l e v e l . 42 F i n a l l y , the r e g r e s s i o n was r e c a l c u l a t e d o m i t t i n g t h e h e i g h t , age, and s i t e i n d e x v a r i a b l e s . The r e s u l t i n g r e g r e s s i o n e q u a t i o n was: b = 0.179551+1)^ 5 - 0.6083 The c o r r e l a t i o n c o e f f i c i e n t , " r " , was e q u a l t o + O.967I w h i c h was s i g n i f i c a n t at the 0.1 p e r ce n t , l e v e l . The a n a l y s i s o f v a r i a n c e t o show which v a r i a b l e s remove a s i g n i f i c a n t amount o f v a r i a t i o n i n the r e g r e s s i o n ( Q u e n o u i l l e , 1950) i s shown i n t a b l e IV. TABLE IV: A n a l y s i s o f v a r i a n c e f o r the r e g r e s s i o n o f "b" on d i a m e t e r a t b r e a s t h e i g h t , (D^ t o t a l h e i g h t ( H ) , age ( A ) , and s i t e i n d e x (S). Source o f V a r i a t i o n D.f. S.s. M. s. V.r. Due t o r e g r e s s i o n o f "b" on 4.5 1 7.560240 7.560240 200.012K K K E x t r a v a r i a t i o n due t o 1 0.259805 0.259805 6.873* E x t r a v a r i a t i o n due t o A and S. 2 0.079186 0.039593 1.047 N S ' T o t a l due t o r e g r e s s i o n <4 7.899231 1.974808 52.245 K K K R e s i d u a l 5 0.188997 0.037799 T o t a l 9 8.088228 S i g n i f i c a n t a t t h e 0.1 p e r c e n t , l e v e l , x S i g n i f i c a n t a t the 5 p e r c e n t , l e v e l . N S Not s i g n i f i c a n t . From the above a n a l y s i s o f v a r i a n c e i t c a n be seen t h a t the m u l t i p l e r e g r e s s i o n o f "b" on d.b.h., t o t a l h e i g h t , age, and s i t e i n d e x , was h i g h l y s i g n i f i c a n t as was shown by the m u l t i p l e 1+3 c o r r e l a t i o n c o e f f i c i e n t . The regression of "h" on d.b.h. was highly s i g n i f i c a n t and the extra v a r i a t i o n removed by including t o t a l height i n the regression was probably s i g n i f i c a n t . The extra v a r i a t i o n removed by including age and s i t e index was not s i g n i f i c a n t . The f i n a l regression should therefore include the variables d.b.h. and t o t a l height. In i t s simplest form the equation would be as that above, namely: b = 0.261+1 + 0.259507D, c - 0.018558H. but the i n c l u s i o n of higher powers of D^ ^  and or the i n c l u s i o n of a combined variable, such as D^ ^  / Ht» m i S k t give a more sa t i s f a c t o r y r e s u l t . The multiple regression of "b." on diameter at breast height  (D^ 5), t o t a l height (H t), and D^ ^ /E^. f o r Douglas f i r . o The regression of "b" on D^ 5 f ° r the 10 Douglas-fir trees used i n the above test i s shown i n f i g . 8 and as the co r r e l a t i o n was good i t was decided to use t h i s combined variable. As there was l i t t l e evidence of c u r v i l i n e a r i t y from f i g s . 7a and 7b, i t was decided that the i n c l u s i o n of quadratic terms was not necessary. The regression equation was: b = 0.2875 - 0.0200900^ 5 - 0.002877H"t + 1.169878D^ ^ 2 / ^ t The multiple c o r r e l a t i o n c o e f f i c i e n t , "E", was equal to + 0.9886 which f o r 6 degrees of freedom and 1+ variables was s i g n i f i c a n t at the 1 per cent, l e v e l . Omitting the height variable, the regression became: b = 0.21+76 - 0.059050D^ 5 + 1.3161+59D^52/Ht. With R equal to + 0.9885 which i s also s i g n i f i c a n t at the 1 per cent, l e v e l . 45 The regression of "b" on D^ ^  /H t alone was h = 0.G242 + 0.998840 D ^ 2 / ^ The f i n a l analysis of variance i s shown i n table V. TABLE V: Analysis of variance f o r the regression of "b" on diameter at breast height (D^ ^) , t o t a l height (H t), and D^ ^  /H t. Source of v a r i a t i o n D.f. S.s. M. s. V.r. Due to regression of "b" on D u # 5 2 / H t 1 7.883586 7.883586 259.704 x s* Extra v a r i a t i o n due to D, c 4. 5 Extra v a r i a t i o n due to 1 1 0.020527 0.001977 0.020527 0.001977 0.676 N S ' 0.065 N S Total due to regression 3 7.906090 2. 635363 86 .815 K S S Residual 6 0.182138 0.030356 Total 9 8.088138 S i g n i f i c a n t at the 0.1 per cent, l e v e l . Not s i g n i f i c a n t . Prom the above analysis of variance i t can be seen that the i n c l u s i o n of the d.b.h. and height variables did not remove a s i g n i f i c a n t amount of extra v a r i a t i o n above that already removed by the regression of "b" on D^ ^  / H t a l o n e * T n e f i n a l form of the regression i s therefore b = 0.0242 + 0.998840 D^ 5 2 / H t  The regression of "b" on D^ ^  /E^ f o r Western hemlock. The above equation describes the v a r i a t i o n of taper, expressed by the regression c o e f f i c i e n t "b", f o r f i r only. A similar equation was obtained for 10 hemlock trees selected at random from P.S.Ps. 101 and 109. The formula obtained f o r 4 6 hemlock was: D = O.O631 + 1.1587"80D^ 5 2 / H t o The common r e g r e s s i o n o f "b" on D^ ,. /H^. f o r Douglas f i r and  Western hemlock. I t can be s e e n f r o m f i g . 8. t h a t the v a l u e s o f "b" f o r hemlock are g e n e r a l l y h i g h e r t h a n t h e c o r r e s p o n d i n g v a l u e s f o r f i r , t h a t i s , t a p e r i n c r e a s e s more r a p i d l y w i t h i n c r e a s e i n d.b.h. and h e i g h t f o r hemlock t h a n f o r f i r . A n a n a l y s i s o f v a r i a n c e was t h e n c a r r i e d out t o see i f t h e r e was a s i g n i f i c a n t d i f f e -rence i n t h e s l o p e s o f the r e g r e s s i o n l i n e s o r whether, i n f a c t , the two s e t s o f d a t a c o u l d be combined and a common r e g r e s s i o n l i n e f o r f i r and hemlock be c a l c u l a t e d . The method o f a n a l y s i s u s e d was t h a t d e s c r i b e d by F i s h e r (1944) and by W i s h a r t (1950) and t o which r e f e r e n c e has been made b e f o r e . The f i n a l a n a l y s i s i s shown i n t a b l e V I . TABLE V I : A n a l y s i s o f v a r i a n c e f o r d i f f e r e n c e i n t a p e r between Douglas f i r and Western hemlock. Source o f v a r i a t i o n D.f. S. s. M. s. V . r . Due t o common r e g r e s s i o n f o r f i r and hemlock 1 13.870911 13.870911 Due t o d i f f e r e n c e between b f . r and b h e m l o c k 1 0.073306 0.073306 3.121 R e s i d u a l 16 0.375797 0.023487 T o t a l 18 14.320014 The v a r i a n c e r a t i o , 3.121, was s i g n i f i c a n t o n l y a t the 10 p e r c e n t , l e v e l (F ^  f o r 1 and 16 d.f. = 3 .05) . Hence i t appears t h a t the d i f f e r e n c e i n t a p e r between f i r 47 and hemlock was o n l y s m a l l and t h a t a common r e g r e s s i o n o f "b" 2 on ^ /H.J. would have been s a t i s f a c t o r y . T h i s r e g r e s s i o n was h = 0.0758 + 1.056985 D ^ 5 2 / H t The r e g r e s s i o n o f " a w on ^ / H t * H a v i n g e s t a b l i s h e d a f o r m u l a f o r c a l c u l a t i n g the r e g r e s s i o n c o e f f i c i e n t "b" a t any g i v e n d.b.h. and t o t a l h e i g h t , i t was t h e n n e c e s s a r y t o e s t a b l i s h a s i m i l a r f o r m u l a f o r the r e g r e s s i o n c o n s t a n t ".a". I t was d e c i d e d t h a t as the r e g r e s s i o n o f " a " on ^ /H^ ( f i g . 9) gave a good f i t , t h i s v a r i a b l e would be u s e d a g a i n . The r e s u l t i n g r e g r e s s i o n e q u a t i o n was: a = l31 .70I fD^ 5 2 /H t - 34.856. I t c o u l d be see n a l s o t h a t i t was u n n e c e s s a r y t o c a l c u l a t e two se p a r a t e r e g r e s s i o n s f o r f i r and f o r hemlock. T a b l e s V I I and V I I I g i v e v a l u e s o f " a " and "b" f o r t r e e s between 60 and 140 f e e t i n h e i g h t and between 6 and 22 i n c h e s d i a m e t e r a t b r e a s t h e i g h t . They a p p l y o n l y t o the mixed f i r -h e mlock-cedar s t a n d s , as e x e m p l i f i e d by P. S.Ps. 101-109, on the U n i v e r s i t y R e s e a r c h F o r e s t a t Haney, b u t they c o u l d a l s o be used f o r s i m i l a r s t a n d s where these o c c u r ; The t a b l e s a l s o o n l y a p p l y t o t h a t p a r t o f the stem between a p p r o x i m a t e l y 15 and 80 p e r ce n t , o f the t o t a l h e i g h t o f the t r e e . I n t e r m e d i a t e v a l u e s o f "a " and "b" not shown i n t h e t a b l e s , may be o b t a i n e d e i t h e r by d i r e c t c a l c u l a t i o n from the b a s i c e q u a t i o n s o r e l s e f r o m the a l i n e m e n t c h a r t s shown i n f i g s . 10 and 11. TABLE V I I : V a l u e s o f r e g r e s s i o n const-ant " a " , f o r 50-80 y e a r Douglas f i r and W e s t e r n hemlock. Diameter a t "breast h e i g h t o.h. ( i n c h e s ) 60 70 80 T o t a l H e i g h t ( f e e t ) 90 100 110 120 • 130 • 140 • B a s i s No. o f t r e e s 6 44.2 32.9 24.4 17.8 3 8 105. 6 85.5 70.5 58.8 49.4 4 10 153.2 129.7 111.4 96.8 84.9 74.8 4 12 202. 2 175.8 154.8 137.5 123.2 111.0 100. 6 2 14 252.0 223. 2 199-8 180 .2 163.7 149.5 2 16 339.7 302. 2 271.7 246.0 224.4 205.9 3 18 391.8 353-0 320.7 293.4 269.9 1 20 443.9 404.0 370.3 341.3 22 496.2 455.-4 420.4 1 B a s i s No. o f t r e e s 1 i 3 2 5 2 3" 1 3 20 a = 131 . 704 D k # 5 2 / H t " 34.856 • TABLE V I I I : Values o f r e g r e s s i o n c o e f f i c i e n t "b", f o r 50-80 y e a r s Douglas f i r and W e s t e r n hemlock. Diameter a t b r e a s t h e i g h t o. "b. ( i n c h e s ) 60 70 80 T o t a l H e i g h t ( f e e t ) 90 100 110 120 130 140 B a s i s No. of t r e e s 6 0.710 0. 620 0.551 0.k99 3 8 1.203 1. 0k2 0. 921 0.827 0.752 4 10 1.837 1.586 1.397 1.250 1.133 1.036 0.956 4 12 1.978 1.766 1.598 1.459 1.345 1.246 1.162 2 1k 2.378 2.11+7 1.959 1.802 1.669 1.555 2 16 3.082 2. 781 2.536 2.331 2.157 2.008 3 18 3.500 3.189 2.936 2. 710 2.521 1 20 3.918 3.598 3.332 3.095 22 4.338 4.009 3.730 1 B a s i s No, of t r e e s 1 3 2 5 2 3 1 3 20 D = 0.0758 + 1.056. 985 D k # 5 2 / H t NOTE: Table v a l u e s a re a p p r o x i m a t e l y 9.4 p e r c e n t , h i g h f o r f i r , 9.4 p e r cent, low f o r hemlock. 51 52 • • 1 EIGHT ——I • / id i to — . 22 / / 21 - / \ \ 65 It -It - , / — ... ^  —--s \ ! • , / \ \ 7 0 l i - / 1-4 \ : 1 l t -M -j / _ / j 7S / r t r / 1 \ / : / / | -2 - \ / - / to j /"; / \ AC : F -1 : -t \ *i \ • - 4 \*:- - 7 > A i \ ; V *** - -4 m / / pi y / / t j 7 35 • -1 — w t i i J 1 2 9 4 LOGARITHMIC SCALE 1 i 4 7 1 FIG. 1 1 : Alinement chart f o r t = 0 . 0 7 5 8 + 1 . 0 5 6 9 8 5 D ^ ^2/E 53 Use o f t a b l e s V I I and V I I I To o b t a i n the volume o f s t a n d i n g t r e e s u s i n g t a b l e s V I I and V I I I the f o l l o w i n g procedure i s adopted: (1) the d i a m e t e r a t b r e a s t h e i g h t and t o t a l h e i g h t o f the s t a n d i n g t r e e are measured; (2) the d i a m e t e r s o u t s i d e b a r k a t stump l e v e l and a t i n t e r v a l s t o as h i g h as can he c o n v e n i e n t l y measured ( a t l e a s t 10 p e r c e n t , o f t o t a l h e i g h t ) are r e c o r d e d . The volume o f t h i s s e c t i o n o f the stem i s computed u s i n g the S m a l i a n f o r m u l a ; (3) from t a b l e s V I I and V I I I the v a l u e s o f " a " and o f "b" are o b t a i n e d f o r the d.b.h. and h e i g h t o b t a i n e d i n ( 1 ) ; (k) the d i a m e t e r s o.t>. a t a p p r o x i m a t e l y 15 p e r cen t . o f and 80 p e r c e n t , o f t o t a l h e i g h t are o b t a i n e d f rom the 2 e q u a t i o n D = a - "bH; (5) the volume of the c e n t r a l s e c t i o n o f the stem i s t h e n computed u s i n g the S m a l i a n f o r m u l a ; (6) the volume o f the top i s computed as a cone. (7) a more r a p i d method, b u t one w h i c h i s l i a b l e t o u n d e r - e s t i m a t e the t r u e volume, i s t o c a l c u l a t e the volume o f the t r e e as a p a r a b o l o i d . The f o r m u l a f o r t h i s volume i s : a 2 V = 0.002728 x § The f o l l o w i n g example i l l u s t r a t e s the use o f these t a b l e s : 52+ Example. P.S.P. 108. Tree No. 281. Species: Hemlock. D.b.h. o.b. =11. Total height =100 feet. D.o.b. at stump l e v e l (1.2 feet) = 13.3 inches D. o.b. at 9.33 feet = 11.0 inches Prom basic equations a = 12+2.3, o = 1.2+98. Therefore D 2 = 12+2.3 - 1.2+98H. Diameter o.b. at 15 feet = ^2+2.3 - (1.498 x 15) = 11.0 inches. Diameter l.b. at 80 feet =^ 12+2.3 - (1.2+98 x 80) = 2+. 7 inches. Volume to 15 feet = A., 2 x 1.2 + ( A i• 2 * 9,35) x g > 1 5 A~ ,,- + A-2 + ( - 9 . 3 5 - "15) x 5 > 6 5 where A. 9 = cross-sectional area at 1.2 feet i n square feet A _ tt tl tt Q 7 C It tt tt tt A 9 . 3 5 " y o 5 etc. Volume to 15 feet = 1.16 + 6.62+3.73 = 11.51 cubic feet. Volume from 15 to 80 feet = A 15 + A80 x 6 5 2 = 25.38 cubic feet. Volume of top = Ag Q x 20 = 0.81 cubic feet. Total volume (outside bark) = 11.51 + 25.38 + 0.81 = 37.70 cubic feet. 55 a 2 Volume computed as a p a r a b o l o i d = 0.002728 x ^  = 0.002728 x 1U2 . 3 2 1.2+98' = 36.88 c u b i c f e e t . A c t u a l volume ( o u t s i d e b a r k ) = 32+.86 c u b i c f e e t . The v a r i a t i o n o f '"a" and "b" i n the young cedar-hemlock s t a n d s . By a s i m i l a r method t o t h a t u s e d f o r the o l d e r mixed f i r -h emlock-cedar s t a n d s d e s c r i b e d e a r l i e r i n t h i s t h e s i s , e q u a t i o n s o f the r e g r e s s i o n o f " a " and "b" on d.b.h. and t o t a l h e i g h t were c a l c u l a t e d f o r P.S.Ps. 115 and 118, w h i c h are l o c a t e d i n young mixed s t a n d s o f p r e d o m i n a n t l y hemlock and cedar. A g a i n i t was found t h a t the combined v a r i a b l e , was most s a t i s f a c t o r y and, as can be see n f r o m f i g s . 12 and 13, i t was p o s s i b l e t o c a l c u l a t e common r e g r e s s i o n l i n e s i n b o t h cases. The r e g r e s s i o n e q u a t i o n s were a = 69.6033 5 /H^ -5-553 and b = 0 . 02+0 66 + 1.77831 j 2 / ^ B o t h r e g r e s s i o n s were s i g n i f i c a n t a t the 0.1 p e r ce n t , l e v e l . I t c an he seen f r o m the two r e g r e s s i o n e q u a t i o n s and f r o m f i g s . 12 and 13» t h a t t a p e r i n c r e a s e d more r a p i d l y w i t h i n c r e a s e i n d.b.h. and h e i g h t i n these young s t a n d s t h a n i n the o l d e r s t a n d s ( f i g s . 7 and 8). A t p r e s e n t i t i s not p o s s i b l e t o g i v e the reas o n s f o r t h i s . T a b l e s IX and X g i v e v a l u e s o f " a " and "b" f o r b o t h hem-l o c k and ceda r i n the young cedar-hemlock s t a n d s on the U n i v e r s i t y R e s e a r c h F o r e s t . F i g s . 12+ and 15 are the c o r r e -s p o n d i n g a l i n e m e n t c h a r t s . The V a r i a t i o n o f t a p e r w i t h age. A l t h o u g h age was i n c l u d e d as one o f the independent v a r i a b l e s i n the m u l t i p l e r e g r e s s i o n d e s c r i b e d e a r l i e r and was 56 p:7^:;-r ^it-far' IIPSF EEEE H-H :t -••.-•-r-j-.HtrfT S i i i i TE^IJEEEEOE-UJ i o ±-EB; s 11 r r f i j i - r t r ' ' " iH • - W t±*. t h #1 EEfttbfi B. £E rrt-Ii-Si iff} i+1 htSj i HEEcb '£ m EE5 Eff ±3 : p t t f f i is i p l p STrLJi W t )4+TLt SiEE! EE "qpt IS IP Irarttt; -EEti m 1 i F E E ^ H i St ffl . -FT HK-c S i t t ^ . tEEFE/'EE^-f^ajS^ I n-h-fri-EPi-' ^1 a "IX I m FSEE HrtxEEi 57 TABLE I X : V a l u e s o f r e g r e s s i o n c o n s t a n t , " a " , f o r 20-30 y e a r W estern hemlock and Western r e d cedar. D i a m e t e r a t b r e a s t h e i g h t o.b. ( i n c h e s ) 20 30 T o t a l h e i g h t ( f e e t ) 40 50 60 70 B a s i s No. o f t r e e s 2 8 . 3 7 3.71 1.U1 2 3 25 .8 1 5 . 3 10.1 6 .98 4 4 3 1 . 6 2 2 . 3 16.7 1 3 . 0 7 5 3 8 . 0 2 9 . 3 2 3 . 4 1 9 . 3 3 6 57.1 4 4 . 5 3 6 . 2 3 0 . 2 1 7 6 2 . 6 5 1 . 2 43.1 2 8 8 3 . 5 68 .6 5 8 . 0 1 B a s i s No. o f t r e e s 1 1 9 6 2 1 20 a = 69.6033 \ . 5 2 / H t - 5.553 TABLE X: V a l u e s o f r e g r e s s i o n c o e f f i c i e n t , "b", f o r 20-30 y e a r Western hemlock and W e s t e r n r e d cedar. D iameter a t b r e a s t h e i g h t o. b. ( i n c h e s ) 20 T o t a l h e i g h t ( f e e t ) 30 40 50 60 70 B a s i s No. o f t r e e s 2 0.397 0.277 0.218 2 3 0.841 0.574 0.441 0.361 4 4 0.989 ( O.762 0. 610 0.515 7 5 1.153 0.930 0.782 0.676 3 6 1.641 1.321 1.108 0.956 1 7 1.784 1.494 1.286 2 8 2.318 1.938 1. 668 1 B a s i s No. o f t r e e s 1 1 9 6 y 2 1 20 b = 0.04066 + 1.77831 D. ^/H 60 N-t E . 4 J : : HEIGHT feet \ . Li r he 1 inches \ \ 2 \ 1 1 25 A— 3 a tS > ^ ! — A i—-V i _ 3 0 c \ — -\ V , — \ * -38 • • 4- i \ r 2 0 Y \ • - / \ \ 1 \ 45 »-h ,— SO / \ ss p 6 0 * 65 i d Mi • 1 1 • i E 1 1 ! i 2 3 4 ! _ 1 J O C A R I T H M I C S C A L E s PIG. 14: Alinement chart f o r a = 69.6033D^ 5 /H t - 5.553 61 r r D.b.KobL 8 ^ \ \ \ • \ b \ HEIGHT feet r - 2 0 P M - 2 5 / L 3 / I t / / / - 3 0 / / / / -1 3 1 _ 0 4 2 I N>8 P / • / i / /  / / y y y y-/ \ 4 0 45 - 50 -60 D i 3 4 S LOGARITHMIC S C A L E \ \ t 6 5 \ H i " T O 6 7 Kd FIG. 15 : Alinement chart f o r b = 0.01+066 + 1.77831L, 2/H 4 . 5 t 62 found to "be non-significant, there was some reason to believe that the v a r i a t i o n of taper with age within a p a r t i c u l a r tree followed a prescribed pattern. Gray (1956) found that f o r a dominant tree the amount of taper increased steadily with age. For a tree i n the lower portion of the codominant crown class or i n the upper portion of the intermediate crown class the amount of taper remained f a i r l y steady, and f o r a suppressed tree the amount of taper usually decreased with increase i n age. The available data f o r studying the v a r i a t i o n of taper with age was somewhat li m i t e d . However i t was possible to use four hemlock trees from P.S.P. 115. The taper-line inside bark was calculated and, from the stem analysis of each tree, taper l i n e s were constructed at five-year i n t e r v a l s throughout the l i f e of the tree. The values of "b", the regression co-e f f i c i e n t , have been plotted over the ages at which they were obtained i n f i g . 16. As an increase i n the value of "b" implies an increase i n the amount of taper, the following conclusions may be drawn: (1) the taper of the dominant tree increased with age, and was s t i l l increasing when i t was f e l l e d ; (2) the taper of the codominant tree increased with 7 age u n t i l 22 years when i t began to decrease, presumably because i t was changing from the dominant to the co-dominant class; (3) the taper of. the intermediate tree increased steadily, but more slowly than the dominant and codominant trees, u n t i l 22 years and then began to decrease as i t passed into the intermediate crown class; iWa:b^;M:liK«^Tn mm . 6k (k) the t a p e r o f the s u p p r e s s e d t r e e , a f t e r a p e r i o d o f q u i t e r a p i d i n c r e a s e f r o m 12 t o 17 y e a r s , d e c r e a s e d f r o m t h e n on as i t p a s s e d through the codominant and i n t e r m e d i a t e crown c l a s s e s t o the s u p p r e s s e d crown c l a s s . A l t h o u g h i t i s i m p o s s i b l e t o draw d e f i n i t e c o n c l u s i o n s > f r om such s p a r s e d a t a , c e r t a i n f a c t s are apparent and worthy o f f u r t h e r study. I t appears t h a t a l l t r e e s have a p e r i o d o f marked i n c r e a s e i n t a p e r . T h i s i n c r e a s e i n t a p e r becomes l e s s marked as age i n c r e a s e s , and e v e n t u a l l y the t a p e r b e g i n s t o d e c r ease as the t r e e becomes dominated by s u r r o u n d i n g t r e e s . The decrease i n the amount o f t a p e r i s f i r s t n o t i c e d i n s u p p r e s s e d t r e e s and l a s t n o t i c e d i n codominant t r e e s . Taper c o n t i n u e s i n c r e a s i n g t h r o u g h o u t the l i f e o f dominant t r e e s . The v a r i a t i o n i n t a p e r among open-grown t r e e s . Prom p a r t I I o f t h i s t h e s i s i t w i l l be remembered t h a t the f o r m o f the open-grown t r e e s from the c u t - o v e r l a n d r e s e m b l e d a cone r a t h e r t h a n a q u a d r a t i c p a r a b o l o i d . The e q u a t i d e s c r i b i n g the stem p r o f i l e o f these t r e e s i s D = a - bH and, as w i t h the p a r a b o l o i d a l t a p e r - l i n e , i t i s p o s s i b l e t o t r a c e the v a r i a t i o n o f the r e g r e s s i o n c o n s t a n t , " a " , and the r e g r e s s i o n c o e f f i c i e n t , "b", w i t h d i a m e t e r o r h e i g h t . The v a r i a t i o n o f " a " and o f "b" w i t h d i a m e t e r a t b r e a s t h e i g h t . The v a r i a t i o n o f " a " and o f "b" w i t h d i a m e t e r a t b r e a s t h e i g h t i s shown i n f i g s . 17a and 17b. As would be e x p e c t e d , " a " i n c r e a s e d w i t h i n c r e a s e i n d.b.h. and f o r these d a t a the r e l a t i o n s h i p was l i n e a r . The v a r i a t i o n o f "b" w i t h d.b.h. was not so c l e a r . P o r hemlock and c e d a r , "b" appeared to i n c r e a s e s l o w l y w i t h i n c r e a s e i n d.b.h. b u t w i t h f i r the v a l u e s ' o f "b" 12 .1' 38 EE 3F! fflliiiiiiiiipis Be ft1 MM1 : ''cc1ia&!:^:Ti :^t T0:»5. • . u. UJ. 4 -66 were too scattered to give a d e f i n i t e pattern. I t appears that there i s greater v a r i a t i o n of taper among open-grown trees than among close-grown trees and. that the pattern of t h i s v a r i a t i o n i s not as well defined. 67 CONCLUSIONS. A knowledge o f the form o f t r e e stems, and o f the manner i n which such stems v a r y i n t a p e r , are o f im p o r t a n c e . i n the d e t e r m i n a t i o n o f volume and i n the c o n s t r u c t i o n o f volume t a b l e s . A t p r e s e n t t h e r e a re two t h e o r i e s r e l a t i n g t o stem form, those o f Metzger and Gray. Metzger c l a i m e d t h a t the f o r m o f the f o r e s t t r e e stem depends on c e r t a i n f o r c e s a c t i n g on i t , o f wh i c h wind i s the most i m p o r t a n t . He d e s c r i b e d the t r e e stem as a "beam o f u n i f o r m r e s i s t a n c e " w h i c h , a c c o r d i n g t o the laws o f s t a t i c s , i s a c u b i c p a r a b o l o i d . More r e c e n t l y , Gray has c l a i m e d t h a t the c u b i c p a r a b o l o i d i s t o o s t r o n g as the base o f the s t e m i s not f i x e d i n a s o l i d s t r a t u m as Metzger supposed. He s u g g e s t s t h a t the q u a d r a t i c p a r a b o l o i d i s the most e c o n o m i c a l stem-form. The q u a d r a t i c p a r a b o l o i d was t e s t e d on 73 t r e e s f rom the U n i v e r s i t y R e s e a r c h F o r e s t . I n each case i t was f o u n d t o be c l o s e l y c o r r e l a t e d w i t h the a c t u a l stem p r o f i l e , e x c e p t a t the b u t t due to b u t t - s w e l l , and a t the top where the stem resembled a cone. The c u b i c p a r a b o l o i d was f o u n d t o g i v e a good f i t i n the l o w e r p a r t o f the stem b u t u n d e r - e s t i m a t e d the d i a m e t e r i n the upper p a r t o f the stem. A f o r m u l a w h i c h was a l s o t e s t e d was t h a t o f C E . Behre which d e s c r i b e s a h y p e r b o l a . When us e d i n t h i s s t u d y , Behre's f o r m u l a was found to g i v e a s l i g h t l y b e t t e r f i t t h a n the q u a d r a t i c p a r a b o l o i d . The f o r m u l a d e s c r i b i n g the stem p r o f i l e o f a q u a d r a t i c 2 p a r a b o l o i d i s o f the form D = a - bH, where D i s the d i a m e t e r o f the stem a t a h e i g h t H, " a " i s the r e g r e s s i o n c o n s t a n t , and 68 "b" the r e g r e s s i o n c o e f f i c i e n t . The r e g r e s s i o n c o e f f i c i e n t i s an i n d e x o f t a p e r and can be u s e d t o t r a c e the p a t t e r n o f t a p e r v a r i a t i o n w i t h v a r i o u s f a c t o r s w h i c h are thought t o be r e l a t e d t o t a p e r . By u s i n g m u l t i p l e r e g r e s s i o n t e c h n i q u e s i t i s p o s s i b l e t o reduce t h e s e f a c t o r s t o one o r two. I n one such t e s t u s i n g 10 Douglas f i r t r e e s f r o m a mixed f i r , hemlock, and c e d a r s t a n d , average age about 65 y e a r s , i t was f o u n d t h a t age and s i t e i n d e x were n o t s i g n i f i c a n t b u t t h a t d i a m e t e r a t b r e a s t h e i g h t , D^ ^, and t o t a l h e i g h t , H^ ., were. The f i n a l r e g r e s s i o n was b = 0.021+2 + 0.99881+0 D^ I t was f o u n d t h a t t h e r e was no s i g n i f i c a n t d i f f e r e n c e between the r e g r e s s i o n o f "b" on 2 2 D^ ^ /H.j. f o r f i r and the r e g r e s s i o n o f "b" on D^ ^ /H^ f o r hemlock, so t h a t a common r e g r e s s i o n f o r b o t h s p e c i e s was o b t a i n e d . A s i m i l a r r e g r e s s i o n was o b t a i n e d f o r " a " , the r e g r e s s i o n con-s t a n t , and t a b l e s o f " a " and o f "b" were c o n s t r u c t e d ( t a b l e s V I I and V I I I ) . From th e s e t a b l e s i t i s p o s s i b l e t o d e r i v e the stem p r o f i l e , o r " t a p e r - l i n e " , f o r any t r e e o f known d.b.h. and t o t a l h e i g h t . By t h i s means i t i s p o s s i b l e t o c a l c u l a t e the volume o f s t a n d i n g t r e e s . S i m i l a r t a b l e s ( t a b l e s I X and X) have been c o n s t r u c t e d f o r W estern hemlock and Western r e d cedar i n the younger mixed s t a n d s o f p r e d o m i n a n t l y hemlock and cedar on the U n i v e r s i t y R e s e a r c h F o r e s t . A b r i e f s t u d y o f the v a r i a t i o n o f t a p e r w i t h the age o f the t r e e i n d i c a t e s t h a t the amount o f t a p e r i n c r e a s e s t h r o u g h o u t the l i f e o f the t r e e as l o n g as the t r e e r e m a i n s i n the dominant crown c l a s s . As soon as the t r e e p a s s e s i n t o the codominant c l a s s the r a t e o f i n c r e a s e i n t a p e r f a l l s o f f and when the t r e e p a sses i n t o the i n t e r m e d i a t e and s u p p r e s s e d c l a s s e s the amount 69 of taper decreases with increasing age. The stem-form of open-grown trees d i f f e r s from that of. close-grown or forest trees. The stem generally has the form of a cone or, i n some cases, a n e i l o i d . 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