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UBC Theses and Dissertations

Composite system for estimating tree taper and merchantable volume Segaran, Sandy 1975

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C O M P O S I T E S Y S T E M FOR E S T I M A T I N G T R E E T A P E R AND MERCHANTABLE VOLUME by SANDY SEGARAN C e r t , i n F o r . Biom, (Oxon), F . I . S . , F.S.S. A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n t h e Department o f F o r e s t r y 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 t h e r e q u i r e d s t a n d a r d . (Dr. A. Kozak - S u p e r v i s o r ) (Dr. D.D. Munro) (Dr. W.G. Warren) THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1975 In presenting t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e for reference and study. I further agree that permission for extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s representatives. I t i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my written permission. 3. SnQOJxavi formerly (K. Sandrasegaran) Department of Forestry, The U n i v e r s i t y of B r i t i s h Columbia, Vancouver, B.C., Canada, V6T 1W5 i i i ABSTRACT A composite system f o r estimating taper and merchantable volume up to 5 inches top diameter i s described f o r p l a n t a t i o n grown trees of Eucalyptus robusta over the D range of 10 to 24 inches and heights of 73 to 122 fe e t . The stem p r o f i l e could be described by the model g = a - bh where g; i s the s e c t i o n a l area i n square feet at s e c t i o n a l height h_ fee t , a_ i s the regression constant and b the regression c o e f f i c i e n t . Relations are derived between the e a s i l y measured tree parameters of diameter at breast height (D 4 5) , diameter at 10 feet high ( D 1 Q ) and t o t a l height (H_) with both a_ and b. As a r e s u l t , the regression c o e f f i c i e n t b, which i s a measure of taper, v a r i e s d i r e c t l y as the square of the diameter at breast height, the product of the diameter at breast height and t o t a l height, as well as, the product of the square of the diameter at breast height and t o t a l height. Various formulae are summarised to derive the standing volumes for desired heights or nominated diameters, and diameters at s p e c i f i c heights or v i c e versa, and c o l l a t e d i n Section 8. The r e l a t i o n s of a_ on H, b on H and form f a c t o r f o r t o t a l volume (Fv by classes, are graphed and inferences are drawn. Bias i n height or diameter estimation i s small, similar f o r each class and not s u b s t a n t i a l . Further the height bias i s i n opposite sign to diameter bias. Bias, i n estimating merchantable volumes to 5 inches top diameter by the taper model, as compared with that of the volume model f o r merchantable volume i s small and n e g l i g i b l e . The o v e r a l l bias f o r the sample trees i s 0.16 cubic fee t , that i s , 0.13 per cent of the t o t a l sample merchantable volume estimated by the taper model. For the range of basic data investigated, t h i s taper model was found to adequately describe the stem p r o f i l e . i v TABLE OF CONTENTS Page TITLE PAGE i RIGHTS OF PUBLICATIONS AND LOAN i i ABSTRACT i i i TABLE OF CONTENTS Iv LIST OF TABLES v i i LIST OF FIGURES v i i i ACKNOWLEDGEMENTS i x SYMBOLS AND ABBREVIATIONS X THE SPECIES x i i l (a) D i s t r i b u t i o n x i i i (b) Introduction to Peninsular. Malaysia . . . . . . . . . . . . . . xiv (c) General Performance i n Malaysia x i v (d) Description of. Sample.Area xv (e) C l i m a t i c Factors x v i (f) Geology and S o i l s x v i CHAPTER 1. Introduction 1 2. An a n a l y t i c a l survey of tree volume properties, transformations and taper 2 2.1 Tree volume properties and t h e i r e f f e c t on the error term of the regression model 2 2.2 Transformation of var i a b l e s 3 2.3 Process of weighting 5 2. 4 Form and taper 7 2.5 C r i t e r i a f o r s e l e c t i o n of regression models 8 V Page 3. L i t e r a t u r e review 10 4.1 Sampling technique . . 15 4.2 Parameters measured . 15 5. The taper l i n e 16 5.1 Derivation 16 5.2 C h a r a c t e r i s t i c properties of the taper l i n e 17 6. Computational d e t a i l s 18 6.1 Subset of s i g n i f i c a n t v a r i a b l e s 18 6.2 a. expressed as a function of tree growth parameters . 19 6.3 b expressed as a function of tree growth parameters . 19 6.4 Percentage u t i l i s a b l e volume 20 6.5 Computation of a, b_ and Fv 20 6.6 The advantages of a l l combination procedure 21 7. Discussion of analysis 21 7.1 a_ and H and b on H by one-inch D^ classes 21 7.2 Form f a c t o r f o r t o t a l volume (F ) by one-inch D. _ v 4.5 classes 22 7.3 Check on assumption 22 7.4 Tests of bias and standard er r o r of estimate of merchantable volume estimated by the volume and taper models up to 5 inches top diameter by diameter s i z e classes 23 7.5 Diameter estimation t e s t s f o r bias and standard error f o r measured heights from 10 to 90 feet . . . . 23 7.6 Bias i n height estimation by one-inch D^ classes . . 24 v i Page 8. Summary of formulae to be used i n p r a c t i c a l a p p l i c a t i o n . . 24 9. Conclusions 26 10. LITERATURE CITED 28 V l l Table I I . LIST OF TABLES Page I. Range of basic data s t r a t i f i e d by one-inch 37 D, classes 4. 5 C l i m a t o l o g i c a l data 38 I I I . Merchantable volume estimation t e s t s of bias and standard error f o r sample trees i n the 10 to 20 D, r inches s i z e classes 39 4.5 IV. Diameter estimation t e s t of bias (AB ) and c standard e r r o r (SEE ) f o r measured s e c t i o n a l c heights from 10-90 feet 40 V. Bias (AB ) i n s e c t i o n a l height estimation 41 c v i i i LIST OF FIGURES F i g u r e Page 1. D e f i n i t i o n o f t h e t a p e r l i n e t o d e s c r i b e t h e stem p r o f i l e 42 2. P l o t o f a on H by o n e - i n c h D„ ,_ c l a s s e s 43 — — 4.5 3. R e l a t i o n o f b and H by o n e - i n c h D^ . c l a s s e s 44 4. R e l a t i o n between form f a c t o r f o r t o t a l volume (Fv) and o n e - i n c h D c l a s s e s 45 4.5 5. S c a t t e r g r a m o f average r e s i d u a l mean square o f t h e number o f independent v a r i a b l e s f o r a_ . . 46 6. S c a t t e r g r a m o f average r e s i d u a l mean square on t h e number o f independent v a r i a b l e s f o r b 47 i x ACKNOWLEDGEMENT A t t h e o u t s e t I must s p e c i a l l y thank t h e t h r e e members o f the committee, Drs. A. Kozak, D.D. Munro and W.G. Warren f o r the m e t i c u l o u s review, u s e f u l a d v i c e , f r u i t f u l comments, c o n s t r u c t i v e s u g g e s t i o n s and g e n e r a l l y i n t e r e s t i n g d i s c u s s i o n s on t h e s u b j e c t - m a t t e r o f t h e t h e s i s . In f a c t , t h e i r c o n t r i b u t i o n a t a l l s t a g e s o f development d e s e r v e s worthy mention. A l s o I must add an e x t r a word o f thanks t o my s u p e r v i s o r , Dr. A. Kozak f o r h i s u n f a i l i n g a s s i s t a n c e a t a l l times and c o n s i s t e n t encouragement d u r i n g t h e t e n u r e o f my s t u d i e s a t t h e U n i v e r s i t y . Indeed i t was a g r e a t s o u r c e o f pleasure,.' i n s p i r a t i o n and c o n f i d e n c e t o work w i t h him, f o r which I ext e n d my s i n c e r e g r a t i t u d e . F u r t h e r a p p r e c i a t i o n must a l s o be a c c o r d e d t o ( i ) D r s . Kozak and Munro f o r o b t a i n i n g f i n a n c i a l s u p p o r t from t h e N a t i o n a l R e s e a r c h C o u n c i l d u r i n g t h e summer o f 1974, ( i i ) t h e F a c u l t y o f F o r e s t r y , U n i v e r s i t y o f B r i t i s h Columbia f o r the T e a c h i n g A s s i s t a n t s h i p d u r i n g t h e S e s s i o n 1974/75 and, ( i i i ) M a c M i l l a n B l o e d e l L i m i t e d f o r ~ i t s F e l l o w s h i p i n F o r e s t M e n s u r a t i o n , a l s o f o r the S e s s i o n 1974/75; a l l o f which were c o l l e c t i v e l y i n s t r u m e n t a l f o r the p u r s u i t o f my s t u d i e s a t t h e U n i v e r s i t y o f B r i t i s h Columbia. F o r t h e d a t a which were c o l l a t e d w h i l e the a u t h o r was i n charge o f t h e F o r e s t M e n s u r a t i o n and F o r e s t S t a t i s t i c s D i v i s i o n s i n P e n i n s u l a r M a l a y s i a u n t i l 22-2-74, g r a t e f u l thanks t o t h e F o r e s t Department o f P e n i n s u l a r M a l a y s i a a r e a l s o r e c o r d e d . X SYMBOLS AND ABBREVIATIONS Symbols o r A b b r e v i a t i o n s Meaning a r e g r e s s i o n c o n s t a n t b r e g r e s s i o n c o e f f i c i e n t C , C , ...C r e g r e s s i o n c o e f f i c i e n t s o 1 n D o r D^ j . d i a m e t e r o v e r b a r k a t b r e a s t h e i g h t i n i n c h e s D d i a m e t e r i n s i d e b a r k a t b r e a s t h e i g h t i n i n c h e s u D^ j _ , D y D 1 ( y D 3Q ' d i a m e t e r o u t s i d e b a r k i n i n c h e s a t 1-1/2, 3, D c o , D_ . Dnn 10, 30, 50, 70 and 90 f e e t h e i g h t . t>u /o yo D^ . d iameter i n i n c h e s a t 6-9 metres h e i g h t 6-9 D, upper stem d i a m e t e r o u t s i d e b a r k i n i n c h e s a t h a p o i n t h g r e a t e r than 90 f e e t above the ground where the n o t i c e a b l e change i n stem t a p e r o c c u r s . D^ d i a m e t e r o u t s i d e b a r k i n i n c h e s a t i _ f e e t h e i g h t d d i a m e t e r i n s i d e b a r k i n i n c h e s a t any g i v e n h e i g h t o r d i s t a n c e from the t i p . d. d i a m e t e r i n s i d e b a r k i n i n c h e s a t i f e e t h e i g h t l — dp ^ d i a m e t e r underbark i n i n c h e s a t 0.1 o f t o t a l h e i g h t . d i a m e t e r i n i n c h e s a t t h e s m a l l end o f t h e f i r s t 16 l o g o f 16 f e e t l e n g t h , pom p o i n t o f measurement bh b r e a s t h e i g h t g s e c t i o n a l a r e a i n square f e e t g^ s e c t i o n a l a r e a i n square f e e t a t jL f e e t h e i g h t g b a s a l a r e a i n square f e e t a t b r e a s t h e i g h t bh o u t s i d e b a r k x i Symbols o r A b b r e v i a t i o n s Meaning G g i r t h o u t s i d e b a r k a t b r e a s t h e i g h t i n i n c h e s H t o t a l h e i g h t i n f e e t hp p a r a b o l i c h e i g h t i n f e e t h s e c t i o n a l h e i g h t i n f e e t h^ s e c t i o n a l h e i g h t a t _ i f e e t above h i g h e s t ground l e v e l . h, . h e i g h t i n f e e t above b r e a s t h e i g h t bn V t o t a l volume i n c u b i c f e e t r V . u t i l i s a b l e volume p e r c e n t a g e a t d i a m e t e r i Dip * V. u t i l i s a b l e volume o u t s i d e b a r k i n c u b i c f e e t a t i 1 — f e e t h e i g h t . u t i l i s a b l e volume p e r c e n t a g e a t h e i g h t x_ f e e t f rom h i g h e s t ground l e v e l VD. volume i n c u b i c f e e t a t d i a m e t e r i 1 V p a r a b o l i c volume i n c u b i c f e e t P V. u t i l i s a b l e volume p e r c e n t a g e i p V volume i n c u b i c f e e t a t h e i g h t 1 f e e t from h i g h e s t ground l e v e l F form f a c t o r f o r t o t a l volume K crown r a t i o r crown l e n g t h B p a r a b o l i c base P V , volume i n c u b i c f e e t above b r e a s t h e i g h t bh L d i s t a n c e i n f e e t from t r e e t i p f f r e q u e n c y o f independent v a r i a b l e s n v a l u e o f 3 .14159 o r C^L 22/7 Kd average crown d i a m e t e r i n f e e t Symbols o r A b b r e v i a t i o n s -ve n e g a t i v e 2 a p o p u l a t i o n v a r i a n c e o f t h e dependent v a r i a b l e l o g l o g a r i t h m t o base t e n l n l o g a r i t h m t o base e where e = 2.71828 SEE s t a n d a r d e r r o r o f e s t i m a t e 2 r c o e f f i c i e n t o f d e t e r m i n a t i o n x i i i The S p e c i e s (a) D i s t r i b u t i o n Eucalyptus rdbusta Sm. (Syn. E. m u l t i f l o r a P o i r . ) i s commonly known as Swamp mahogany and t o g e t h e r w i t h E. saligna, E. gvandis, E. botryoi-d.es and E. .deanei, i s r e f e r r e d as E a s t e r n B l u e Gum. I t s n a t u r a l h a b i t a t i s c o n f i n e d t o a narrow b e l t i n A u s t r a l i a , s t r e t c h i n g from t h e n o r t h o f F r a s e r I s l a n d i n Queensland t o so u t h o f Bega i n New South Wales w i t h a s u b - t r o p i c a l d i s t r i b u t i o n . The a n n u a l v a r i a t i o n i n r a i n f a l l w i t h i n t h i s a r e a i s 40 t o 60 i n c h e s w i t h a w e l l - d e f i n e d summer maximum i n t h e n o r t h e r n range b u t becoming u n i f o r m towards t h e s o u t h ( F o r e s t r y and Timber Bureau, A u s t r a l i a , 1957). The s o i l s , i n p l a n t a t i o n s o f E. vobusta, a r e t y p i c a l l y heavy i n swamp l o c a l i t i e s though i t w i l l grow on l i g h t s o i l s p r o v i d e d t h a t they a r e n o t t o o d r y ( F o r e s t r y and Timber Bureau, A u s t r a l i a , 1953), b u t area s near the c o a s t a r e s u b - s a l i n e ( B l a k e l y , 1955). I t i s a w e l l - t r i e d e x o t i c and has g i v e n s p e c t a c u l a r r e s u l t s i n many c o u n t r i e s ( S t r e e t s , 1962). The r a p i d growth i s due t o i t s p a r t i c u l a r branching and l e a f p r o d u c t i o n habits (Jacob, 1954). In C e y l o n i t i s grown f o r f u e l p r o d u c t i o n and i s grown on d r y Patana g r a s s l a n d a t e l e v a t i o n s o f 3,500 t o 5,000 f e e t (Streets-, 1962) and i n I n d i a i t has grown w e l l a t e l e v a t i o n s o f 4,000 f e e t i n t h e sub-Himalayan t r a c t (Troup, 1921). The s p e c i e s has a l s o been found t o be p r o m i s i n g i n B r a z i l and P u e r t o R i c o (Wadsworth e t - a l . , 1953) and being an accommodating s p e c i e s , w i l l grow on a much w i d e r range o f s o i l s t h a n i s i n d i c a t e d from i t s n a t u r a l d i s t r i b u t i o n ( F o r e s t r y and Timber Bureau, 1953) .' I n T r o p i c a l A s i a , o p t i m a l c o n d i t i o n s i n c l u d e a l t i t u d e s between 1,000 t o 1,500 metres, r a i n f a l l g r e a t e r t h a n 60 i n c h e s p e r annum and s o i l s which a r e m o i s t and heavy (F.A.O., 1957). I t i s a l s o a v e r y e x a c t i n g s p e c i e s t o l i g h t (FAO, 1957). From p u l p i n g t r i a l s i n I n d i a (Guha e t a l . , 1965), i t i s i n f e r r e d t h a t good q u a l i t y p a p e r i s x i v produced by some admixture of a su i t a b l e proportion of long-fibred pulp with that of the s h o r t - f i b r e d pulp of . E. vobusta. (b) Introduction to Peninsular Malaysia Various species of Eucalypts were introduced i n 1876 and planted at the Singapore Botanic Gardens, though the e a r l y growth rates were disappointing due to unfavourable c l i m a t i c conditions (Anderson, 1912). This was followed by pla n t i n g E. citviodova, E. gomphocephala and E. vobusta between 1906-1907 and of the three, only E. vobusta was promising (Anderson, 1912).. In 1924, various species of Eucalypts were planted i n the h i l l s tations i n Malaysia as ornamentals and the e a r l i e s t recorded introduction by the Forest Department was i n 1927 (Anon., 1927). Thereafter, plantations of various Eucalypts were established, at Cameron Highlands with two d i s t i n c t s p e l l s of p l a n t i n g a c t i v i t y ; 1931-34. and 1938-41. From 1952, plantations were i n c r e a s i n g l y established f o r pulpwood and fuelwood. The species' t r i a l s comprised the following, (a) E. vobusta, (b) E. salignat (c) E. gvandis, (d) E. bicostata, (e) E. covymbosa, (f) E. deglupta, (g) E. globulus, (h) E. maculata, (i) E. mellidova, (j) vacemosa3 (k) E. sidevoxylon, (1) E. umbellata3 (m) E. citvioda, (n) E. paneculata3 (o) E. pell-ita, (p) E. vesiniflova and (q) E. tovelViana. Of these, (a), (b) and (c) are promising, (e), ( f ) , (g), (h), ( i ) , ( j ) , (k), (1) are d e f i n i t e f a i l u r e s and (m), (n), (o), (p), (q) are inconclusive (Anon. 1960). A recent assessment of the three most promising species indicates- that E. vobusta has a better growth performance than E. saligna or E. gvandis (Sandrasegaran, 1966i). (c) General Performances i n Malaysia The plantations of E. vobusta were established on cleared montane oak f o r e s t at elevations of 5,000 feet above mean sea- l e v e l , with o r i g i n a l seeds obtained from Queensland, A u s t r a l i a but subsequent plantations were X V from l o c a l s e e ds. V i a b l e seeds are p r o d u c e d a f t e r f o u r y e a r s (Barnard e t , a l . , 195.7) and t h i s i s c o n s i d e r e d r e l a t i v e l y e a r l y when compared t o I n d i a and J a m a i c a where v i a b l e seeds a r e p r o d u c e d a f t e r twenty and e i g h t y e a r s r e s p e c t i v e l y ( S t r e e t , 1962). The c u l m i n a t i o n o f t h e C u r r e n t .Annual Increment i s 500 c u b i c , f e e t p e r a c r e p e r y e a r w i t h an a n n u a l average h e i g h t increment o f n i n e f e e t ( F r e e z a i l l a h e t a l . , 1966). Due t o t h e l i g h t demanding p r o p e r t i e s o f t h e s p e c i e s , c r o p development i s g r e a t l y i n f l u e n c e d by p l a n t i n g s p a c i n g . S t u d i e s on t h e growth o f t h e s p e c i e s a l s o i n d i c a t e t h e c r o p development i s i n f l u e n c e d by an e f f e c t i v e t h i n n i n g regime a t th e o n s e t o f c o m p e t i t i o n ( F r e e z a i l l a h e t a l . , 1966). The crown d i a m e t e r (Kd) stem d i a m e t e r (D) r a t i o o f the s p e c i e s conforms t o a Type Two r e l a t i o n (Dawkins, 1963), t h a t i s , an i n c r e a s e i n stem d i a m e t e r must be f o l l o w e d w i t h a c o n c o m i t a n t i n c r e a s e i n crown s i z e . The Kd;D r a t i o d e c r e a s e s from 18.98 a t 6 i n c h e s D t o 17.80 a t 24 i n c h e s p_, which i n f o r m a t i o n i s u t i l i z e d t o d e t e r m i n e i n t e r - r o w and w i t h i n - r o w p l a n t i n g d i s t a n c e s , as a g u i d e t o w o r k i n g p l a n s and management p r a c t i c e f o r t h e s p e c i e s i n M a l a y s i a (Sandrasegaran, 1 9 6 6 i i ) . (d) D e s c r i p t i o n o f Sample A r e a The a r e a i s s i t u a t e d a t an e l e v a t i o n o f 5,000 f e e t above mean s e a -l e v e l on a s o u t h - f a c i n g , r a t h e r s t e e p s l o p e i n U l u Bertam F o r e s t Reserve, Cameron H i g h l a n d s . I t was p l a n t e d w i t h E. robusta s e e d l i n g s a t a p l a n t i n g s p a c i n g o f 12 x 12 f e e t i n J u l y 1952 and a t time o f measurement i n J u l y 1972 was 20 y e a r s o f c r o p age. The o r i g i n a l f o r e s t was worked f o r f i r e w o o d i n 1951 and thence c l e a r e d and b u r n t i n 1952 p r i o r t o p l a n t i n g . T r e e s a r e o f good form and o f u n i f o r m development w i t h s m a l l and narrow crowns t e n d i n g t o be whippy. The b r a n c h e s a r e l i g h t and s m a l l . The d r a i n a g e i s good. A t time o f measurement f o r t h i s t a p e r s t u d y , t h e undergrowth con-s i s t e d o f h e r b s c o m p r i s i n g Mikania soandens, Impatients spp.3 Begonia spp. xvi w i t h some Zingibevaceae spp., Melastoma spp. and Musa spp. (e) C l i m a t i c F a c t o r s The c l i m a t i c d a t a a r e g i v e n i n T a b l e 2. D i f f e r e n c e s between the c o n d i t i o n s i n t h e sample a r e a and w i t h i n t h e n a t u r a l h a b i t a t a r e ; ( i ) In the sample a r e a , t h e mean annua l r a i n f a l l i s more than 100 i n c h e s w h i l e i n i t s n a t u r a l h a b i t a t , i t v a r i e s from 40 t o 6 0 . i n c h e s , ( i i ) There i s a r e l a t i v e l y d r y s p e l l (June t o August) w i t h two r a i n f a l l peaks (March t o May and September t o December) whereas i n i t s n a t u r a l range, t h e r a i n f a l l o c c u r s i n summer i n t h e n o r t h b u t t e n d i n g t o be u n i f o r m e l s e w h e r e , ( i i i ) The h i g h e s t and l o w e s t • t e m p e r a t u r e s i n t h e sample a r e a a r e 74.3°F and 53.9°F r e s p e c t i v e l y , hence c o m p l e t e l y f r o s t f r e e , w h i l e i n t h e n a t u r a l l y o c c u r r i n g a r e a s , i t i s r e s p e c t i v e l y 85.4°F and 30.5°F. On a c l i m a t i c b a s i s , t h e c o n d i t i o n i s f a v o u r a b l e as E. robusta i s f r o s t t e n d e r . (f) Geology and S o i l s The s o i l s a r e d e r i v e d from s o l i d i n t r u s i v e igneous r o c k s (Dennet, 1930) and b e l o n g t o t h e B u k i t Temiang s e r i e s , a s e r i e s t h a t i s d e r i v e d from g r a n i t e w i t h a c h a r a c t e r i s t i c s h a l l o w p r o f i l e c o m p r i s i n g o f ( i ) decomposing, l o o s e humus w i t h an i n t e n s e r o o t i n g i n t h e 0"-2" l a y e r , ( i i ) humus w i t h some admixture o f m i n e r a l m a t t e r and good r o o t i n g i n t h e 2"-4" l a y e r , ( i i i ) m i n e r a l l a y e r , l i g h t c o l o u r e d i n t h e 4"-48" l a y e r , w i t h i n c r e a s i n g p r o p o r t i o n o f c l a y and s i l t towards bottom o f p i t and weathered g r a n i t e a t about 3 f e e t 6 i n c h e s . There was a l s o a s t e a d y d e c r e a s e o f f i n e and c o a r s e sand, as w e l l as g r a v e l , from t h e t o p t o t h e bottom o f t h e p i t . 1 1. I n t r o d u c t i o n Volume i s the q u a n t i t a t i v e b a s i s o f p r o f e s s i o n a l f o r e s t r y and a p p l i e d f o r e s t r e s e a r c h (Young e t a l . , 1967) and i s the most m e a n i n g f u l i n d e x o f p r o d u c t i v i t y (Greaves, 1971). I t i s t h e b a s i c s t a n d a r d adopted f o r t h e u n i v e r s a l s a l e o f p r i m a r y p r o d u c t s and q u a l i t y t i m b e r . Hence the c o n s t r u c t i o n o f t r e e volume t a b l e s f o r (a) e s t i m a t i n g the volume o f the growing s t o c k i n f o r e s t i n v e n t o r y s u r v e y s and t i m b e r - c r u i s i n g work, (b) c a l c u l a t i n g p e r i o d i c i n c r e m e n t s t o determine growth and y i e l d , (c) formu-l a t i n g f o r e s t management p o l i c i e s , (d) a s s e s s i n g t h e t i m b e r volume i n a f o r e s t a r e a t o p l a n l o g g i n g o p e r a t i o n s and a l s o t o s e r v e as a b a s i s f o r a s c e r t a i n i n g the a l l o w a b l e c u t and (e) comparing the t r e a t m e n t e f f e c t s o f d i f f e r e n t s i l v i c u l t u r a l o p e r a t i o n s , have been the s u b j e c t o f a g r e a t d e a l o f r e s e a r c h and e x t e n s i v e s t u d y from the dawn o f f o r e s t r y . The commence-ment o f modern f o r e s t r y a l s o saw e f f o r t s b e i n g d i r e c t e d towards t h e d e f i -n i t i o n o f i n d i v i d u a l t r e e volume i n m a t h e m a t i c a l terms as w e l l as q u a n t i -f y i n g t h e measure o f t r e e form. From the e a r l y p a r t o f the t w e n t i e t h c e n t u r y , e x t e n s i v e s t u d i e s were c a r r i e d o u t on t r e e form and t a p e r t o d e s c r i b e the stem p r o f i l e , r e s u l t i n g i n a w e l t e r o f form f a c t o r and t a p e r e q u a t i o n s , some o f them b e i n g s o p h i s t i c a t e d and r a t h e r f o r m i d a b l e , though s i m u l a t i n g t h e s e complex models i s c o m p a r a t i v e l y easy w i t h p r e s e n t - d a y e l e c t r o n i c d a t a p r o c e s s i n g t e c h n i q u e s . T h i s t h e s i s shows a r e l a t i v e l y easy and p r a c t i c a b l e composite e s t i m a t i n g system o f t a p e r and volume o f i n d i v i d u a l s t a n d i n g t r e e s i n terms o f the e a s i l y measured t r e e growth p a r a m e t e r s o f ,., D-^Q a n d H_ a p p l i c a b l e t o , a t l e a s t , p l a n t a t i o n - c r o p s o f Eucalyptus vobusta, where and a r e d i a m e t e r s i n i n c h e s a t 4.5 and 10 f e e t above h i g h e s t ground l e v e l and H t h e t o t a l h e i g h t o f the t r e e i n f e e t . B r i e f l y , t h e o b j e c t i v e s o f t h i s study are: 1. To demonstrate the computation o f m e r c h a n t a b l e t r e e volume, and u t i l i s a b l e 2 volume t o any demanded t o p d i a m e t e r o r h e i g h t , 2. To e s t a b l i s h a composite e x t i m a t i o n system f o r t h e p r e d i c t i o n o f d i a m e t e r s a t s p e c i f i e d h e i g h t s o r h e i g h t s f o r any demanded t o p d i a m e t e r s and volume e x p e c t a t i o n s t o any o f t h e d e s i r e d s i z e l i m i t s . No volume t a b l e s f o r t o t a l volume o r u t i l i s a b l e volumes have been com p i l e d by d i a m e t e r and h e i g h t c l a s s e s , as g r o u p i n g t e nds t o produce e r r o r s i n volume e s t i m a t e s . I t has been shown t h a t 15% o f t h e v a r i a n c e o f the e s t i m a t e d mean volume w i l l not be a c c o u n t e d f o r i f g r o u p i n g e r r o r i s i g n o r e d (Meng, 1972). In t h i s d i s s e r t a t i o n , form o f a t r e e i s d e f i n e d as t h e "shape" o f a s o l i d , t h e D-H c u r v e o f which i s d e t e r m i n e d by t h e power i n d e x o f D, t h a t i s , i f t h e f o r m u l a o f t h e stem curve i s r e p r e s e n t e d by t h e p a r a b o l a , 2 H = k D , k_ r e p r e s e n t s a measure o f t a p e r , o r the r a t e o f t a p e r . T r e e t a p e r i s d e f i n e d as t h e r a t e o f n a r r o w i n g i n d i a m e t e r i n r e l a t i o n t o i n c r e a s e i n h e i g h t o f a g i v e n form o r shape. A low v a l u e o f k would be c h a r a c t e r i s t i c o f a r a p i d r a t e o f n a r r o w i n g i n c o n t r a s t w i t h a h i g h v a l u e which c o r r e s p o n d s t o a slow r a t e o f n a r r o w i n g . 2. An a n a l y t i c a l s u r v e y o f t r e e volume p r o p e r t i e s , t r a n s f o r m a t i o n s and t a p e r 2.1 Tree volume p r o p e r t i e s and t h e i r e f f e c t on the e r r o r term o f th e r e g r e s s i o n model  Tr e e volume f o r a g i v e n D i s h i g h l y skewed and hence not n o r m a l l y d i s t r i b u t e d . A l s o t h e v a r i a n c e o f t r e e volume i s heterogeneous i n each diameter and h e i g h t c l a s s . The b i g g e r t r e e s v a r y i n volume t o a much g r e a t e r e x t e n t than s m a l l t r e e s . A r e s i d u a l o f 0.2 c u b i c f e e t i s v e r y l a r g e f o r a t r e e o f 6 i n c h e s D whereas i t i s i n s i g n i f i c a n t f o r a D o f 30 i n c h e s . Hence the r e s i d u a l s o f l a r g e t r e e s from t h e t r u e model a r e l a r g e compared t o the r e s i d u a l s from the s m a l l e r t r e e s , g i v i n g e x p e c t a t i o n s t h a t a r e not p r e c i s e , though u n b i a s e d , i n t h e p r e s e n c e o f pronounced h e t e r o s c e d a s t i c i t y . 3 The a p p l i c a t i o n o f the method o f l e a s t s q uares assumes t h a t t h e s t o c h a s t i c e r r o r term i n t h e r e g r e s s i o n model i s an independent random v a r i a b l e and i s d i s t r i b u t e d w i t h u n i f o r m o r homoscedacic v a r i a n c e . Hence, m i n i m i z i n g t h e r e s i d u a l sum o f sq u a r e s , t h e r e s i d u a l s f o r t h e l a r g e r t r e e s would markedly a f f e c t t h e c h o i c e o f t h e l e a s t square e s t i m a t o r , and would o f f s e t t h e r e s i d u a l s o f t h e s m a l l e r t r e e s , t h e r e b y i g n o r i n g t h e r e l a t i v e v a l u e o f t h e d e v i a t i o n s o f the s m a l l e r t r e e s and t h e i r c o n t r i b u t i o n t o the c h o i c e o f t h e r e g r e s s i o n e s t i m a t o r . F u r t h e r i n the case o f extreme e x t r a p o -l a t i o n , c o n f i d e n c e o f t h e volume v a l u e s would be c o n s i d e r a b l y low when t h e c o n s t a n t s a r e dete r m i n e d w i t h o u t any c o n s i d e r a t i o n o f h e t e r o g e n e i t y o f v a r i a n c e (Ohtomo, 1956). A l s o n e g a t i v e volume e x p e c t a t i o n s a r e o b t a i n e d a t the lower end o f t h e d a t a ( V i n c e n t and Sandrasegaran, 1965; V i n c e n t et.: a l . , 1956; Sandrasegaran, 1971) and have t o be c o n s i d e r e d as z e r o volumes which can g i v e r i s e t o f a l s e i n g r o w t h o f volumes as t h e s t a n d grows. T h i s l e a d s t o t h e development o f a r e g r e s s i o n f u n c t i o n w i t h o u t h a v i n g t o s a c r i f i c e p r e c i s i o n i n t h e r e g r e s s i o n e s t i m a t e s . 2.2 T r a n s f o r m a t i o n o f v a r i a b l e s T h i s p r o c e d u r e has been commonly a p p l i e d t o a c h i e v e v a r i a n c e homogeneity o f t h e new dependent v a r i a b l e s . N e v e r t h e l e s s i t s use a l t e r s t h e i m p l i e d r e l a t i o n between t h e dependent and independent v a r i a b l e s , f o r example ( i ) a square r o o t t r a n s f o r m a t i o n V = a Q + a^ D, i m p l i e s t h a t the r e l a t i o n 2 i s q u a d r a t i c o f t h e form V = a^ + a^ D + a^ D o r ( i i ) a l o g a r i t h m i c t r a n s -f o r m a t i o n Log V = b Q + b ^ D, i n f e r s an e x p o n e n t i a l r e l a t i o n , V = c b D , where c = a n t i - l o g b^ and b = a n t i - l o g b^. There i s a l s o some tendency t o use t r a n s f o r m a t i o n s w i t h o u t r e a l l y knowing what happens t o t h e v a r i a n c e ( F r e e s e , 1964). F u r t h e r , when a n a l y s i s o f t r a n s f o r m e d v a r i a b l e s i s complete, t h e s o l u t i o n s g i v e r i s e t o problems o f i n t e r p r e t a t i o n and p r e s e n t a t i o n , as 4 u n b i a s e d e s t i m a t o r s on t h e t r a n s f o r m e d s c a l e would not be n e c e s s a r i l y b i a s -f r e e when r e t r a n s l a t e d t o the o r i g i n a l s c a l e . Hence i t i s n e c e s s a r y t o r e s o r t t o c o r r e c t i o n s f o r t r a n s f o r m a t i o n b i a s (Meyer, 1938; 1944; Q u e n o u i l l e , 1956; Neyman and S c o t t , 1960; B a s k e r v i l l e , 1972). Though v a r i a n c e - s t a b i -l i z i n g t r a n s f o r m a t i o n s commonly n o r m a l i z e as a b y - p r o d u c t ( K e n d a l l and S t u a r t , 1966), they a r e o f t e n e x p e d i e n t s and are a t b e s t approximate i n t h e i r a c t i o n (Pearce, 1965). T r a n s f o r m a t i o n t o l o g a r i t h m s i s p r i m a r i l y an e x p e d i e n t t o f a c i l i t a t e f i t t i n g , which does n o t n e c e s s a r i l y r e s u l t i n h o m o s c e d a s t i c i t y o r n o r m a l i t y ( C u r t i s , 1967). T h i s can be seen from t h e Cn C o r e g r e s s i o n f u n c t i o n , V = C A D H recommended f o r use i n t r e e volume con-s t r u c t i o n (Schumacher and H a l l , 1933). T a k i n g l o g a r i t h m s o f b o t h s i d e s o f the r e g r e s s i o n f u n c t i o n , t h e new f u n c t i o n i s o b t a i n e d , Log V = ( l o g C n) •+ C 1 ( l o g D) + C 2 ( l o g H) where t h e new v a r i a b l e ( l o g V) has a c o n d i t i o n a l d i s t r i b u t i o n which i s a p p r o x i m a t e l y normal w i t h homogeneous v a r i a n c e . The two r e g r e s s i o n f u n c t i o n s , t h e f i r s t i n terms o f V and t h e second i n terms o f ( l o g V) a r e n o t e q u i v a l e n t , t h a t i s ; E(V/D, H) = C D C l H C2 i s n o t t h e same as a n t i l o g E ( l o g V/D, H) = a n t i l o g ( l o g C Q + ^ l o g D + C 2 l o g H) = a n t i l o g l o g C Q D 1 H 2 = C Q D C1 H C2 Thus i t c o u l d be seen, t h a t i n t h e f i r s t r e g r e s s i o n , t h e r e g r e s s i o n f u n c t i o n i s t h e l o c u s o f t h e c o n d i t i o n a l mean o f V, b u t i n t h e second, t h e r e g r e s s i o n f u n c t i o n i s the l o c u s o f i t s c o n d i t i o n a l g e o m e t r i c mean. Hence t h e c o e f -f i c i e n t s o f t h e two f u n c t i o n s a r e g e n e r a l l y d i f f e r e n t , t h e i n f e r e n c e b e i n g t h a t l e a s t s q uares e s t i m a t o r s u s i n g t h e l o g t r a n s f o r m a t i o n s o f t h e v a r i a b l e s a r e u n b i a s e d e s t i m a t o r s o f t h e r e g r e s s i o n c o e f f i c i e n t s o f t h e second f u n c t i o n . Thus t h e l e a s t s q u a r e s e s t i m a t o r s o f C , and by t h e method o f t r a n s -5 formations are biased estimators of the regression c o e f f i c i e n t s of the f i r s t regression function, the regression where we are p r i m a r i l y concerned. The transformation of the independent v a r i a b l e s does not bring any bias i n the estimation process of the reqression function, but, however i t i s apparent, that one i s t a l k i n g about d i f f e r e n t reqression models as the independent v a r i a b l e s mav be d i f f e r e n t . Thouqh the bias of the sample estimators can be removed, yet the adjusted l e a s t squares estimator of the reqression function mav no lonqer be the best i n the s t a t i s t i c a l sense and thus there mav be other l i n e a r estimators for which the variance i s smaller. 2.3 Process o f weighting Weighting of the dependent v a r i a b l e can be c l a s s i f i e d as another type of transformation. This procedure s t a b i l i z e s the expected values of the squared deviations y i e l d i n g homoscedacic variance, as the smaller, l e s s v a r i a b l e trees are weighted heavily than i n unweighted regression. Though over a range of species, the volume variance tends to be proportional to ( D 2 ) 2 (Cunia, 1964; Wright, 1964) or ( p 2 H ) 2 (Haack, 1963; Munro, 1964; Gregory and Haack, 1964; Evert, 1969; Sandrasegaran, 1968; 1969i, 1970i, 1971, 1972, 19731; Smalley and Beck, 1971), yet i t has also been shown that the variance i s not proportional to the functions above and consequently 2 an exponential function of the form, log a = c + C D + H has been derived (Gerrard, 1966). Also volume variance was found to be exponentially 2 r e l a t e d to D H (Moser and Beers, 1969). Other hypotheses have also been 2 2 V made using d i f f e r e n t weights, D H and (DH) to obtain an improved f i t i n the smaller diameter classes and a c o r r e c t i o n of the tendency to conform too c l o s e l y to the data of the large s i z e classes, r e s u l t i n g from large absolute s i z e of the deviations encountered i n these classes (Sandrasegaran, 1974i). The form factor method of volume table compilation was one other 6 method o f w e i g h t i n g t r e e volume r e s i d u a l s (Meyer, 1953). Hence t h e e f f e c t i v e n e s s : o f w e i g h t i n g i s r e l i a n t on a pviori knowledge o f t h e v a r i a t i o n o f t h e p o p u l a t i o n v a r i a n c e . T h e o r e t i c a l l y , w eights s h o u l d be used t h a t a r e i n v e r s e l y p r o p o r t i o n a l t o t h e square o f the r e s i d u a l s , b u t i n p r a c t i c e i t may be d i f f i c u l t t o determine t h e most a p p r o p r i a t e way t o weight a p a r t i c u l a r r e g r e s s i o n f u n c t i o n . Y e t a v e r y r e c e n t s t u d y o f r e g r e s s i o n t e c h n i q u e s f o r p r o d u c i n g a volume t a b l e showed t h a t w e i g h t e d r e g r e s s i o n g i v e s the b e s t e s t i m a t e s (Mendiboure and P r o n i e r , 1972; Bury e t . a l . , 1972). However o t h e r s have come t o t h e c o n c l u s i o n t h a t t h e r e e x i s t s no w h o l l y s a t i s f a c t o r y method o f w e i g h t i n g o r t r a n s f o r m a t i o n (Smith and Munro, 1965). To sum up on w e i g h t i n g s and t r a n s f o r m a t i o n s , i t must be s t r e s s e d t h a t any model b u i l d i n g i s s u b j e c t i v e and t o a c e r t a i n e x t e n t a r b i t r a r y . There i s an i n t u i t i v e f e e l i n g t h a t i f , f o r a g i v e n p h y s i c a l o r b i o l o g i c a l problem, t h e model f i t s w e l l , t h e n t h e r e s u l t s , f o r a l l p r a c t i c a l p u rposes, a r e v a l i d . The t e s t s o f s i g n i f i c a n c e i n d i c a t e whether an i n t u i t i v e f e e l i n g j seems r i g h t o r wrong b u t do n o t p r o v e o r d i s p r o v e a n y t h i n g , t h e y o n l y s u p p l y e v i d e n c e . The models need not be c l o s e l y r e l a t e d t o t h e a c t u a l s i t u - -a t i o n t o y i e l d approximate and u s e f u l r e s u l t s . Most o f t h e t i m e , i t i s s u f f i c i e n t t h a t t h e model assumptions be a p p r o x i m a t e l y f u l f i l l e d . Such c a s e s o c c u r when t h e c o n d i t i o n a l d i s t r i b u t i o n i s n o t normal, t h e c o n d i t i o n a l v a r i a n c e i s n o t t o o heterogeneous o r t h e sample u n i t s a r e n o t drawn by a t r u l y s i m p l e random s a m p l i n g t e c h n i q u e . A l s o , i n a b s o l u t e terms, t h e t r a n s -f o r m a t i o n s do n o t s t a b i l i z e t h e v a r i a n c e , and f o r each p a r t i c u l a r p o p u l a t i o n , t h e v a r i a n c e f u n c t i o n o f t h e dependent v a r i a b l e i s a b s o l u t e l y u n ique, as w e l l as never.known, and hence f o r c e d t o approximate i t i n t h e b e s t p o s s i b l e p r a c t i c a l way. C o n s e q u e n t l y , t h e s e pose the f o l l o w i n g q u e s t i o n s : (1) I s a p a r t i c u l a r t r a n s f o r m a t i o n s u f f i c i e n t l y s u c c e s s f u l i n homogenizing t h e 7 t r e e volume v a r i a n c e , so t h a t t h e r e g r e s s i o n a n a l y s i s r e s u l t s a r e s u f f i c i e n t l y v a l i d ? , ( i i ) Knowing t h a t r e l a t i v e l y s m a l l d e p a r t u r e s from h o m o s c e d a s t i c i t y do n o t a p p r e c i a b l y a f f e c t t h e v a l i d i t y o f t h e f i n a l c o n -c l u s i o n s , can we use s i m p l e t r a n s f o r m a t i o n s and o b t a i n m e a n i n g f u l r e s u l t s ? , ( i i i ) Or must we use complex methods and i n s u r e o u r s e l v e s a g a i n s t f a u l t y c o n c l u s i o n s ? Answers t o t h e s e q u e s t i o n s can be o n l y o b t a i n e d from an e x p e r i m e n t a l s e n s i t i v i t y a n a l y s i s . D i f f e r e n t s e t s o f wei g h t s and v a r i a n c e f u n c t i o n s must be t r i e d on a l a r g e v a r i e t y o f sample d a t a , t o see t h e e x t e n t o f t h e i r e f f e c t on t h e e s t i m a t i n g p r o c e s s o f t h e r e g r e s s i o n a n a l y s i s . F i n a l l y one can say, t h a t as t r e e volume v a r i a n c e i n c r e a s e s w i t h D, a d d i t i o n a l v a r i a b l e s s uch as t r e e h e i g h t , d i a m e t e r a t stump h e i g h t , d i a m e t e r a t m i d d l e t r e e h e i g h t , some form f a c t o r s t o g e t h e r w i t h some o f t h e i r s q u a r e s , c r o s s - p r o d u c t s , r a t i o s , e t c . may g i v e sample d a t a f o r which, f o r a l l p r a c t i c a l p u r p o s e s , t h e v a r i a n c e i s no l o n g e r non-homogeneous, b u t the p r o c e d u r e o f t a k i n g a d d i t i o n a l measurements i s n o t f e a s i b l e i n terms o f c o s t s and l a b o u r . 2.4 Form and t a p e r The concept o f form f a c t o r i s a b s t r a c t and t h e q u a n t i t a t i v e v a l u e o f form f a c t o r i s dependent on t r e e s p e c i e s , i t s g e n e t i c i n h e r i t a n c e and s u r r o u n d i n g s , t h e e x p r e s s i o n o f h e i g h t used, t h e e x p r e s s i o n o f volume r e q u i r e d and t h e p l a c e where t h e s e c t i o n a l a r e a i s measured. The form o f i n d i v i d u a l t r e e s v a r i e s c o n s i d e r a b l y i n a ti m b e r s t a n d (Wright, 1923) and the measure o f t a p e r i n terms o f the form q u o t i e n t i s not v e r y r e l i a b l e as t r e e s o f s i m i l a r form q u o t i e n t v e r y o f t e n have v a r i a n t b o l e forms. An average v a l u e o f form f a c t o r does not s u f f i c i e n t l y p r o v i d e a measure o f t a p e r and form q u o t i e n t s do not g i v e s a t i s f a c t o r y volume e x p e c t a t i o n s . I t must be s t r e s s e d t h a t i n t h e c o m p i l a t i o n o f volume t a b l e s , u s i n g an e q u a t i o n based on G and H o n l y , a s a t i s f a c t o r y compromise between maximum a c c u r a c y 8 and r e a s o n a b l e ease and time o f measuring i s a c h i e v e d ( S i m p f e n d o r f e r , 1959) and a comparison o f volume e q u a t i o n s f o r Eucalyptus sali-gna showed t h a t e q u a t i o n s w i t h form f a c t o r a r e c o n s i d e r e d more e x a c t , b u t t h e use o f e q u a t i o n s w i t h o u t form f a c t o r i s o f t e n p r e f e r a b l e , s i n c e l e s s f i e l d work i s i n v o l v e d (Vega, 1972). The average-end-area method o f S m a l i a n ' s f o r m u l a t o compute the t r u e volume o f a t r e e by summation o f s h o r t l o g s e c t i o n s from stump h e i g h t t o t h e t r e e t o p assumes t h a t th e t r e e i s composed o f a number o f t r u n c a t e d p a r a b o l o i d s , and hence each s h o r t s e c t i o n i s a f r u s t u m o f a p a r a b o l o i d w i t h t h e c y l i n d e r as t h e l i m i t i n g c a s e . A l t h o u g h the r e s u l t w i l l be b i a s e d t o t h e e x t e n t t h a t t a p e r s i n t h e s e s h o r t s e c t i o n s f a i l t o c o r r e s p o n d t o t a p e r s i n t r u n c a t e d p a r a b o l o i d s , the b i a s becomes s m a l l e r as s h o r t e r s e c t i o n a l l e n g t h i s used (Johnson, 1957; Sandrasegaran, 1 9 7 3 i i ) , and thus t a k i n g t r e e form i n d i r e c t l y i n t o a c c o u n t . I n a t t e m p t i n g t o q u a n t i f y t r e e form, a r e l a t i o n s h o u l d be d e v e l o p e d i n terms o f t h e e a s i l y measured t r e e p a r a m e t e r s , p_, and H_, w i t h a s i m p l e and a c c e p t a b l e e s t i m a t i n g system f o r p r a c t i c a l p u r p o s e s . The use o f s i m p l e f u n c t i o n s , s o r t i n g and g r a p h i c a l methods i s adequate f o r many uses i n o p e r a t i o n a l r e s e a r c h (Kozak and Smith, 1966), a c o n c l u s i o n r e a c h e d subsequent t o v a r i o u s t e s t s o f m u l t i v a r i a t e t e c h n i q u e s i n t r e e t a p e r a n a l y s i s . 2.5 C r i t e r i a f o r s e l e c t i o n o f r e g r e s s i o n models The v a r i o u s c r i t e r i a o f comparison o f models, t o s e l e c t - t h e " b e s t " one, f o r m u l a t e d t o d a t e , comprise t h e f o l l o w i n g : 2 ( i ) The c o e f f i c i e n t o f d e t e r m i n a t i o n , r , f o r a p a r t i c u l a r d a t a s e t i s n o t d i s t r i b u t i o n dependent, b u t i t s m e a n i n g f u l use as a b a s i s o f comparison i s dependent on the same 9 number o f independent v a r i a b l e s , as i t changes m o n o t o n i c a l l y w i t h an i n c r e a s e i n the number o f v a r i a b l e s , ( i i ) The mean squared r e s i d u a l which o n l y changes w i t h the number o f degrees o f freedom. I t s v a l i d use i s o n l y between d i f f e r e n t a r i t h m e t i c o r l o g a r i t h m i c d e v i a t i o n s but not between e i t h e r a r i t h m e t i c and l o g a r i t h m i c f u n c t i o n s (Spurr, (1952), ( i i i ) The s t a n d a r d i z e d t o t a l squared e r r o r (Mallows, 1966), known as C , which i s a s t a t i s t i c used as a c r i t e r i o n o f goodness J? o f f i t and measures t h e s q u a r e d b i a s e s p l u s s q u a r e d random e r r o r i n f i t t i n g an e q u a t i o n ( D a n i e l and Wood, 1971). (i v ) An i n d e x o f r e l a t i v e p r e c i s i o n u s i n g t h e c o n c e p t o f maximum l i k e l i h o o d ( F u r n i v a l , 1961), and t h i s i n d e x cannot be c a l c u l a t e d f o r 1/V, u n l e s s t h e a b s o l u t e v a l u e i s t a k e n s i n c e t h e i n d e x i s a n e g a t i v e q u a n t i t y , as t h e f i r s t d e r i v a t i v e i s n e g a t i v e . A l s o i f V i s n o r m a l l y d i s t r i b u t e d , i t does not i m p l y t h a t Log V i s a l s o d i s t r i b u t e d n o r m a l l y . A l s o subsequent t o a s e r i e s o f t e s t s i n v o l v i n g a b s o l u t e and r e l a t i v e s t a n d a r d e r r o r s o f e s t i m a t e f o r t r e e volume, i t has been c o n c l u d e d t h a t no s i n g l e s t a t i s t i c c o u l d determine the b e s t e q u a t i o n ( H e j j a s , 1967). Though o b j e c t i v i t y i s t h e essence o f s t a t i s t i c a l r e s e a r c h , d i s c o u n t i n g p e r s o n a l b i a s and p r e f e r e n c e i n t h e s e l e c t i o n o f f u n c t i o n s , i t i s t h e o p i n i o n o f the w r i t e r t h a t e q u a t i o n s s h o u l d n o t be f o r m u l a t e d from a p r e - d e t e r m i n e d s e t o f v a r i a b l e s and t e s t e d f o r a s e e m i n g l y s u i t a b l e one. I n a c t u a l i t y , t h e " f i s h i n g " approach o f t r y i n g , say, y = a + bx, l o g y = a + bx, l o g y = a + b l o g x, y = a + b l o g x, y = a + b/x, e t c . s h o u l d . n o t be i n d u l g e d i n as i t seems t o suggest t h a t t h e u s e r has l o s t t h e a b i l i t y t o t h i n k , o r i s w o e f u l l y i g n o r a n t o f the n a t u r e o f h i s m a t e r i a l (Warren, 1974). 10 F i n a l l y , mention must be made o f t h r e e p a p e r s i n the f i e l d o f f o r e s t r y p e r t a i n i n g t o t e s t s o f p a r a l l e l i s m and c o n c u r r e n c e o r c o i n c i d e n c e i n comparing r e g r e s s i o n s from two o r more s e t s o f d a t a (Cunia, 1968; Kozak, 1970; Warren, 1974). 3. L i t e r a t u r e r e v i e w T h i s t h e s i s approaches t h e p r o b l e m from a d i f f e r e n t v i e w p o i n t and hence no r e v i e w s on d i f f e r e n t t r e e form t h e o r i e s o r v a r i a n t volume models a r e made. Adequate coverage o f t h e former (Gray, 1956; Mergen, 1954; Odgers, 1956; Newnham, 1958; Schniewind, 1962; L a r s o n , 1963; Doerner, 1964, 1965; Heger, 1965; S h i n o z a k i e t a l . , 1965) and a l l a s p e c t s o f work w i t h r e g a r d t o t h e l a t t e r ( L o e t s c h and H a l l e r , 1964; Prodan, 1965; Husch e t ;•; a l . , 1972; Demaerschalk, 1973; Sandrasegaran, 1973i) have been under-t a k e n o r c o l l a t e d . Hence t h i s r e v i e w c o n s i s t s o f p r o p o s e d t a p e r c u r v e s d e s c r i b i n g stem p r o f i l e s and an o u t l i n e o f a d d i t i o n a l v a r i a b l e s used i n o b t a i n i n g b e t t e r volume e x p e c t a t i o n s , as w e l l as some r e l a t i o n o f growth v a r i a b l e s on form. The f i r s t e v e r s t u d y taken t o d e v e l o p a q u a n t i t a t i v e measure o f the t r e e p r o f i l e was by a Swedish c i v i l e n g i n e e r when he p r o p o s e d t h e m a t h e m a t i c a l f u n c t i o n , d/Du = £n < (Hojer, 1903), where t h e r e g r e s s i o n c o n s t a n t s C and had been d e f i n e d f o r each form c l a s s . T h i s was f o l l o w e d by t h e i n t r o d u c t i o n o f a ' b i o l o g i c a l c o n s t a n t ' , C 3 , i n t o t h a t o f H o j e r ' s e q u a t i o n (Jonson, 1910; 1911; 1926-27, quoted by Behre, 1923), t h e format b e i n g , C + L 2 (100/H) 11 d/Du = C Zn which, no doubt, was b e s t adapted f o r use i n second-growth s t a n d s , e x p e c i a l l y i f th e y a r e even-aged and can be used t o g r e a t advantage i n p r e p a r i n g l o c a l volume t a b l e s i n v i r g i n s t a n d s o f v e r y l a r g e t i m b e r ( C l a u g h t o n - W a l l i n and M c V i c k e r , 1920) . The advantage o f Jonson's method i n e x p r e s s i n g the form v a r i a t i o n i s t h a t t h e c l a s s i f i c a t i o n i s made i n d e -pendent o f h e i g h t , t h e two f o r m - d e t e r m i n i n g d i a m e t e r s always b e i n g i n t h e same r e l a t i o n t o each o t h e r ( C l a u g h t o n - W a l l i n , 1918). I t has a l s o been used i n e s t i m a t i n g t i m b e r , g i v i n g average r e s u l t s (Wickenden, 1921). A n o t h e r e q u a t i o n d e v e l o p e d f o r t h e stem cu r v e was d/Du = (L/H)/ ( C Q + C L/H) (Behre, 1923; 1927; 1935) and some t r a n s f o r m a t i o n s o f t h e s e were g i v e n l a t e r (Bruce, 1972). 2 2 The stem p r o f i l e above bh was a l s o d e s c r i b e d by d /D u = C^ ( L / H ) 2 + C 1 ( L / H ) 3 + C 2 ( L / H ) 4 where C Q + C± + 0.^ = 1 and C Q , C 2 a r e f u n c t i o n s o f Du and H (Matte, 1949). Form f a c t o r was the n c a l c u l a t e d as £ c 2 + L (100/H) - C^j C 2 which m u l t i p l i e d by the p r o d u c t o f b a s a l a r e a and h e i g h t above b r e a s t h e i g h t gave c u b i c volume above b r e a s t h e i g h t . 2 Another e q u a t i o n f o r m u l a t e d was d/Du = C Q (L/H) + (L/H) + 3 C (L/H) though a f o u r t h degree p o l y n o m i a l would have g i v e n b e t t e r 12 e x p e c t a t i o n s (Osumi, 1959). A f u r t h e r approach u s i n g a 15th degree p o l y n o m i a l , the r e l a t i o n s h i p o f d / d Q t o r e l a t i v e h e i g h t , d Q ^ was e x p r e s s e d as a f u n c t i o n o f D and.H and i n t e g r a t i n g t h e e q u a t i o n t o o b t a i n t a p e r , volume and assortment t a b l e s ( G u i r g i u , 1963). A f u r t h e r s a t i s f a c t o r y equation of taper was suggested, d/Du = (h/H) / C 0 + C l (i> + °2 ( (Prodan, 1965). A composite e s t i m a t i n g s e t o f t a p e r and volume e q u a t i o n s 2 3 were d e r i v e d , e x p r e s s i n g t h e v a l u e o f (d/D) as a f u n c t i o n o f D, H and — , 3rd., 32nd., and 40th. powers o f r e l a t i v e h e i g h t whereby an e s t i m a t i n g system o f t a b l e s were o b t a i n e d above b r e a s t h e i g h t f o r v a r i o u s c o m b i n a t i o n s o f p r o d u c t u n i t s , u t i l i z a t i o n l i m i t s and s i z e c l a s s e s o f m a t e r i a l (Bruce e t a l . , 1968). The n e c e s s i t y o f u s i n g such h i g h powers was a t t r i b u t e d t o the b u t t s w e l l and t i p . The e q u a t i o n f i t t e d was: ( d / D ) 2 = C Q ( M ) 3 / 2 + (M 3 /2 - M 3 ) ( C 1 D + C 2 H) + (M 3 /2 - M 3 2) ( C 3 HD + C 4 H 5) + (M 3/2 . M 4 0 } ( H 2 } where M = L/(H - 4'5) A s i m p l e p r e d i c t i o n model f o r shape (taper) above bh f o r p l a n t e d s l a s h p i n e t r e e s was f i r s t c o n s t r u c t e d as f o l l o w s ; d = C Q D L/(H - 4'5) >J + C ; L L(H - 4'5) + C, L ( h - 4'5) (H + h + 4'5) and s i n c e the p r e d i c t e d d i a m e t e r s were n o t always m o n o t o n i c a l l y d e c r e a s i n g j u s t above 8.5 f e e t f o r c o m b i n a t i o n s o f s m a l l d i a m e t e r s and t a l l h e i g h t s , t h e e s t i m a t 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 s from t h e i n d i v i d u a l t r e e r e g r e s s i o n s were p l o t t e d a g a i n s t D, H_ and number o f t r e e s p e r a c r e , w i t h t h e r e s u l t o f an a p p a r e n t and l i n e a r r e l a t i o n o f t o R which was s u b s t i t u t e d i n t h e above e q u a t i o n and renumbered t o g i v e , d - C Q D L/(H - 4.5) + C, 13 L ( h - 4.5) + C 2 H L ( h - 4.5) + C 3 L ( h - 4.5) (H + h + 4.5) and was found t o f i t t h e d a t a w e l l from bh t o t r e e t o p w i t h m o n o t o n i c a l l y d e c r e a s i n g f u n c t i o n s (Bennett and S w i n d e l , 1972). 2 A f u r t h e r e q u a t i o n p r o p o s e d was (d/D) = C Q + (h/H) + 2 (h/H) w i t h a c o n s t r a i n t C Q + C + = 0 imposed so as t o a l l o w t h e di a m e t e r s i n s i d e b a r k a t the t o p .to be s e t a t z e r o ; (Kozak e t a l . , 1969i, i i ) A d d i t i o n a l c o n d i t i o n i n g was w a r r a n t e d so as n o t t o o b t a i n n e g a t i v e d i a m e t e r s around t h e p r o x i m i t y o f t h e t o p f o r t h e two s p e c i e s o f r e d c e d a r and s p r u c e . V a r i o u s t e s t s c a r r i e d o u t w i t h t h e s e f u n c t i o n s i n d i c a t e d a s t a b l e system o f e s t i m a t i o n (Kozak e t - a l . , 1969i; Smith and Kozak, 1971). V e r y r e c e n t l y r e s e a r c h f i n d i n g s showed t h a t c o m p a t i b l e t a p e r e q u a t i o n s d e r i v e d from volume e q u a t i o n s f i t t e d on t r e e volume d a t a and com p a t i b l e volume e q u a t i o n s d e r i v e d from t a p e r e q u a t i o n s f i t t e d on t h e t a p e r d a t a gave i d e n t i c a l e s t i m a t e s o f t o t a l volume o f t r e e s (Demaerschalk, 1973) . In t h e p r e d i c t i o n o f t r e e volume, v a r i o u s a u t h o r s have used a t h i r d i n d ependent parameter t o o b t a i n b e t t e r volume e s t i m a t e s , namely, Kr (Nasulund, 1947), D ( I l v e s s a l o , 1947; Van S o e s t , 1959) d., /D (Mesavage — 16 and G i r a r d , 1946), D,. n (Schmid e t , a l . , 1971), D (Van L a a r , 1968), dn , 6-9 — — 25 0.1 (Kuusela, 1965). G r a p h i c a l t e c h n i q u e s were a l s o used t o d e v e l o p t a p e r and volume t a b l e s (Duff and B u r s t a l l , 1955) and t o r e l a t e t h e ' p e r c e n t a g e o f t o t a l volume t o t h e p e r c e n t a g e o f t o t a l t r e e h e i g h t ( S p e i d e l , 1957). A p a r a b o l a was f i t t e d t o d e s c r i b e a g r e a t e r p o r t i o n o f t h e b o l e shape , (Newnham, 1958). P o l y n o m i a l a n a l y s i s i n t r o d u c i n g some new tre e - m e a s u r e -ment i d e a s (Grosenbaugh, 1954; 1966) and m u l t i v a r i a t e t e c h n i q u e s ( F r i e s , 1965; F r i e s and Matern, 1965) were a l s o t e s t e d . P o s s i b l e c o n v e r s i o n o f o u t s i d e bark v a l u e s o f t a p e r and form t o i n s i d e bark v a l u e s have a l s o been 14 i n v e s t i g a t e d (Smith and Kozak, 1967). The stem p r o f i l e was d e s c r i b e d by two s e p a r a t e l o g a r i t h m i c c u r v e s , one f o r the main stem and the o t h e r f o r the top p a r t ( P e t t e r s o n , 1927) w h i l e a c o m b i n a t i o n o f t h r e e v a r i a n t models f o r the v a r i o u s p a r t s o f the stem was t e s t e d ( H e i j b e l , 1928) . Butt-?taper e q u a t i o n s were f i t t e d on p l o t t e d a v e r a g e s , so as t o e s t i m a t e D form measurements o f d i a m e t e r a t v a r i o u s h e i g h t s (Breadon, 1957). Three mathe-m a t i c a l models t o d e s c r i b e t h e volume d i s t r i b u t i o n t o any s t a n d a r d o f u t i l i z a t i o n were a l s o d e v e l o p e d (Honer, 1964; 1965i; 1 9 6 5 i i ) . In an attempt t o e s t i m a t e t r e e volume p r o p o r t i o n s by l o g p o s i t i o n , s e p a r a t e e q u a t i o n s were f i t t e d f o r each p e e l e r l o g (Burkhart- e t a l . , 1971). Based on sub-j e c t i v e l y drawn harmonized t a p e r c u r v e s , l o g p o s i t i o n volume t a b l e s were co m p i l e d ( F l i g g and Breadon, 1959) . I t was found n e c e s s a r y t o use t o t a l h e i g h t and g i r t h a t 1/4 t o t a l h e i g h t t o e l i m i n a t e the p r u n i n g f a c t o r i n d e v e l o p i n g form e q u a t i o n s f o r Populus and t h i s e x p l a i n e d a l l t h e v a r i a b i l i t y due t o form (Mendiboure, 1972) . In a n o t h e r i n v e s t i g a t i o n on the stem form o f Pinus sylvestus, H, D and K — -— L i had a s i g n i f i c a n t e f f e c t on form b u t f a c t o r s such as s i t e c o n d i t i o n s , s t a n d d e n s i t y , and t r e e age d i d n o t themselves s i g n i f i c a n t l y a f f e c t form ( K u l e s h i s , 1972). The average e f f e c t s o f age, s i t e and crown c l a s s a r e s m a l l i n r e l a t i o n t o t h o s e o f D, H, form and bark t h i c k n e s s (Demaerschalk and Smith, 1972). F u r t h e r , a method o f p r e p a r i n g a l i g n m e n t c h a r t s g i v i n g t h e d i s t r i -b u t i o n o f m e r c h a n t a b l e volume by l o g - d i a m e t e r c l a s s e s f o r e x o t i c c o n i f e r o u s s t a n d s has a l s o been d e s c r i b e d (Warren, 1959). To sum up t h e l i t e r a t u r e r e v i e w on t h e development o f m a t h e m a t i c a l models t o d e s c r i b e the stem p r o f i l e , i t i s apparent t h a t a b e w i l d e r i n g number o f v a r i a n t t r e e t a p e r models by a w e l t e r o f t e c h n i q u e s have been , c o n s t r u c t e d s i n c e 1903. From the number o f models s u g g e s t e d , i t i s f a i r l y o b v i o u s t h a t t h e c o n s t r u c t i o n o f a t a p e r model i s h e a v i l y dependent on t h e s p e c i e s , t r e e form, l o c a t i o n and s t a n d s t r u c t u r e . Hence each s e t o f d a t a must be t r e a t e d as a s e p a r a t e e n t i t y . and t h e r e l a t i o n d e v e l o p e d c h a r a c t e r i s t i c f o r t h a t l o c a t i o n , s p e c i e s o r t r e e form. In f a c t , one wonders whether a s i n g l e g l o b a l model c o u l d be d e v e l o p e d t o a d e q u a t e l y d e s c r i b e t h e stem p r o f i l e , e s p e c i a l l y f o r v e r y much o l d e r t r e e s w i t h e x c e s s i v e b u t t f l a r e s v a r y i n g from r e g i o n t o r e g i o n . 4. Data c o l l a t i o n 4.1 Sampling t e c h n i q u e A t o t a l o f 178 t r e e s was . measured f o r D^ and H r a n g i n g from 10.2 to 24.8 i n c h e s and from 73 t o 132 f e e t , r e s p e c t i v e l y . The p o p u l a t i o n i n t h i s a r e a was d i v i d e d i n t o o n e - i n c h ^ c l a s s e s and the f r e q u e n c y o f t r e e s sampled i n each D^ ^ c l a s s was s e l e c t e d i n such a manner t h a t i t s d i s t r i b u t i o n v a r i e s a p p r o x i m a t e l y p r o p o r t i o n a l l y t o the t r e e p o p u l a t i o n o f t h e a r e a . The d a t a a r e p r e s e n t e d i n T a b l e 1. The completed system o f measurement was used i n r e c o r d i n g q u a n t i t a t i v e measurements o f H, t h a t i s , say any v a l u e from 16,0 t o 16,9 f e e t i s r e c o r d e d as 16 f e e t (Sandrasegaran, 1 9 6 6 i i i ) , 4.2 Parameters measured The growth parameters measured on each sampled t r e e were; U ) f i l l ' ^3,0' V 5 ' ( — > ' ^10' ^30, ^50' ^TO; ^90 ± U ± n C h e S ' a p p l i c a b l e depending on th e h e i g h t o f t h e t r e e , and s c a l e d s t a n d i n g u s i n g s t e e l l a d d e r s , (b) D^ f o r h>90 i n c h e s w i t h t h e h e l p o f a B a r r and S t r o u d FP 12 Dendrometer (Sandrasegaran, 1 9 6 9 i i ) and (c) H i n f e e t w i t h a Haga hypsometer (Sandrasegaran 1 9 7 0 i i ) . V a r i a b l e s D_ _ t o D,.- were measured by a s t e e l t a p e which was 1.5 90 so g r a d u a t e d t h a t t h e D s e r i e s o f measurements (at a p o i n t c o r r e s p o n d i n g t o t h e c i r c u m f e r e n c e ) can be r e a d d i r e c t l y hy. wrapping the t a p e around the 16 c i r c u l a r o b j e c t . Dh was e s t i m a t e d a t a p o i n t o f n o t i c e a b l e change i n stem t a p e r and where such a p o i n t can be d i s t i n g u i s h e d w i t h ease d u r i n g f i e l d measurements. 5. The t a p e r l i n e 5.1 D e r i v a t i o n The t a p e r l i n e f o r an i n d i v i d u a l t r e e i s d e f i n e d by t h e l i n e a r r e l a t i o n g = a - bh where g i s t h e s e c t i o n a l a r e a i n square f e e t a t s e c t i o n a l h e i g h t h f e e t ( F i g u r e 1 ) . T h i s s t u d y was i n d e p e n d e n t l y d e v e l o p e d from t h a t o f Gray (1956) and Newnham (1958). The p a r a b o l i c base, B , i s g i v e n by the v a l u e o f a_, w h i l e t h a t o f the p a r a b o l i c h e i g h t , h , by a/b, t h a t i s , t h e t a p e r l i n e o f a p a r t i c u l a r —E. t r e e i s d e f i n e d , once t h e v a l u e s o f a_ and b are known. From t h e above r e l a t i o n , i t i s a p p a r e n t t h a t g = a when h = 0, and h = a/b when g = 0. F o r a g i v e n h e i g h t , I K ; h. = (a - g j / b . The t o t a l volume o f an i n d i v i d u a l t r e e i s , V T = 1/2 (Bp x hp) = a /2b, The u t i l i s a b l e volume t o D^ i s V Q = (g^ + g)/2 x h^, t h a t i s , — i V D. (a + g )/2 (a - g )/b V D = ( a 2 - g 2 . ) / 2 b i The u t i l i s a b l e volume t o h i , v = £i h. 2 I 2a - bh. The p e r c e n t a g e o f volume o f wood a v a i l a b l e i n the t r e e a t d i a m e t e r D^ i s , Di p i ( a 2 - g 2 . ) / 2 b [ / a _ 2 ' ' 2b x 100 17 t .-,2 1 0 0 The p e r c e n t a g e o f volume o f wood a v a i l a b l e t o h e i g h t , hi_ i s , V. h i p 2 -, a - bh^ 100 1 0 0 bh. 2a - bh. 5.2 C h a r a c t e r i s t i c p r o p e r t i e s o f t h e t a p e r l i n e I t i s now a p p a r e n t from 5.1 t h a t a p l o t o f s e c t i o n a l a r e a on s e c t i o n a l h e i g h t i n t h e main stem w i l l d e p i c t a d e c r e a s i n g l i n e a r r e l a t i o n o f g w i t h i n c r e a s i n g s e c t i o n a l h e i g h t . T h i s r e l a t i o n , denoted as the t a p e r l i n e , can be p r o j e c t e d t o i n t e r c e p t t h e v e r t i c a l and h o r i z o n t a l axes a t p a r a b o l i c h e i g h t and p a r a b o l i c base r e s p e c t i v e l y . The t r i a n g l e demarcated by the t a p e r l i n e , p a r a b o l i c base and p a r a b o l i c h e i g h t (the l a s t two p arameters a r e d i r e c t f u n c t i o n s o f t r e e v o l u m e ) e n a b l e s t h e p a r a b o l i c volume t o be computed by t h e f o r m u l a , 1/2 ( p a r a b o l i c base X p a r a b o l i c h e i g h t ) . The above p a r a b o l i c volume i s a c l o s e a p p r o x i m a t i o n t o t h e stem volume, e s p e c i a l l y i n p l a n t a t i o n e s t a b l i s h e d c o n i f e r s and second growth c r o p s , where the c u r v e s o f b u t t - s w e l l and t i p volume do n o t d i f f e r s i g n i f i c a n t l y from the t a p e r l i n e . T h i s method o f c o n s t r u c t i n g the t a p e r l i n e i s dependent o n , o n l y a minimum number o f p o i n t s , u s u a l l y , t h r e e o r f o u r , p r e f e r a b l y w e l l - s p a c e d a l o n g t h e stem above t h e r e g i o n o f b u t t - s w e l l and below the c o n i c a l t o p . A l s o two o r t h r e e measurements s p r e a d o v e r the t o p and b u t t r e g i o n s a r e s u f f i c i e n t enough t o o b t a i n a curve o f a h i g h 18 degree o f a c c u r a c y . Hence t h e t o t a l volume computation i n v o l v e s c a l c u l a t i n g from r e a d i n g s from the graph, the volume a t the t o p end and a t t h e b u t t r e g i o n c o r r e s p o n d i n g t o t h e a r e a s o u t s i d e o r i n s i d e t h e p a r a b o l i c t r i a n g l e and summing o r s u b t r a c t i n g t h e s e a r e a s from the p a r a b o l i c volume. I f t h e t o p l i m i t t o which volume i s d e s i r e d i s l e s s t h a n t h e p a r a b o l i c h e i g h t , t h e volume r e p r e s e n t e d by the a p p r o p r i a t e t r i a n g l e i s s u b t r a c t e d from t h e p a r a b o l i c volume, and i n i n s t a n c e s , where the t o t a l h e i g h t i s l e s s than the p a r a b o l i c h e i g h t , t h e r e q u i r e d t o t a l volume i s d e r i v e d by drawing a c u r v e down from t o t a l h e i g h t t o j o i n t h e t a p e r l i n e a t an a p p r o p r i a t e p o s i t i o n and t h e volume computed. In the same way, the u t i l i s a b l e volume t o any d e s i r e d t o p l i m i t s on the stem can be a s c e r t a i n e d . U s i n g t h i s method, an a s s o r t m e n t o f volume can be c a r r i e d out f o r each t r e e , t h a t i s , a f o r e c a s t o f t h e volume m a t e r i a l by number o f l o g s o f v a r y i n g s i z e s from a f o r e s t s t a n d can be determined, .. Such i n f o r m a t i o n would g r e a t l y a s s i s t i n u t i l i z a t i o n w i t h r e g a r d t o m i l l d e s i g n s , s a l e s p l a n n i n g , s i l v i c u l t u r a l s t u d i e s p e r t a i n i n g t o t h e o u t t u r n from d i f f e r e n t i a l p l a n t i n g s p a c i n g s , t h i n n i n g regimes, e t c . , as w e l l as u t i l i z a t i o n s t a n d a r d s depending on market c o n d i t i o n s and commercial r e q u i r e m e n t s p r e v a i l i n g a t a g i v e n , p a r t i c u l a r p e r i o d . 6. C o m p u t a t i o n a l d e t a i l s 6.1 Subset o f s i g n i f i c a n t v a r i a b l e s From e x p e r i e n c e g a i n e d i n p r e v i o u s s t u d i e s on t h e stem p r o f i l e o f c o n i f e r s (Sandrasegaran, 1973), s i x independent v a r i a b l e s i n c l u d i n g 2 2 s e l e c t e d c o m b i n a t i o n s , D , D , H, D , D t - ' , H a n d ® A t- H were con-s i d e r e d u s i n g t h e a l l c o m b i n a t i o n t e c h n i q u e , f o r r e g r e s s i n g each o f the dependent v a r i a b l e s a_ and b on t h e above measured t r e e growth p a r a m e t e r s . 2 As a p r e l i m i n a r y g u i d e , the average o f the r e s i d u a l mean square, S , f o r s e t s o f m number o f groups o f i n dependent v a r i a b l e s was computed and a 19 2 s c a t t e r g r a m o f S on m was drawn t o determine t h e b e s t c u t - o f f p o i n t . F o r a_, t h e c u t - o f f p o i n t was 4 independent v a r i a b l e s , beyond which, the graph was more o r l e s s h o r i z o n t a l t o t h e X - a x i s ( F i g u r e 4 ) . In t h e p l o t o f 2 S on m f o r b, t h e c u t - o f f was 3 independent v a r i a b l e s ( F i g u r e 5 ) . The p r a c t i c a l r o l e o f t h e independent v a r i a b l e s , t h e knowledge and n a t u r e o f the e x p e r i m e n t a l m a t e r i a l and t h e magnitude o f t h e s t a n d a r d e r r o r o f e s t i m a t e were s t u d i e d t o g e t h e r t o d e c i d e on t h e f i n a l i s e d form o f the r e q u i r e d r e g r e s s i o n s f o r a_ and b. 6.2 a e x p r e s s e d as a f u n c t i o n o f t r e e growth parameters In t h e p r e d i c t i o n o f a_ as a f u n c t i o n o f D and H, t h a t i s , 2 a = f (D. _ D__, H), i t was o b s e r v e d , t h a t b o t h t h e i n c r e a s e o f R and t h e 4. o, 10 d e c r e a s e o f t h e s t a n d a r d e r r o r o f e s t i m a t e , (SEE), were minor, beyond f o u r i ndependent v a r i a b l e s . Hence t h e a c c e p t a b l e model i s a = 0.2918 -0.107651D, r + 0.081758D n„ + 0.007029D 2, r - 0.000004D2, _ H, w i t h a SEE o f 4.5 10 4.5 4.5 2 0.00832, and an R o f 0.9875, t h a t i s , 98.75 p e r c e n t o f t h e v a r i a t i o n i s e x p l a i n e d by t h e r e g r e s s i o n . F o r 3 independent v a r i a b l e s , SEE was 0.00846 2 w i t h an R o f 0.9773 and f o r 5 independent v a r i a b l e s , SEE was 0,00829 and 2 an R o f 0.9877, hence t h e c h o i c e o f t h e s u b s e t o f f o u r v a r i a b l e s . 6.3 b e x p r e s s e d as a f u n c t i o n o f t r e e growth p a r a m e t e r s In d e v e l o p i n g t h e model, b = f (D^ H) , t h e r e l a t i v e v a l u e s o f the mean squared r e s i d u a l and t h e m u l t i p l e c o e f f i c i e n t o f v a r i a t i o n . "2 i n d i c a t e d t h a t t h e model t o be s e l e c t e d was b = 0.009245 + 0.000067D 4 5 2 2 - 0.000017D,, H + 0.000001D. _ H, w i t h a SEE and R o f 0.000782 and 0,6812 4.5 4.5 2 r e s p e c t i v e l y . F o r two independent v a r i a b l e s , SEE and R were r e s p e c t i v e l y 0.000809 and 0.6436 and f o r f o u r independent v a r i a b l e s , SEE was 0.000778 2 w i t h an R o f 0.6819. Hence t h e s u b s e t o f t h r e e independent v a r i a b l e s was 20 s e l e c t e d . Thus b which i s a measure o f t h e r a t e o f . t a p e r , v a r i e s d i r e c t l y as the square o f t h e d i a m e t e r a t b r e a s t h e i g h t , t h e p r o d u c t o f 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 as w e l l as the p r o d u c t o f the square o f 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 . 6.4 P e r c e n t a g e u t i l i s a b l e volume From t h e f o r m u l a o u t l i n e d i n 5.1, t h a t i s , V D i p 100, t h e p e r c e n t a g e u t i l i s a b l e volume t o v a r i o u s nominated diameters, o u t s i d e b a r k l i m i t s f o r v a r i o u s o n e - i n c h D^ c l a s s e s can be c o m p i l e d and used f o r the e s t i m a t i o n o f the p r o p o r t i o n o f u t i l i s a b l e volume o f wood a v a i l a b l e i f the upper stem l i m i t i s s p e c i f i e d depending on t h e u t i l i z a t i o n s t a n d a r d s . S i m i l a r l y t h e p e r c e n t a g e u t i l i s a b l e volume t o h^ i s dete r m i n e d from = 100 bh. _ h i p l 6.5 Computation o f a_, b and Fv From t h e models o u t l i n e d i n 6.2 and 6.3, a s e t o f v a l u e s o f <i and b can be computed by o n e - i n c h D c l a s s e s and p l o t t e d o v e r t o t a l h e i g h t — 4.5 ( F i g u r e s 1 and 2) t o deduce v a r i o u s r e l a t i o n s on t h e r a t e o f t a p e r , and hence the s t a n d i n g volumes o f t r e e s can be det e r m i n e d . A l s o t h e form f a c t o r f o r t o t a l volume (Fv) f o r each o n e - i n c h D„ r. c l a s s i s d e t e r m i n e d f r o m 4.5 2 a 2b 0 — . — .005454 x D x H and p l o t t e d i n F i g u r e 3 f o r purposes o f r e f e r e n c e . I n f e r e n c e s on t a p e r v a r i a t i o n and c h a r a c t e r i s t i c f e a t u r e s o f a_, b and Fv a r e d e s c r i b e d i n S e c t i o n s 7.1 and 7.2. 21 6.6 The advantages o f a l l c o m b i n a t i o n p r o c e d u r e A l l c o m b i n a t i o n p r o c e d u r e e x p l o r e s a l l p o s s i b l e l i n e a r c o m b i n a t i o n s o f n_ independent v a r i a b l e s , t h a t i s , i f t h e r e a r e n independent v a r i a b l e s , t h i s t e c h n i q u e examines ( 2 n - 1) p o s s i b l e c o m b i n a t i o n s , and hence t h e r e i s no n e c e s s i t y o f p r e f i x i n g c o m b i n a t i o n s , c r o s s - p r o d u c t s , r a t i o s , e t c . o f v a r i a b l e s i n a model. Thus a l l p o s s i b l e c o m b i n a t i o n s can be i n v e s t i g a t e d s e p a r a t e l y t o a s c e r t a i n the " b e s t " s e t o f n v a r i a b l e s , as w e l l as, any o t h e r c o m b i n a t i o n s u g g e s t e d by t h e o r e t i c a l o r p r a c t i c a l c o n s i d e r a t i o n s . O f t e n a s i m p l e model can be chosen, i n v o l v i n g j u s t a few v a r i a b l e s w i t h a min i m a l s a c r i f i c e o f the " e x p l a i n e d " v a r i a t i o n o f t h e dependent v a r i a b l e , as i t appears t h a t the r e l a t i v e mean squared r e s i d u a l t e n ds t o d e c r e a s e a s y m p t o t i c a l l y w i t h an i n c r e a s e i n t h e number o f v a r i a b l e s (Rudra and F i l m e r , 1970). The c r i t e r i o n used i n c o m b i n a t i o n a l s c r e e n i n g i s t h e magnitude o f the r e l a t i v e mean squared r e s i d u a l which i s a f u n c t i o n o f the number o f degrees o f freedom. The computer programme used i n t h i s s t u d y i s t h e "Rex Programme" and coded i n F o r t r a n IV language (Grosenbaugh, 1967) and p r o v i d e s the o p t i o n o f f i x i n g o r f o r c i n g v a r i a b l e s and combining them i n t o s e t s o f groups as t h i s i s u s e f u l i n k e e p i n g dimensions w i t h i n r e a s o n a b l e bounds (Rudra and F i l m e r , 1970). T h i s " a l l c o m b i n a t i o n " p r o c e d u r e may be p r a c t i c a l up t o 10 independent v a r i a b l e s (Kozak, 1969). 7. D i s c u s s i o n o f a n a l y s i s 7.1 a_ on H and b on H by o n e - i n c h D^ ^ c l a s s e s The p l o t s o f a on H and b on H a r e p r e s e n t e d i n F i g u r e s 1 and 2, r e s p e c t i v e l y . The p a t t e r n o f t a p e r v a r i a t i o n i s apparent from t h e two graphs and c o n s e q u e n t l y t h e r e i s a " f a n n i n g o u t " o f the l i n e s as the'.D c l a s s e s i n c r e a s e i n F i g u r e 2. By c o n s t r u c t i n g t a b l e s o f a and b, the stem p r o f i l e o r t a p e r l i n e o f any t r e e c o u l d be d e r i v e d and c o n s e q u e n t l y t h e 22 s t a n d i n g volume o f the t r e e s i n a s t a n d . As a_ i n c l u d e d the v a r i a b l e D 1 Q , t h e v a l u e o f was o b t a i n e d from the r e g r e s s i o n , D = -3.6124 + 0.7589D 4 5 + 2.1184£n T>4 g (R 2 = 0.9655, SEE = 0.0589). 7.2 Form f a c t o r f o r t o t a l volume (F ) by o n e - i n c h D„ ,_ c l a s s e s v 4.5 U s i n g t h e f o r m u l a o u t l i n e d i n 6.5, F was c a l c u l a t e d and graphed v by o n e - i n c h D c l a s s e s , and p r e s e n t e d i n F i g u r e 3. I t i s seen t h a t F f t * 3 V i n c r e a s e s from 10 t o a p p r o x i m a t e l y 12 i n c h e s t i l l a maximum o f 0.4479 i s r e a c h e d and s u b s e q u e n t l y d e c r e a s e s p r o g r e s s i v e l y t o a v a l u e o f 0.2989 a t the 2 4 - i n c h D c l a s s , i n f e r r i n g t h a t t h e r e i s l e s s t a p e r up t o 12 i n c h e s f t • 3 D. and t h e r e a f t e r i t i n c r e a s e s t i l l 24 i n c h e s D . Hence t h e d a t a c o n f i r m t h a t t h e s m a l l e r stems a r e more c y l i n d r i c a l t h a n b i g g e r stems. 7.3 Check on assumption As a check on th e assumption t h a t the u t i l i s a b l e volumes are s u f f i c i e n t l y c l o s e a p p r o x i m a t i o n s t o t h e t r u e u t i l i s a b l e volumes up t o 5 i n c h e s t o p d i a m e t e r , g r a p h i c a l a n a l y s e s were under t a k e n . One t r e e was randomly s e l e c t e d from each o f th e c l a s s group, 15 t r e e s i n a l l , and f o r each i n d i v i d u a l t r e e , t h e d i f f e r i n g nominated h e i g h t s a l o n g the stem were p l o t t e d a g a i n s t t h e r e s p e c t i v e s e c t i o n a l a r e a s o u t s i d e b a r k . These p o i n t s were j o i n e d t o g e t h e r by a s t r a i g h t l i n e and i t s r e s p e c t i v e l i n e , by t h e method o f l e a s t squares was t h e n superimposed on i t , t o compare the u t i l i s a b l e volume e s t i m a t e d by t h e t a p e r l i n e method, w i t h t h a t by a c t u a l p l o t t i n g . I t was found t h a t the u t i l i s a b l e volume o u t s i d e b a r k o v e r -e s t i m a t e d o r u n d e r - e s t i m a t e d by t h e r e g r e s s i o n l i n e l i e s w i t h i n t h e l i m i t o f - 3.8 p e r c e n t o f i t s t r u e volume, which i s more than a c c e p t a b l e f o r p r a c t i c a l p u r p o s e s . 23 7.4 T e s t s o f b i a s and s t a n d a r d e r r o r o f e s t i m a t e o f merchantable volume e s t i m a t e d by t h e volume and t a p e r models up t o 5 i n c h e s top d i a m e t e r i n s i d e b a r k by d i a m e t e r s i z e c l a s s e s  The b i a s and s t a n d a r d e r r o r between the merchantable volume e s t i m a t e d by the volume model (Sandrasegaran, 1973) and the t a p e r e q u a t i o n i n t h i s s t u d y was computed f o r th e sample t r e e s and p r e s e n t e d i n T a b l e 3. In each s i z e c l a s s t h e mean b i a s i s n o t s u b s t a n t i a l and t h e o v e r a l l b i a s o f 0.16 c u b i c f e e t ( t h a t i s , 0.13 p e r c e n t o f merchantable volume by t a p e r model) i s n e g l i g i b l e . The v a r i a t i o n i n the p a t t e r n o f b i a s i s a l s o v e r y s m a l l from one s i z e c l a s s t o a n o t h e r . Of c o u r s e , a c o m p a t i b l e e s t i m a t i n g system where i d e n t i c a l volumes are d e r i v e d from b o t h the volume and t a p e r e q u a t i o n s i s p r e f e r r e d (Munro and Demaerschalk, 1974), y e t f o r p r a c t i c a l p u r p oses i n e s t i m a t i n g merchantable volume f o r a p l a n t a t i o n c r o p o f Eucalyptus vobusta, t h i s n e g l i g i b l e d i s c r e p a n c y i s a c c e p t a b l e f o r p r a c t i c a l p u r p o s e s , and hence the use o f t h e t a p e r model i s j u s t i f i a b l e . The square o f SEE , t h a t i s , t h e mean square e r r o r , comprises o f two components, one c i s t h e p u r e e r r o r o r t h e v a r i a n c e which i s a measure o f the v a r i a t i o n o f t h e d a t a and the o t h e r i s the l a c k o f f i t which i s a measure o f the square o f t h e b i a s . Thus the d i f f e r e n c e i n SEE from one c l a s s t o a n o t h e r can be - c_ a t t r i b u t e d t o t h e d i f f e r e n c e i n t h e average b i a s A B ^ 7.5 Diameter e s t i m a t i o n t e s t s f o r b i a s and s t a n d a r d e r r o r f o r measured h e i g h t s from 10 t o 90 f e e t ' ^ _ The d i a m t e r s d e r i v e d by t h e t a p e r e q u a t i o n f o r h e i g h t s 10 t o 90 f e e t were compared w i t h t h e o b s e r v e d d i a m e t e r s i n terms o f b i a s i n i n c h e s and s t a n d a r d e r r o r by d i a m e t e r c l a s s e s . The r e s u l t s a r e p r e s e n t e d i n T a b l e 4. The p a t t e r n o f v a r i a t i o n i n b i a s i s more o r l e s s c o n s i s t e n t between s i z e c l a s s e s and t h e range o f v a r i o u s h e i g h t s c o n s i d e r e d . A l s o t h e r e i s no s u b s t a n t i a l b i a s i n each s i z e c l a s s . The o v e r a l l 24 b i a s f o r each o f t h e h e i g h t s i s a l s o c o n s t a n t . The s t a n d a r d e r r o r s a r e low f o r t h e s m a l l e s t s i z e c l a s s e s and l a r g e f o r t h e l a r g e s i z e c l a s s e s , b u t i n terms o f magnitude, can be c o n s i d e r e d as r e l a t i v e l y i n s i g n i f i c a n t . 7.6 B i a s i n h e i g h t e s t i m a t i o n by o n e - i n c h c l a s s e s The b i a s e s i n f e e t r e s u l t i n g from t h e d i f f e r e n c e o f t h e e s t i m a t e d h e i g h t s by t h e t a p e r model and the o b s e r v e d h e i g h t s a r e t a b u l a t e d i n T a b l e 5. A p e r u s a l b r i n g s o u t two s a l i e n t p o i n t s , namely t h a t d i a m e t e r b i a s (from T a b l e 5) and h e i g h t b i a s appear t o be o f the o p p o s i t e s i g n and a b i a s o f 0.1 i n c h i n d i a m e t e r i s e q u i v a l e n t t o a b i a s o f a p p r o x i m a t e l y one f o o t i n h e i g h t . T h i s s t a t e o f o p p o s i t e s i g n s i n t h e b i a s e s o f d i a m e t e r and h e i g h t would p r o b a b l y n e u t r a l i s e t h e e f f e c t s when s e c t i o n a l volume i s d e t e r -mined. The o v e r a l l b i a s i s more o r l e s s c o n s i s t e n t . 8. Summary o f formulae t o be used i n p r a c t i c a l a p p l i c a t i o n In p r a c t i c a l a p p l i c a t i o n s o f e s t i m a t i n g t h e s t a n d i n g volume o f t r e e s o f Eucalyptus vobusta, o v e r the range o f 10 t o 24 i n c h e s t h e f o l l o w i n g models a r e t o be used: 1. a = 0.2918 - 0.107651D, _ + 0/081758D, r t 4.5 10 + 0.007209D 2„ r - 0.000004D 2 CH 4.5 4.5 2. b = 0.009245 + 0.000067D 2 _ - 0.000017D,, _H 4.5 4.5 + 0.000001D 2 CH 4.5 2 3. T o t a l t r e e volume; V T = a /2b h. 4. Volume t o h e i g h t i ; V, = (2a - bh.) n. z l l a 2 - a 2 . 2 5. Volume a t d i a m e t e r i ; V = • 1 where g. = 0.005454D . D i 2b 1 1 25 6. P e r c e n t a g e o f u t i l i s a b l e wood volume t o h e i g h t i ; 100 bh. V. h. 2 rp a 2a - bh. l 7. P e r c e n t a g e o f u t i l i s a b l e wood volume a t d i a m e t e r i ; v:;. = 100 Dip a - g. H e i g h t t o d i a m e t e r i ; h ^ = — - — -Diameter a t h e i g h t i ; g. = a - bh. where D. = 13.54 g.' The above e x p r e s s i o n s would p r e d i c t t r e e and s t a n d volumes t o any demanded h e i g h t o r t o p d i a m e t e r , h e i g h t f o r any d i a m e t e r o r d i a m e t e r a t any h e i g h t . The f i e l d measurements r e q u i r e d a r e D^ ,_, D ^ and H. I t i s r e a s o n a b l e t o assume, based on a priori knowledge t h a t the independent v a r i a b l e , D^ 0, w i l l overcome t h e v a r i a b i l i t y i n t h e measurement o f d i a m e t e r s a t b r e a s t h e i g h t , p o s s i b l y due t o b u t t - s w e l l i n g a t t h i s p o i n t , i n t h e o l d e r t r e e s . 26 9. C o n c l u s i o n s In t h i s s t u d y , i t has been shown t h a t t h e stem p r o f i l e o f p l a n t a t i o n t r e e s o f Eucalyptus vobusta c o u l d be a d e q u a t e l y d e s c r i b e d by t h e r e l a t i o n , g = a - bh where g_ i s t h e s e c t i o n a l a r e a o f the t r e e b o l e i n square f e e t a t s e c t i o n a l h e i g h t h f e e t , a_ i s t h e r e g r e s s i o n c o n s t a n t and b t h e r e g r e s s i o n c o e f f i c i e n t . A measure o f t a p e r b,„ i s r e l a t e d t o v a r i o u s t r e e growth p a r a m e t e r s and a r e g r e s s i o n i s d e v e l o p e d o f t h o s e s i g n i f i c a n t f a c t o r s which a r e r e l a t e d t o t a p e r v a r i a t i o n . S i m i l a r l y t h e r e l a t i o n o f a_ and d i f f e r i n g growth para m e t e r s a r e a l s o d e t e r m i n e d . As a r e s u l t o f t h e s e , i t was found t h a t ( i ) b v a r i e s d i r e c t l y as t h e square o f th e d i a m e t e r a t b r e a s t h e i g h t , t h e p r o d u c t o f t h e 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 , as w e l l as t h e p r o d u c t o f t h e square o f t h e di 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 and ( i i ) a_ v a r i e s as the d i a m e t e r a t b r e a s t h e i g h t as w e l l as a t 10 f e e t h i g h , t h e p r o d u c t o f t h e square o f t h e diameter 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 , and t h e square o f t h e d i a m e t e r a t b r e a s t h e i g h t . The s e c t i o n a l a r e a , g^, can be dete r m i n e d from the r e l a t i o n g^ = a - bh., when h. i s known and s u b s e q u e n t l y e q u i v a l e n t t o g . can be r e a d o f f from t h e p r e - p r e p a r e d d i a m e t e r - s e c t i o n a l a r e a t a b l e s . S i m i l a r l y knowing D^, g^ , can be r e a d o f f t h e p r e - p r e p a r e d s e c t i o n a l a r e a t a b l e s and h. a t D. can be computed from t h e e q u a t i o n , tu = (a - g^)/b. The volume a t 2 2 di a m e t e r , D^, i s d e t e r m i n e d from = (a - g J / 2 b and t h e volume t o h i , — i i s o b t a i n e d from t h e r e l a t i o n , = ( h i / 2 ) ( 2 a - b h . ) . The t o t a l volume n. l 2 i s a /2b. The p e r c e n t a g e o f u t i l i s a b l e wood volume a t i s 100 and t o h. i s I 27 The p a t t e r n o f t a p e r v a r i a t i o n i s apparent from the graphs o f a on H and k^on H by ,- c l a s s e s , where a h i g h e r v a l u e b i s a s s o c i a t e d w i t h a g r e a t e r r a t e o f t a p e r and a tendency o f the l i n e s b e i n g "fanned o u t " w i t h an i n c r e a s e i n D. c l a s s e s . A p l o t o f form f a c t o r f o r t o t a l volume 4 . 0 by D, _ c l a s s e s showed t h a t the l e a s t t a p e r e d t r e e , which was a maximum 4 . 0 "~ v a l u e o f 0.4479 f o r F , was l o c a t e d a t t h e 12 i n c h D„ _ c l a s s . The stems v 4.5 a r e more c y l i n d r i c a l i n s m a l l e r t r e e s (D c l a s s e s from 10 t o 12 i n c h e s ) t h a n i n l a r g e r t r e e s (D c l a s s e s from 12 t o 24 i n c h e s ) . The mean b i a s i n e s t i m a t i n g t h e m erchantable volume t o 5 i n c h e s top d i a m e t e r i n s i d e b a r k i s n o t s u b s t a n t i a l i n each s i z e c l a s s , and the o v e r a l l b i a s f o r t h e sample t r e e s i s 0.16 c u b i c f e e t , a n e g l i g i b l e amount. The p a t t e r n o f v a r i a t i o n i n b i a s between the s i z e c l a s s e s i s a l s o s m a l l . The same i s t r u e f o r d i a m e t e r e s t i m a t i o n a t s p e c i f i c h e i g h t s where t h e v a r i a t i o n i n b i a s i s s m a l l and c o n s i s t e n t and t h e b i a s i s not s u b s t a n t i a l i n each s i z e c l a s s . The b i a s between o b s e r v e d and e s t i m a t e d h e i g h t s i s i n o p p o s i t e s i g n t o t h a t o f d i a m e t e r b i a s . The o v e r a l l b i a s i n h e i g h t e s t i m a t i o n i s more o r l e s s c o n s i s t e n t . F i n a l l y t h i s s t u d y shows f o r p l a n t a t i o n grown t r e e s o f Eucalyptus robusta o v e r the range o f s i z e c l a s s e s i n v e s t i g a t e d : the model g = a - bh a d e q u a t e l y d e s c r i b e s the stem p r o f i l e and can be used t o e s t i m a t e t h e s e c t i o n a l h e i g h t a t a g i v e n d i a m e t e r o r v i c e - v e r s a ; t h e 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 * H can be d e r i v e d from a knowledge o f t h e v a l u e s o f a_ and b and c o n s e q u e n t l y the s t a n d i n g t r e e volumes a t nominated s e c t i o n a l d i a m e t e r s o r t o d e s i r e d s e c t i o n a l h e i g h t s u s i n g the r e s p e c t i v e formulae i n S e c t i o n 8. 28 LITERATURE CITED Anderson, J.W. 1912. Gardens B u l l e t i n , S t r a i t s "Times "and F e d e r a t e d - — -'Malay S t a t e s . V o l . 1, 145-150. Anon. 1927. 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J o u r n a l o f F o r e s t r y , 67, 126-128. F.A.0. 1957. Tree p l a n t i n g p r a c t i c e s i n t r o p i c a l A s i a . F o r e s t r y D e v e l o p -ment Paper No. 11. F l i g g , D.M. and R.E. Breadon. 1959. Log p o s i t i o n volume t a b l e s . F o r . Surv. Note No. 4, B r i t i s h Columbia F o r e s t S e r v i c e . F o r e s t r y and Timber Bureau, C a n b e r r a . 1953. The n a t u r a l o c c u r r e n c e o f t h e E u c a l y p t s . L e a f l e t No. 65., 26 pp. F o r e s t r y and Timber Bureau. i957.;:"Forest t r e e s o f A u s t r a l i a , C a n b e r r a , 28pp. F r e e s e , F. 1964. L i n e a r r e g r e s s i o n models f o r f o r e s t r e s e a r c h , U.S. F o r e s t S e r v i c e R e s e a r c h Paper No. FPL 17, U.S. Department o f A g r i c u l t u r e , 136 pp. 30 F r e e z a i l l a h b i n Che Yeom, K, Sandrasegaran and S,S, Singham, 1966, Permanent sample p l o t i n f o r m a t i o n on t h e s t o c k i n g , growth and y i e l d o f Eucalyptus robusta Sm, grown i n Cameron H i g h l a n d s , Malaya. Malayan R e s e a r c h Pamphlet No. 48, Forest Research: I n s t i t u t e , Kepong, M a l a y s i a , , 38 pp. F r i e s , J . 1965. E i g e n v e c t o r a n a l y s e s show t h a t b i r c h and p i n e have s i m i l a r form i n Sweden and B r i t i s h Columbia, F o r e s t C h r o n i c l e , 4 1 ( 1 ) , 135-=-139. F r i e s , J . and B. Matern. 1965. On the use o f m u l t i v a r i a t e methods f o r t h e c o n s t r u c t i o n o f t r e e t a p e r c u r v e s , P a p e r No. 9 p r e s e n t e d a t the c o n f e r e n c e o f IUFR0, s e c t i o n 25, Stockholm, 32 pp, F u r n i v a l , G.M. 1961. An i n d e x f o r comparing e q u a t i o n s used i n c o n s t r u c t i n g volume t a b l e s . F o r e s t S c i e n c e , 7 ( 4 ) , 337-341. G e r r a r d , D.J. 1966. The c o n s t r u c t i o n o f s t a n d a r d t r e e volume t a b l e s by we i g h t e d m u l t i p l e r e g r e s s i o n . U n i v e r s i t y o f T o r o n t o , F a c u l t y o f F o r e s t r y , T e c h n i c a l R e p o r t No. 6, 35 p . A l s o as Department o f Lands and F o r e s t s R e s e a r c h Report No. 61, O n t a r i o , Canada. G u i r g i u , V. 1963. 0 metoda a n a l i t i c a do i n t o c m i r e a t a b e l e o r d e n d r o m e t r i c e l a c a l c u l a t o a r e l e e l e c t r o m i c e (An a n a l y t i c a l method o f c o n s t r u c t i n g d e n d r o m e t r i c a l t a b l e s w i t h t h e a i d o f e l e c t r o n i c computers), Rev. P a d u r i l o r 7 8 ( 7 ) , 369-374. Gray, H.R. 1956. The form and t a p e r o f f o r e s t t r e e stems. I n s t i t u t e Paper No. 32, I m p e r i a l F o r e s t r y I n s t i t u t e , O x f o r d . 79 pp. Greaves, A. 1971. An i n t e r i m r e p o r t on t h e s i t e s t u d i e s and assessment o f t h e a s s o c i a t e d growth and p r o d u c t i v i t y o f p l a n t a t i o n s o f Gmelina a v b o r e a i n the mid-western s t a t e o f N i g e r i a . B u l l e t i n o f the N i g e r i a n F o r e s t r y Department, V o l . 31, No. 4, F e d e r a l Department,of F o r e s t r y , Ibadan, N i g e r i a , 10 pp. Gregory, R.A. and P.M. 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Volume t a b l e s f o r hemlock and S i t k a s p r u c e on t h e Ehugach N a t i o n a l F o r e s t , A l a s k a . USDA Research Note N0R-4, 4\>pp, 31 Heger, L. 1965. Morphogenesis o f stems o f D o u g l a s - f i r , U n i v e r s i t y o f o f B.C., F a c u l t y o f F o r e s t r y , Ph,D t t h e s i s , 176 pp, H e i j b e l , I . 1928. A system o f e q u a t i o n s f o r d e t e r m i n i n g stem form i n p i n e Svensk S k o g s v f o r e n . T i d s k r . , 3-4, 393-422. H e j j a s , J . 1967. Comparison o f a b s o l u t e and r e l a t i v e s t a n d a r d e r r o r s and e s t i m a t e s o f t r e e volumes. U n i v e r s i t y o f B r i t i s h Columbia, F a c u l t y o f F o r e s t r y , M.F. t h e s i s , 58 pp. H o j e r , A.G. 1903. T a l l e n s och granens t i l l v a x t . B i h a n g t i l l F r . Loven. Om v a r a b a r r s k o g a r , Stockholm. (Quoted by Jonson and s u b s e q u e n t l y t o o by B e h r e ) . Honer, T.G. 1964. 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Optimum p l a n t i n g d i s t a n c e s and c r o p d e n s i t i e s o f t h e t e n e x o t i c s p e c i e s i n Malaya u t i l i s i n g t r i a n g u l a r s p a c i n g based on a c o n s i d e r a t i o n o f crown d i a m e t e r t o stem d i a m e t e r r e l a t i o n s h i p s . Malayan R e s e a r c h Pamphlet No. 51, 44 pp. 34 Sandrasegaran, K. 1 9 6 6 i i i . Q u a n t i t a t i v e assessment o f v a r i o u s crown parameters o f s i n g l e t r e e s . Malayan F o r e s t e r XXIX, 20-27. Paper p r e s e n t e d a t the P a n - M a l a y s i a n F o r e s t r y C o n f e r e n c e h e l d i n September 1966. . 1968. A g e n e r a l volume t a b l e f o r Pinus cavibaea Mor. Malayan F o r e s t e r XXXI, 20-27. . 1969i. A g e n e r a l volume t a b l e f o r Tectona grandis L i n n . f (teak) grown i n North-West Malaya. Malayan F o r e s t e r XXXII, 187-200. . 1 9 6 9 i i . I n t r o d u c t i o n t o t h e use o f t h e FP12 B a r r and S t r o u d Dendrometer. Malayan F o r e s t e r XXXII, 279-286. . 1970i. A s t a n d a r d volume t a b l e f o r Pinus mevkusii Jungh and de Vriese grown i n the F o r e s t R e s e a r c h I n s t i t u t e p l a n t a t i o n s i n Malaya. Malayan F o r e s t e r XXXIII, 80-91. . 1 9 7 0 i i . Perbandingan s u k a t a n pokok-pokok yang d i t e b a n d dengan t a k s i r a n k e t i n g g i a n n y a . (In Malay language) Comparison o f measured and e s t i m a t e d h e i g h t o f f e l l e d t r e e s . Rimba Muda, J i l i d V, B i l a n g a n 4, 126-130. . 1971. S t o c k i n g and y i e l d o f tiup-tiup f o r e s t (Adinandra dumosa Jack) i n J o h o r e , Malaya. Malayan Research Pamphlet No. 60, 55 pages. F o r e s t R e s e a r c h I n s t i t u t e , Kepong, P e n i n s u l a r M a l a y s i a . 1972. A g e n e r a l volume t a b l e f o r Khizophova apiculata B l . (Syn. R. 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Paper p r e s e n t e d a t t h e A n n u a l M e e t i n g o f t h e Northwest S c i e n t i f i c A s s o c i a t i o n , U n i v e r s i t y o f Idaho i n A p r i l . Mimeo, 8 pp. S p i e d e l , G. 1957. D i e r e c h n e r i s c h e n g r u n d l a g e n d e r l e i s t u n g s k o n t r o l l e und h i r e p r a k t i s c h e d u r c h f u r u n g i n der f o r e s t e i n r i c h t a u n g . S c h r i f t e n r e i h e d e r F o r s t l i c h e n F a k u l t a t , U n i v e r s i t a t G o t t i n g e n , No. 19, 118 pp. S p u r r , S.H. 1952. F o r e s t I n v e n t o r y , The Ronald P r e s s Company, New York, 476 pp. S t r e e t s , R.J. 1962. E x o t i c t r e e s i n t h e B r i t i s h Commonwealth, . C l a r e n d o n P r e s s , O x f o r d , 315-320. Troup, R.S. 1921. The s i l v i c u l t u r e o f I n d i a n t r e e s , C l a r e n d o n P r e s s , Oxford, ,212 pp. Van L a a r , A. 1968. 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J o u r n a l o f F o r e s t r y , 21, 564-581. Young, H.E., W.C. Robbins and W. Sue. 1967. E r r o r s i n volume d e t e r m i n a t i o n o f p r i m a r y f o r e s t r y p r o d u c t s . P r o c e e d i n g s o f th e XIV IUFRO Congress, S e c t i o n 25, Volume V I , 546-562. 37 TABLE I . Range o f b a s i c d a t a s t r a t i f i e d by o n e - i n c h D c l a s s e s D4.5 c l a s s e s i n i n c h e s Frequency o f t r e e s D 4 . _ i n i n c h e s b H i n f e e t Lowest H i g h e s t Mean • Lowest H e i g h t s Mean 10 14 10.2 10.6 10. 3 73 91 81 11 17 11.1 11.8 11.4 78 96 86 12 9 12.2 12.6 12.3 83 99 90 13 12 13.3 13.6 13.4 87 104 96 14 14 14.2 14.7 14.5 89 109 100 15 16 15.3 15.7 15.6 93 113 102 is 17 16.2 16.8 16.5 96 116 104 17 17 17.4 17.8 17.6 99 120 106 18 19 18.3 18.9 18.7 100 121 108 19 16 19.5 19.8 19.6 101 123 110' 20 13 20.4 20.7 20.6 103 125 113 21 4 21.4 21.9 21.7 104 127 115 22 4 22.3 22.7 22.6 106 129 117 23 3 23.4 23.8 23.5 107 131 120 24 3 24.3 24.8 24.7 110 132 122 N.B. The v a l u e s were s e l e c t e d c l o s e t o t h e mean, from 890 t r e e s . 38 TABLE I I . C l i m a t e - l o g i c a l d a t a Mean Temperature ( ° F ) + Month * i R a i n f a l l i n i n c h e s Maximum \ Minimum J anuary 6.29 70.9 55.2 F e b r u a r y 5.04 72.7 53.9 March 8.60 73.9 54. 3 A p r i l 12.32 74.3 56.3 May 9.59 • 74.2 57.1 June 5.57 73.4 55.4 J u l y 5.09 73.3 55.4 August 6.90 72.8 55.6 September 10.15 73.2 55.4 October 13.30 72.0 56.2 November •13.20 72.1 56.8 December 9.45 71.0 56.2 Summation/ 105.50 72.8 55.7 Mean 1927 - 1957 — r e c o r d s from H y d r o l o g i c a l Data, D r a i n a g e and I r r i g a t i o n Department, F e d e r a t i o n o f Malaya. +1954 - 1963 — r e c o r d s from the F e d e r a l E x p e r i m e n t a l S t a t i o n a t Cameron H i g h l a n d , F e d e r a t i o n o f Malaya. 39 TABLE I I I . M e r c h a n t a b l e volume e s t i m a t i o n t e s t s o f b i a s and s t a n d a r d e r r o r f o r sample t r e e s ' i n the"10.to 20 i n c h e s s i z e c l a s s D. n I n c h 4.5 C l a s s e s Average B i a s (AB ) c S t a n d a r d E r r o r o f E s t i m a t e (SEE ) c 10 o.oi 0.12 11 0.04 0.. 12 12 0.05 0.14 13 0.02 0.15 14 0.03 0.16 15 - 0.03 0.18 16 - 0.04 0. 20 17 - 0.06 0.21 18 0.05 0.15 19 0.06 0.16 20 0.07 0.19 O v e r a l l 0.16 = AB 0.13 = S E E Q n A B C = i = l ( VMR " V ) W h e r S V V t h S m e r c h a n t a b l e n volumes up t o 5 i n c h e s top d i a m e t e r e s t i m a t e d by the volume and t a p e r e q u a t i o n s r e s p e c t i v e l y f o r each d i a m e t e r s i z e c l a s s and n i s t h e f r e q u e n c y o f t r e e s i n each s i z e c l a s s SEE = c n I i = l (V - V ) MR MT 2 / N s t a n d a r d e r r o r o f e s t i m a t e f o r each s i z e c l a s s . AB = I (V - V )/N, o v e r a l l measure o f b i a s ; N i s t h e t o t a l 0 i = l MR MT , ,. . number o f t r e e s . SEE = 0 n I 11=1 (V, MR v 2/ " V M T ) N o v e r a l l s t a n d a r d e r r o r o f e s t i m a t e i n terms o f m e r c h a n t a b l e volume. 40 TABLE IV. Diameter e s t i m a t i o n t e s t o f b i a s (AB ) and s t a n d a r d e r r o r c (SEE ) f o r measured s e c t i o n a l h e i g h t s from 10-90 f e e t S e c t i o n a l h e i g h t s i n f e e t °4.5 10 30 50 70 90 c l a s s e s i n i n c h e s AB c SEE c AB c SEE c AB c SEE c AB c SEE c AB c SEE c 10 0.1 0.4 0.2 0.3 -0.2 0.5 -0.1 0.4 -0.1 0.4 11 0.1 0.5 0.2 0.4 -0.3 0.4 -0.1 0.4 0.1 0.3 12 0.1 0.7 0.2 0. 6 -0.2 0.2 0.2 0.5 0.1 0.5 13 0.2 0.6 0. 3 0.8 -0.1 0.3 0.1 0.6 0.1 0.4 14 0.2 0.7 0.2 0.6 0.1 0.5 -0.3 0.5 0.2 0.6 15 0.4 0.8 0.3 0.9 0.1 0.7 -0.4 0.7 0.4 0.5 16 0.5 1.1 0.4 1.2 0.2 0.6 -0.2 0.8 0.3 0.7 17 0. 2 1.3 0.3 1.1 0. 3 0.8 0.3 0.7 0.2 0.8 18 0. 2 1.2 0. 3 1.4 0.2 0.5 0.4 1.1 0.4 0.9 19 0.1 1.3 0.4 1.2 0. 3 0.3 0.5 1.3 0.3 1.2 20 0.1 1.4 0.4 1.3 0.2 0.5 0.4 1.2 0.3 1.5 O v e r a l l 0. 3 0.8 0.3 0.9 0.2 0.6 0.3 0.7 0.3 0.5 where D. and D. a r e the o b s e r v e d and l l e s t i m a t e d s e c t i o n a l d i a m e t e r s i n i n c h e s a t t h e s p e c i f i e d h e i g h t s ; n i s t h e number o f t r e e s i n each s i z e c l a s s . h , s t a n d a r d e r r o r o f e s t i m a t e f o r each s i z e c l a s s . n AB = 1 (D.- D.)/n c . i x 1=1 SEE n Z i = l (D. D . ) 2 / n x 41 TABLE V. B i a s (AB ) i n s e c t i o n a l h e i g h t e s t i m a t i o n B i a s between measured and o b s e r v e d s e c t i o n a l h e i g h t s :.. i n i n c h e s 10 30 50 70 90 10 -0.9 -1.8 1.9 1.1 1.0 11 -1.1 -1.9 2.7 0.9 -0.8 12 -1 = 0 -1.8 2.2 -1.8 -0.9 13 -2.2 -2.8 1.1 -0.9 -1.1 14 -2.1 -2.1 -0.9 2.8 -1.9 15 -3.9 -2.9 -1.1 3.9 -3.8 16 -4.8 -3.8 -1.8 1.9 -2.9 17 -2.2 -2.9 -2.8 -2.9 -1.8 18 -2.1 -2.8 -1.9 -3.8 -4.1 19 -0.8 -3.8 -2.9 -4.8 -3.1 20 -0..9 -2.2 -2.2 • -3.9 -3.2 O v e r a l l -2.8 -2.9 -3.0 -3.2 -2.8 n AB = (h. - h . ) / n where i = l 1 1 h. and h. l _ i a r e t h e o b s e r v e d and e s t i m a t e d s e c t i o n a l h e i g h t s i n f e e t ; n i s the f r e q u e n c y o f t r e e s i n each s i z e c l a s s . 42 Figure! Definition of the taper line to describe the stem profile (Descriptions in sections 5.1 and 5.2) Sectional height in feet 1. g = a - bh where D = 13'54 g r 2. h = ( a - g ) / b where g = 0-005454 D 2 3. V D j = ( a 2 - g ? ) / 2 b 4. Vhj = ( h s / 2 ) ( 2 a - bhi) 5. V j = a 2 / 2 b , that is,4r [ the product of the parabolic base which is equal to a and parabolic height which is given by a / b j 4 3 Figure 2. Graph of a on H by one-inch classes 3 5 3-3 3-1 •9 2-7 H 2 5 2-3 - 21 5 1-9 o ID <D S* 1-5 cc 1-3 1-1 -0 9 -0 7 -0-5 70 23" - •22 ' ,21" 20" — 19' 18" -•17" 16' - 1 5 ' •14" 13" - 1 2 " - 1 1 " — i — 80 10" 90 100 110 Total height in feet (H) 120 130 140 Figure 3. Relation of b and H by one-inch D±5 classes c 0) 0 0 2 2 0-018 § 0-014 K •010 A 0 - 0 0 6 +-7 0 •16" •15" •14" •10" •13' .12" •11" 0-033 0-029 0-025 0-021 0-017 0-046 0-042 A 0-038 H 0-034 H •030 9 0 110 130 9 0 110 130 100 Total height in feet (H) 45 Figure 4. Relation between form factor for total volume (Fy) and one-inch D 4 5 classes 0-45 4 0-40 i I 0-35 o > 1 2 0-30 o u t 0-25 0-20 0*15 i 0'10 14 16 18 2 0 One inch D 4 5 classes 10 12 2 2 24 46 Figure 5. Scattergram of Average Residual Mean Square, s_2, on the number of independent variables for a 10 _ 4 x 7 i d 4 * 6 10 * 5 H II = , 0 x 4 (A « 1 0 _ 4 x 3 •o tn <D CC I 10" 4 *2 (a i d 4 x i o \ \ \ \ \ \ \ —r~ 3 Number of variables 47 Figure 6. Scattergram of Average Residual Mean Square, s_2, on the number of independent variables for b -6 i 10 * * \ T 1 1 1 1 -0 1 2 3 4 5 Number of variables = m 

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