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Some methods of sampling triangle based probability polygons for forestry applications Errico, Darrell 1981

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SOME METHODS OF SAMPLING TRIANGLE BASED PROBABILITY POLYGONS FOR B.S.F. UNIVERSITY OF BRITISH COLUMBIA, 1976 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the Department of F o r e s t r y We accept t h i s t h e s i s as conforming to the required standard FORESTRY APPLICATIONS by DARRELL ERRICO THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1981 D a r r e l l E r r i c o , 1981 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l l m e n t of the requirements f o r an advanced degree at the 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 f o r r e f e r e n c e and study. I f u r t h e r agree that permission f o r e xtensive 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 r e p r e s e n t a t i v e s . 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 w r i t t e n permission. Department of / ^ g ^ s T f ^ y The U n i v e r s i t y of B r i t i s h Columbia Vancouver, B. C. Canada Date &/-Q q ~ a */ PAGE i i ABSTRACT There i s i n t e r e s t i n f o r e s t sampling methods which have the a b i l i t y to provide r e l i a b l e estimates of volume without i n c u r r i n g unreasonable c o s t s . F r a s e r (1977), to t h i s end, d e s c r i b e d an i n d i v i d u a l tree v a r i a b l e p r o b a b i l i t y sampling method which s e l e c t s sample trees with p r o b a b i l i t i e s based on the areas of polygons d e r i v e d from t r i a n g l e s . A comparison of some a l t e r n a t i v e methods of sampling these polygons confirms F r a s e r ' s work and demonstrates that the method proposed by him probably has the g r e a t e s t p o t e n t i a l f o r p r a c t i c a l f o r e s t sampling. TABLE OF CONTENTS PAGE i i i Page LIST OF TABLES i v LIST OF FIGURES v Chapter INTRODUCTION 1 ONE LITERATURE REVIEW 3 TWO METHOD OF ANALYIS 8 THE BASIC METHOD 8 METHOD 1 . . . . 14 METHOD 2 17 METHOD 3 20 METHOD 4 23 METHOD OF COMPARISON 29 THREE RESULTS AND DISCUSSION 39 FOUR SUGGESTIONS FOR FURTHER WORK 49 LITERATURE CITED 54 PAGE i v LIST OF TABLES Table Page 1 Summary of the compilation of the and a^ f o r each method (used i n formulae (1) and (2)) . . 27 2 Summary of measures required per sample for each method 28 3 D e s c r i p t i o n of stands tested 31 4 C o e f f i c i e n t s of v a r i a t i o n (C) f o r each method using the D 2 a weight 41 5 Variance r a t i o s f or each method r e l a t i v e to the Basic Method 43 6 Comparison of average number of measurements required f o r each method to obtain the same p r e c i s i o n as the Basic Method . . . . . . 44 7 C o e f f i c i e n t s of v a r i a t i o n (C) f o r each method using the D 2Ha weight 48 PAGE v LIST OF FIGURES F i g u r e Page 1 Stem map showing l e a s t d i a g o n a l neighbours and polygon of the type used i n the B a s i c Method 10 2 Stem map showing a l e a s t d i a g o n a l neighbour t r i a n g l e and q u a d r i l a t e r a l of the type used i n Method 1 . 16 3 Stem map showing a l e a s t d i a g o n a l neighbour t r i a n g l e of the type used i n Method 2 . . . 1 9 4 Stem map showing l e a s t d i a g o n a l neighbours and one shaded l a r g e polygon of the type used i n Method 3 22 5 Stem map showing l e a s t d i a g o n a l neighbours and one shaded small polygon of the type used i n Method 4 26 6 Frequency histograms of areas of q u a d r i l a t e r a l s , t r i a n g l e s , small polygons, and l a r g e polygons f o r stand HC 32 7 Frequency histograms of areas of q u a d r i l a t e r a l s , t r i a n g l e s , small polygons, and l a r g e polygons f o r stand HB . 33 8 Frequency histograms of areas of q u a d r i l a t e r a l s , t r i a n g l e s , small polygons, and l a r g e polygons f o r stand HCB 34 9 Frequency histograms of q u a d r i l a t e r a l s , t r i a n g l e s , small polygons, and l a r g e polygons f o r stand FPy 35 PAGE v i ACKNOWLEDGEMENT The author wishes to express h i s g r a t i t u d e to A. R. F r a s e r whose p r e l i m i n a r y work on t r i a n g l e based p r o b a b i l i t y polygons and whose ideas and review were i n v a l u a b l e to t h i s study. A p p r e c i a t i o n i s a l s o extended to Dr. A. Kozak, Dr. J . P. Demaerschalk, and Dr. D. D. Munro f o r t h e i r review of t h i s t h e s i s , and to G. Beech f o r her c a r e f u l work i n producing a p r e s e n t a b l e document. F i n a l l y , the Research Branch of the B r i t i s h Columbia M i n i s t r y of F o r e s t s i s g r a t e f u l l y acknowledged f o r i t s f i n a n c i a l support f o r t h i s p r o j e c t . PAGE 1 SOME METHODS OF SAMPLING TRIANGLE BASED PROBABILITY POLYGONS FOR FORESTRY APPLICATIONS INTRODUCTION Of the v a r i o u s l e v e l s of f o r e s t inventory, the o p e r a t i o n a l c r u i s e r e q u i r e s the most p r e c i s e estimate of volume s i n c e i t i s u s u a l l y on the b a s i s of t h i s estimate that investment d e c i s i o n s are made. There i s i n t e r e s t , t h e r e f o r e , i n sampling techniques which provide such estimates without i n c u r r i n g g r e a t l y increased c o s t s . With the a v a i l a b i l i t y of dendrometers and inexpensive data p r o c e s s i n g more a t t e n t i o n has been given to methods whose sample u n i t s are i n d i v i d u a l trees rather than groups of t r e e s . Grosenbaugh (1967) demonstrated that s e l e c t i n g s i n g l e trees with p r o b a b i l i t i e s r e l a t e d to t h e i r s i z e i s more e f f i c i e n t , s t a t i s t i c a l l y , than p o i n t or p l o t sampling. Jack (1967) and F r a s e r (1977) d e s c r i b e d i n d i v i d u a l tree methods which s e l e c t sample trees with p r o b a b i l i t i e s based on the areas of polygons. These area-based methods have the advantage of not r e q u i r i n g a v i s i t to every tree of the t a r g e t p o p u l a t i o n , as do methods which s e l e c t t r e e s from a PAGE 2 l i s t or which are based on ocu l a r estimates of tree s i z e . Hence area-based methods are b e t t e r s u i t e d to the measurement of l a r g e t r a c t s of timber. Furthermore, F r a s e r ' s method, which i s based on the l o c a t i o n of t r i a n g l e s whose v e r t i c e s are p o i n t s on the ground d e f i n e d by tree stems, i s r e l a t i v e l y easy to apply i n the f i e l d and provides a d d i t i o n a l i n f o r m a t i o n on stand d e n s i t y and tree s p a t i a l d i s t r i b u t i o n . The purpose of t h i s study i s to i n v e s t i g a t e some a l t e r n a t i v e methods r e l a t e d to the method o u t l i n e d by F r a s e r i n order to: a) independently confirm h i s work, b) determine i f any improvements i n s t a t i s t i c a l e f f i c i e n c y can be provided by these a l t e r n a t i v e s and c) provide i n s i g h t s f o r f u r t h e r work. I t i s hoped that t h i s i n v e s t i g a t i o n w i l l help to f u r t h e r the development of the use of t r i a n g l e based p r o b a b i l i t y polygons i n f o r e s t measurements. PAGE 3 CHAPTER ONE LITERATURE REVIEW In the context of a l l o c a t i n g areas to i n d i v i d u a l t r e e s , Brown (1965) c o n s t r u c t e d polygons whose s i d e s were p e r p e n d i c u l a r b i s e c t o r s of l i n e segments j o i n i n g t r e e stem p o s i t i o n s . T h i s r e s u l t e d i n a set of polygons ( h i s t o r i c a l l y named e i t h e r D i r i c h l e t c e l l s or Voronoi polygons), one per t r e e , which had no gaps or o v e r l a p s . Brown c a l l e d the area of a polygon "Area P o t e n t i a l l y A v a i l a b l e " (APA) to the tree which i t contained and used i t as a measure of p o i n t d e n s i t y . He demonstrated that using the APA concept one could d e t e c t c o r r e l a t i o n between basal area and tree d e n s i t y more r e a d i l y than with the c o n v e n t i o n a l f i x e d radius p l o t method of determining tree d e n s i t y . He a l s o i n d i c a t e d the u t i l i t y of APA as a competition index. Jack (1967) employed the polygons d e s c r i b e d by Brown i n the development of a s i n g l e tree sampling technique. In t h i s method a tree i s s e l e c t e d as p a r t of a sample when a uniformly d i s t r i b u t e d random coo r d i n a t e p o i n t f a l l s w i t h i n i t s polygon. Thus, trees are s e l e c t e d with p r o b a b i l i t y PAGE 4 p r o p o r t i o n a l to t h e i r APA. Jack concluded from p r e l i m i n a r y t r i a l s that t h i s sampling method w i l l g i v e r e s u l t s having acceptable l i m i t s of accuracy at lower c o s t than other methods which r e q u i r e v i s i t i n g every t r e e , where the sampled area i s reasonably l a r g e . In a d d i t i o n , he give s r e s u l t s of using APA i n the p r e d i c t i o n of tre e volume increment, showing that a s l i g h t improvement i n p r e d i c t i o n can be made with the i n c l u s i o n of APA. In the i n t e r e s t s of o b t a i n i n g b e t t e r c o r r e l a t i o n between APA and tree s i z e , A d l a r d (1974) proposed a d j u s t i n g polygon s i d e s such that they no longer b i s e c t e d the l i n e segments between trees but instead d i v i d e d the segments at a po i n t weighted by tree s i z e . T h i s r e s u l t e d i n a l l o c a t i n g more APA to l a r g e r stems. F r a s e r and van den Driessche (1971) d i s c u s s e d d e s c r i b i n g the l i n e segments which j o i n p o i n t s i n a plane to form a network of non-overlapping t r i a n g l e s . Such networks have c o n s i s t e n t t r a i t s , that i s , a p o p u l a t i o n of N p o i n t s y i e l d s 2N t r i a n g l e s with 3N common s i d e s . A l s o , a s i n g l e p o i n t has an expected value of s i x sid e s r a d i a t i n g from i t . Thus, sampling t r i a n g l e s f o r average t r i a n g l e area enables one to estimate p o p u l a t i o n d e n s i t y and t o t a l p o p u l a t i o n s i z e . In P A G E 5 a d d i t i o n , v a r i a n c e s of t r i a n g l e areas and t r i a n g l e s i d e l e n g t h s can be used to i n d i c a t e r e g u l a r i t y and degree of clump i n g of p o i n t s . C o n s t r u c t i o n of t r i a n g l e s e t s i s f a c i l i t a t e d w i t h the s e l e c t i o n of l e a s t d i a g o n a l neighbour (LDN) p a i r s of p o i n t s . A p a i r of p o i n t s are LDN's p r o v i d e d t h a t no o t h e r p o i n t o c c u r s on the l i n e segment between the p a i r and t h a t the l i n e segment cannot be i n t e r s e c t e d by a s h o r t e r l i n e segment between any o t h e r p a i r of p o i n t s . E x c e p t f o r a few s p e c i a l c a s e s , f o r m i n g t r i a n g l e s from p a i r s of p o i n t s d e f i n e d t h i s way w i l l r e s u l t i n a unique s e t of t r i a n g l e s . F r a s e r (1977) advanced the use of LDN t r i a n g l e networks c o n s t r u c t e d among t r e e stem p o s i t i o n s w i t h the development of a s i n g l e t r e e v a r i a b l e p r o b a b i l i t y sampling method based on such networks. By a l l o c a t i n g a p o r t i o n of the area of a t r i a n g l e t o each of i t s v e r t e x t r e e s a c c o r d i n g to some p r o p o r t i o n i n g scheme one can c o n s t r u c t polygons around t r e e s . Having l o c a t e d the LDN t r i a n g l e i n which a sample p o i n t f a l l s , one c a l c u l a t e s the p r o b a b i l i t i e s of s e l e c t i o n of the t h r e e v e r t e x t r e e s (based on the chosen p r o p o r t i o n i n g scheme) and then s e l e c t s a t r e e by l i s t s a m p l i n g w i t h v a r i a b l e p r o b a b i l i t i e s . F r a s e r compared two formulae f o r p r o p o r t i o n i n g t r i a n g l e a r e a s , one b e i n g a d i r e c t PAGE 6 proportioning according to tree size and the other being a geometric proportioning which "conceptually" divides the triangle into three quadrilaterals resulting from p a r t i t i o n i n g triangle sides according to tree size and joining the p a r t i t i o n i n g point to the opposite triangle vertex. As such, one need not think in terms of physical polygons but only in terms of probabilites. For each formula of area p a r t i t i o n i n g he applied four d i f f e r e n t measures of tree size or proportioning weights. These were 2 1 (or equal weights), tree diameter at breast height (D), D 2 (or basal area), and D a (or portion of basal area found in a triangle) where a i s the angle measure of the triangle vertex. It should be noted that f i e l d measurements require only conventional tree volume measures plus triangle side distances. Angles, areas, and proportions are calculated l a t e r . Fraser found that the geometric proportioning 2 formula using the D a weight resulted in the most precise estimate of volume. Fraser also pointed out that work on APA polygons can be related to triangle based polygons. For example, three trees are vertices of a Delauney triangle provided no other points occur on or within the circumcircle of the triangle. The centre of the c i r c l e is found by the intersection of PAGE 7 pe r p e n d i c u l a r b i s e c t o r s of t r i a n g l e s i d e s . Rogers (1964) proves that the polygons formed by these b i s e c t o r s are the same as the Voronoi polygons. Thus, the polygons used by Brown and Jack may a l s o be de s c r i b e d as t r i a n g l e based polygons where the vertex trees are weighted e q u a l l y and the t r i a n g l e s are p a r t i t i o n e d by the polygons formed by the pe r p e n d i c u l a r b i s e c t o r s of t h e i r s i d e s . F r a s e r suggests that these Voronoi polygons are s t a t i s t i c a l l y i n e f f i c i e n t i n the context of sampling f o r tree volumes. While A d l a r d ' s m o d i f i c a t i o n might improve t h i s e f f i c i e n c y there would be c o n s i d e r a b l e d i f f i c u l t y i n implementing p r a c t i c a l f i e l d procedures i n order to e s t a b l i s h the r e l a t e d Delauney t r i a n g l e s . PAGE 8 CHAPTER TWO METHOD OF ANALYSIS THE BASIC METHOD Fra s e r (1977) o u t l i n e d a sampling method polygons formed from t r i a n g l e s c o n s t r u c t e d among (F i g u r e 1). T h i s method c o n s i d e r s polygons as s i z e d p l o t s , each c o n t a i n i n g one whole t r e e , h e r e a f t e r c a l l e d the B a s i c Method. For a sample (i= 1 to n), i f : y^ = volume of sample t r e e ^ , a^ = area of polygon c o n t a i n i n g sample t r e e ^ , A = t o t a l area covered by tar g e t p o p u l a t i o n , z i = p r o b a b i l i t y of s e l e c t i n g t r e e ^ = a^/A, u t i l i z i n g LDN t r e e s v a r i a b l e and i s s i z e of n PAGE 9 then the estimate of t o t a l volume (probability proportional to estimated size) i s : Yppes = n n I i = l z i = A n n E i = l a. 1 (1) and i t s variance i s : V(yppes) = n (n-1) i = 1 V a. _1_ n n y ± E — — . , a. i = l l (2) PAGE 1 0 PAGE 11 The f i e l d procedure f o r sampling using t h i s type of polygon i s as f o l l o w s : i ) Locate a sample p o i n t and e s t a b l i s h the LDN t r i a n g l e among tr e e s i n which i t f a l l s . i i ) Take the necessary measurements f o r determining p r o p o r t i o n i n g weights from each vertex t r e e of the t r i a n g l e . Using the d e s i r e d weighting and p r o p o r t i o n i n g formula c a l c u l a t e the p r o b a b i l i t i e s of s e l e c t i n g each t r e e : P^, P2» and P 3. Generate a uniformly d i s t r i b u t e d random number between 0 and 1. S e l e c t t r e e 1, i f the random number i s l e s s than or equal to P-^ ; s e l e c t tree 2 i f the random number i s g r e a t e r than P-^  but l e s s than or equal to + P 2; otherwise s e l e c t t r e e 3. i i i ) Measure s e l e c t e d tree f o r volume. iv) Locate the remaining LDN trees of the s e l e c t e d t r e e and record t h e i r weighting measures and s i d e l engths of the t r i a n g l e s PAGE 1 2 which they form. These measurements are necessary f o r the c a l c u l a t i o n of polygon areas. Note that angle measurements are not r e q u i r e d as they are a l s o c a l c u l a t e d from s i d e d i s t a n c e s . In summary t h i s method r e q u i r e s the measurement of: 1 t r e e measured f o r volume (assumed to i n c l u d e measures for p r o p o r t i o n i n g weights), 6 (average) trees measured f o r p r o p o r t i o n i n g weights, 12 (average) d i s t a n c e measures between LDN t r e e s ( t h i s i s based on the f a c t that a p o i n t i n a t r i a n g l e network has an expected value of 6 s i d e s r a d i a t i n g from i t ) , 1 p r o b a b i l i t y c a l c u l a t i o n i n the f i e l d i n order to s e l e c t volume tree. PAGE 13 Four a l t e r n a t i v e methods r e l a t e d to the Basi c Method are now proposed. In these methods the same formulae for Yppes and V(Yppes) apply, however, the c a l c u l a t i o n of the y i and a^ and the f i e l d procedures d i f f e r . In p a r t i c u l a r , three of the methods dispense with any p r o b a b i l i t y c a l c u l a t i o n i n the f i e l d . In the f o l l o w i n g d i s c u s s i o n , i n order to keep notation simple, the same v a r i a b l e names are kept throughout, even though t h e i r meaning may change s l i g h t l y from method to method. I t i s f e l t that t h i s w i l l be more e a s i l y understood than having a completely d i f f e r e n t set of v a r i a b l e names f o r each method. PAGE 14 METHOD 1 T h i s c o n s i s t s of a sample u n i t of only a p a r t of one tre e of a f i e l d s e l e c t e d LDN t r i a n g l e (Figure 2). The s e l e c t e d tree i s chosen with p r o b a b i l i t y p r o p o r t i o n a l to i t s q u a d r i l a t e r a l area. T h i s method c o n s i d e r s q u a d r i l a t e r a l s as v a r i a b l e s i z e d p l o t s , each c o n t a i n i n g the f r a c t i o n a l p a r t of the t r e e which f a l l s w i t h i n the q u a d r i l a t e r a l . In t h i s case, i f a = s i z e of t r i a n g l e vertex angle at which volume tree i s l o c a t e d , v = volume of volume t r e e , then y^ = volume of p o r t i o n of volume tree contained i n t r i a n g l e _ v a 2TT a^ = area of q u a d r i l a t e r a l c o n t a i n i n g volume tree. PAGE 15 Sampling using t h i s method r e q u i r e s per sample u n i t : 1 tr e e measured f o r volume ( i n c l u d e s a weighting measure), 2 t r e e s measured f o r p r o p o r t i o n i n g weights, 3 d i s t a n c e measures betweeen LDN trees (weighting and d i s t a n c e measures are used f o r c a l c u l a t i o n of q u a d r i l a t e r a l a r e a ) , 1 p r o b a b i l i t y c a l c u l a t i o n i n the f i e l d f o r s e l e c t i o n of volume tre e . PAGE 16 Figure 2 Stem map showing a l e a s t diagonal neighbour t r i a n g l e and q u a d r i l a t e r a l of the type used i n Method 1 PAGE 17 METHOD 2 This i s the simplest case of measuring three trees of a f i e l d s e l e c t e d t r i a n g l e (Figure 3). Once a t r i a n g l e i s s e l e c t e d , i t s three vertex trees are a u t o m a t i c a l l y measured. This method considers t r i a n g l e s as v a r i a b l e s i z e d p l o t s , each c o n t a i n i n g the f r a c t i o n a l parts of the three trees which f a l l w i t h i n the t r i a n g l e . I f , f o r the t r i a n g l e v e r t i c e s j = 1 to 3 : a.. = measure of t r i a n g l e vertex angle^, Vj = volume of tree at v e r t e x j , then y^ = sum of volumes of port i o n s of vertex trees contained i n t r i a n g l e = v l a l 2TT . v 2 a 2 2TT V 3 a 3 2TT = 1 2TT 3 Z v . a . j = l ^ 3 PAGE 18 = t r i a n g l e a r e a . T h i s method r e q u i r e s per sample u n i t : 3 t r e e s measured f o r volume, 3 d i s t a n c e measures between t r e e s (used in c a l c u l a t i o n of t r i a n g l e ang les and a r e a ) , no p r o p o r t i o n i n g weight measures , no p r o b a b i l i t y c a l c u l a t i o n i n the f i e l d . J PAGE 19 Figure 3 Stem map showing a l e a s t diagonal neighbour t r i a n g l e of the type used i n Method 2 PAGE 20 METHOD 3 This i s another three tree case using a f i e l d s e l e c t e d t r i a n g l e much l i k e Method 2. The distances to the 9 (average) LDN neighbours of the three vertex trees are measured i n a d d i t i o n to those measurements required i n Method 2 (Figure 4 ) . This method considers t r i a n g l e s as v a r i a b l e s i z e d p l o t s , each c o n t a i n i n g weighted f r a c t i o n a l p o r t i o n s of the three trees at i t s v e r t i c e s . The f r a c t i o n a l p o r t i o n of a tree i s weighted by the r a t i o of the se l e c t e d t r i a n g l e area to the sum of the areas of a l l t r i a n g l e s common to that tree. Thus i t i s not equal to the f r a c t i o n a l volume p o r t i o n f a l l i n g w i t h i n the t r i a n g l e as i n Method 2. For the selecte d t r i a n g l e v e r t i c e s j = 1 to 3 l e t PLj = area of the large polygon which i s the sum of the areas of a l l LDN t r i a n g l e s having tree^ at a vertex, V j = volume of tree at selecte d t r i a n g l e v e r t e x j , t = area of selected t r i a n g l e . The s e l e c t e d t r i a n g l e has a l l o c a t e d to i t s area p o r t i o n of each tree volume v.. p r o p o r t i o n a l to " t " a t/PL . . T h e r e f o r e , PL, PL„ PL„ a . = t. 1 T h i s method r e q u i r e s per t r i a n g l e sampled: 3 trees measured f o r volume, 24 (average) d i s t a n c e measures between LDN t r e e s , no p r o p o r t i o n i n g weight measures, no p r o b a b i l i t y c a l c u l a t i o n s i n the f i e l d . P A G E 22 F i g u r e 4 Stem map showing l e a s t d i a g o n a l neighbours and one shaded l a r g e polygon of the type used i n Method 3 PAGE 23 METHOD 4 This i s a l s o a three tree sample of a f i e l d s e l e c t e d t r i a n g l e as i n Method 3 except t h a t , i n a d d i t i o n , the weighting measures of the nine (average) surrounding LDN trees are taken, and the weights and distances are used to c a l c u l a t e polygon areas i n the same manner as i n Fraser*s Basic Method (Figure 5). This method considers t r i a n g l e s as v a r i a b l e s i z e d p l o t s , each co n t a i n i n g weighted f r a c t i o n a l p o r t i o n s of the three trees at i t s v e r t i c e s as with Method 3. However, u n l i k e Method 3, the f r a c t i o n a l p o r t i o n of a tree i s weighted by the r a t i o of the area of i t s q u a d r i l a t e r a l i n the selected t r i a n g l e to the sum of the areas of a l l i t s q u a d r i l a t e r a l s . Thus, for the se l e c t e d t r i a n g l e v e r t i c e s j = 1 to 3 l e t PS. = area of the small polygon which i s the sum of a l l q u a d r i l a t e r a l s having tree, at a vertex, q_. = area of the q u a d r i l a t e r a l , i n the sel e c t e d t r i a n g l e , having tree., at a vertex, PAGE 24 v- = volume of t r e e . . 3 3 A p o r t i o n of each t r e e volume V j p r o p o r t i o n a l to q^/PS^ i s a l l o c a t e d to the s e l e c t e d t r i a n g l e a r e a . So, _ v l q l , V 2 q 2 , V 3 q 3 Y i " P S l P S 2 P S 3 3 v .q . = £ s i - 1 . , PS . 3 = 1 3 a. = a r e a of s e l e c t e d t r i a n g l e . i T h i s method r e q u i r e s per t r i a n g l e sampled: 3 t r e e s measured f o r volume ( i n c l u d e s w e i g h t i n g measures), 24 (average) d i s t a n c e measures between LDN t r e e s , 9 (average) t r e e s measured f o r p r o p o r t i o n i n g w e i g h t s ( w e i g h t i n g and d i s t a n c e PAGE 25 measures are necessary f o r c a l c u l a t i o n of q u a d r i l a t e r a l a r e a s ) , no p r o b a b i l i t y c a l c u l a t i o n s i n the f i e l d . T a b l es 1 and 2 summarize the important f e a t u r e s of each of the methods. PAGE 26 F i g u r e 5 Stem map showing l e a s t d i a g o n a l neighbours and one shaded small polygon of the type used i n Method 4 PAGE 27 Table 1 Summary of the c o m p i l a t i o n of the y ^ and a.: f o r each method (used i n formulae (1) and (2)) Method B a s i c : sample tree volume Method 1 : volume of tree p o r t i o n contained i n t r i a n g l e Method 2 : sum of volumes of tre e p o r t i o n s contained i n t r i a n g l e Method 3 : sum o f : ( t r i a n g l e area) (tree volume) l a r g e vertex polygon area area of polygon area of q u a d r i -l a t e r a l c o n t a i n -ing tree area of s e l e c t e d t r i a n g l e area of s e l e c t e d t r i a n g l e Method 4 : sum o f : ( q u a d r i l a t e r a l ) (tree volume) ( area ) area of s e l e c t e d t r i a n g l e small vertex polygon area PAGE 28 Table 2 Summary of measures r e q u i r e d per sample f o r each method Method Number of volume measures Average number of weighting measures Average number of d i s t a n c e measures P r o b a b i l i t y c a l c u l a t i o n r e q u i r e d i n f i e l d B a s i c Method 1 Method 2 Method 3 Method 4 1 1 3 3 3 6 2 none none 9 12 3 3 24 24 yes yes no no no PAGE 29 METHOD OF COMPARISON Four data sets were analyzed i n t h i s study. These are the i d e n t i c a l data sets used by F r a s e r , being stem map and diameter i n f o r m a t i o n f o r four f o r e s t types. (The data f o r a f i f t h type used by F r a s e r had been misplaced and could not be r e c o n s t r u c t e d ) . The type symbols and species composition are: HB - mature western hemlock (Tsuga  h e t e r o p h y l l a (Raf.) Sarg.) and balsam (Abies a m a b i l i s Dougl. ) F o r b e s ) , HC - mature western hemlock and western red cedar (Thuja p l i c a t a Donn), HCB - mature western hemlock, western red cedar and balsam (Abies l a s i o c a r p a (Hook. ) Nutt. ), FPy - mature D o u g l a s - f i r (Pseudotsuga m e n z i e s i i (Mirb.) Franco) and yellow pine (Pinus ponderosa Laws) Information d e s c r i b i n g these stands i s found i n Table 3. F i g u r e s 6, 7, 8, and 9 are frequency histograms of areas of PAGE 30 q u a d r i l a t e r a l s , t r i a n g l e s , s m a l l polygons, and l a r g e polygons f o r each stand. Note that a l l d i s t r i b u t i o n s are s i m i l a r , being skewed r i g h t . These histograms g i v e no in f o r m a t i o n as to s p a t i a l arrangement. As F r a s e r noted the FPy stand appears to be h i g h l y aggregated i n s p a t i a l p a t t e r n while the other three stands show random p a t t e r n . PAGE 31 Tabl e 3 D e s c r i p t i o n of stands te s t e d STAND TYPE Area (m 2) Number of tr e e s Dbh, min (cm) max (cm) Height min (m) max (m) Volume per tree (m 3) Coeff of v a r i a t i o n , percent Volume per 200m 2 p l o t (m 3) Coeff of v a r i a t i o n percent Trees per p l o t (average) HC HB HCB FPy 3520 2378 3176 3567 94 106 76 51 25 22 18 25 93 64 92 73 18 19 15 29 44 40 43 47 2. 80 1.35 2. 48 2. 76 69.6 59.4 75.7 81.0 14.97 12. 07 11. 89 7.90 43.4 32. 3 68.1 76.1 5. 34 8.90 4. 80 2. 86 PAGE 32 quadrilaterals 9 0 triangles 5 0 7 0 9 0 MO 130 150 170 190 small polygons ,5 45 75 105 135 165 19 5 2 2 5 2 5 5 2 8 5 large polygons ^7.5 | |2.5 187.5 2S2.5 337.5 412.5 487 .5 562,5 637.5 712.5 class midpoint F i g u r e 6 Frequency h i s t o g r a m s of areas of q u a d r i l a t e r a l s , t r i a n g l e s , s m a l l p o l y g o n s , and l a r g e polygons f o r stand HC 400 PAGE 33 quadrilaterals > o c cu 3 CT CD 45 75 105 135 165 195 225 255 285 25 75 125 triangles 175 225 275 325 375 425 475 small polygons l l 45 135 225 315 405 595 685 775 865 955 large polygons JZ. 100 300 500 700 900 1100 1300 1500 1700 1900 class midpoint F i g u r e 7 Frequency h i s t o g r a m s of areas of q u a d r i l a t e r a l s , t r i a n g l e s , s m a l l p o l y g o n s , and l a r g e polygons f o r stand HB 400 PAGE 34 quadrilaterals 200 -§ , 5 45 75 105 135 165 195 225 255 285 triangles M 52.5 87.5 122.5 157.5 192 5 227.5 262.5 297.5 332.5 small polygons 1 i 1 2 5 75 l 2 3 175 225 275 325 375 425 475 large polygons io A 3 L" 300 900 class midpoint F i g u r e 8 Frequency h i s t o g r a m s of areas of q u a d r i l a t e r a l s , t r i a n g l e s , s m a l l p o l y g o n s , and l a r g e polygons f o r sta n d HCB & c a> <1> PAGE 35 quadrilaterals 60.5 181.5 302.5 428.5 544.5 665.5 786.5 907.5 1028.5 1149.5 triangles 337.5 562.5 787.5 1012.5 1237.5 1462.5 1687.5 1912.5 2137.5 small polygons 450 750 1050 1350 1650 1950 2250 2550 2850 large polygons i 1 350 1050 1750 2450 3150 3850 4550 5250 5950 6650 class midpoint F i g u r e 9 Frequency h i s t o g r a m s of q u a d r i l a t e r a l s , t r i a n g l e s , s m a l l p o l y g o n s , and l a r g e polygons f o r stand FPy PAGE 36 FORTRAN programs were w r i t t e n to enable the comparison of the methods d e s c r i b e d h e r e i n . These programs i d e n t i f i e d LDN p a i r s ; solved t h e i r r e s u l t a n t t r i a n g l e s ; p a r t i t i o n e d the t r i a n g l e areas i n t o t h e i r polygon p o r t i o n s ; and c a l c u l a t e d the d e s i r e d s t a t i s t i c s . The p o p u l a t i o n forms of equations (1) and (2), that i s , summed over a l l p o s s i b l e samples, were a p p l i e d to these data. The t r i a n g l e p a r t i t i o n i n g formula used was the geometric p r o p o r t i o n i n g formula which c a l c u l a t e s p r o p o r t i o n s (P]_, ? 2 ' a n d p 3 ^ °^ a t r i a r j g l e area i n each q u a d r i l a t e r a l . Thus w l < w l w2 + 2 w 2 w 3 + w l w3> P-L * \rJ(w1 + W 2 ) ( W , + W 3 ) where W 5 " i w2 + " l tf3 + w2 W3 ' and w,, w2 and w^  are tree weights. P 2 and P^ are c a l c u l a t e d s i m i l a r l y f o l l o w i n g symmetry i n s u b s c r i p t s . The p r o p o r t i o n i n g weight used was the D a weight de s c r i b e d by F r a s e r . In a d d i t i o n , a D 2Ha weight, with H being tree height, was tested f o r the FPy, HC, and HCB data s e t s , s i n c e PAGE 3 7 height i n f o r m a t i o n was a v a i l a b l e f o r these types. I t was hoped that t h i s might give a p a r t i t i o n i n g of t o t a l area A i n t o polygon areas which i s more h i g h l y c o r r e l a t e d with tree volumes. Comparisions were made using values of C where c - iooV(n"1) ^ a* n ^ " I 1 n y i n i=l a i and i s the c o e f f i c i e n t of v a r i a t i o n of the estimate of mean volume per u n i t area. As noted by F r a s e r , s i n c e the sample s i z e r e q u i r e d to achieve a given l e v e l of p r e c i s i o n i s d i r e c t l y p r o p o r t i o n a l to C 2, redu c t i o n s i n . C of 1 or 2 percent are important from the standpoint of improving sampling e f f i c i e n c y provided that they can be obtained by change i n p a r t i t i o n i n g formula with no other change i n c o s t s . In order to t e s t sampling e f f i c i e n c y where c o s t items change (that i s where measurements and c a l c u l a t i o n s change) r e l a t i v e v a r i a n c e r a t i o s were c a l c u l a t e d from these c o e f f i c i e n t s of v a r i a t i o n : PAGE 38 C 2 f o r Method X R e l a t i v e v a r i a n c e r a t i o = —* C f o r B a s i c Method i n order to r e l a t e each method to the B a s i c Method. These r e l a t i v e v a r i a n c e r a t i o s g i v e the r e l a t i v e numbers of samples r e q u i r e d f o r equal p r e c i s i o n . That i s , i f n i s used to denote sample s i z e , then: n f o r Method X C 2 f o r Method X = __ j n f o r B a s i c Method C f o r B a s i c Method Thus, one can c a l c u l a t e the average number of measurements re q u i r e d to gain equal p r e c i s i o n f o r each method r e l a t i v e to the B a s i c Method by m u l t i p l y i n g the r e l a t i v e v a r i a n c e r a t i o s by the average number of measurements given i n Table 2. PAGE 39 CHAPTER THREE RESULTS AND DISCUSSION Computed p o p u l a t i o n t o t a l volumes and areas f o r the four stand types were i d e n t i c a l to those r e s u l t s obtained by F r a s e r . S i n c e the two s t u d i e s used d i f f e r e n t methods to c o n s t r u c t LDN t r i a n g l e networks (Fraser found h i s LDN p a i r s manually? the present study was performed using LDN p a i r s i d e n t i f i e d by a FORTRAN program) s l i g h t l y d i f f e r e n t t r i a n g l e s e t s were obtained. The r e s u l t s f o r e i t h e r study are s t i l l , of course, meaningful s i n c e each t r i a n g l e s e t was c o n s i s t e n t l y a p p l i e d throughout each a n a l y s i s ; hence the r e l a t i v e d i f f e r e n c e s of values of C w i l l not change. In the c u r r e n t study, the r e s u l t s f o r the B a s i c Method were judged to be s i m i l a r enough to those r e s u l t s of F r a s e r ( i n terms of t h e i r absolute values and t h e i r behaviour from type to type) as to v e r i f y t h e i r c o r r e c t n e s s . Indeed, when the FPy type was tested with the tree weights of 1 , D, and D that F r a s e r had a p p l i e d , the same trend i n the values of C was observed. Table 4 gives the values of C f o r the four types using PAGE 40 the D a weight. The r e s u l t s are c o n s i s t e n t , showing d e c r e a s i n g values i n order of Method 1, Method 2, the B a s i c Method, Method 3, and Method 4. I t can immediately be seen that improved s t a t i s t i c a l performance i s obtained with Methods 3 and 4 but not with Methods 1 and 2. PAGE 41 Table 4 C o e f f i c i e n t s of v a r i a t i o n (C) f o r each method using the D 2a weight Stand Type Method : HC HB HCB FPy Basi c i 62.4 65.5 79.7 114.6 1 : 130.6 153.1 136. 6 213. 8 2 : 128.0 150. 9 135.2 211.7 3 : 61.0 60. 6 79.2 103. 2 4 : 51.2 53.7 69.8 97.4 PAGE 42 Tabl e 5 gi v e s the values of the r e l a t i v e v a r i a n c e r a t i o s f o r each method compared to the B a s i c Method f o r each stand. As noted p r e v i o u s l y , these are the r a t i o s of samples r e q u i r e d to o b t a i n the same p r e c i s i o n as the B a s i c Method. Thus, f o r example, i t would r e q u i r e 4.38 times more Method 1 samples than B a s i c Method samples i n order to o b t a i n equal p r e c i s i o n with the HC stand type. To see what these r a t i o s mean i n terms of the average number of measurements r e q u i r e d to o b t a i n the same p r e c i s i o n as with the B a s i c Method one needs only to m u l t i p l y the values of Table 5 with the average number of measurements r e q u i r e d i n each method (from Table 2). These values are shown i n Table 6. PAGE 43 Table 5 Variance r a t i o s for each method r e l a t i v e to the Bas i c Method Method Stand Type HC HB HCB FPy 1.00 1.00 1.00 1.00 4.38 5.46 2.94 3. 48 4.21 5.31 2.88 3.41 .96 . 86 .99 .81 .67 .67 .77 .72 Basic 1 2 3 4 PAGE 44 Table 6 Comparison of average number of measurements required for each method to obtain the same p r e c i s i o n as the Basic Method Number of Average number of Stand : volume : weighting : distance : required i n Method : Type : measures: measures : measures : f i e l d B asic : A l l : 1 : 6 : 12 yes 1 : HC : 4.38 : 8.76 : 13.14 : HB : 5.46 : 10.92 : 16.38 : yes HCB : 2.94 : 5.88 : 8.82 : FPy : 3.48 s 6.96 : 10.44 : 2 : HC : 12. 63 : 0 i 12.63 : HB : 15. 93 : 0 : 15.93 : no : HCB : 8.64 : 0 : 8.64 : FPy : 10.23 : 0 : 10.23 3 : HC : 2.88 : 0 : 23.04 : HB : 2.58 : 0 : 20.64 : no : HCB : 2.97 : 0 : 23.76 : FPy : 2.43 : 0 : 19.44 4 : HC : 2.01 : 6.03 : 16.08 : HB : 2.01 : 6.03 : 16.08 : no : HCB : 2.31 : 6.93 : 18.48 : FPy : 2.16 : 6.48 : 17.28 Average number of P r o b a b i l i t y c a l c u l a t i o n PAGE 45 For the data tested here, i t is apparent that none of the new methods offers any advantage over the Basic Method. More s p e c i f i c a l l y , Method 1 requires a probability c a l c u l a t i o n in the f i e l d — in addition to weighting measures — and requires three to five times as many volume measures as the Basic Method. (Volume measures t y p i c a l l y are the most costly as they usually include a diameter measure, a height measure, quality assessment, and sometimes diameter measures up the stem.) Method 2 has the advantage of not requiring a probability c a l c u l a t i o n in the f i e l d nor does i t require any weighting measures; however i t does require eight to sixteen times as many volume measures. Method 3 also dispenses with the probability c a l c u l a t i o n in the f i e l d , and does not require any weighting measures; however, i t needs two to three times as many volume measures and almost twice as many distance measures in order to obtain the same precision as the Basic Method. Method 4 does not require the probability calculation in the f i e l d , but i t does require weighting measures (about the same number as does the Basic Method) and about twice as many volume measures. Table 7 gives the values of C for the Basic Method and 2 Methods 1 and 4 using the D Ha weight (Methods 2 and 3, of PAGE 46 course, show no change s i n c e they make no use of the p r o p o r t i o n i n g weights). There are, once again, the same trends as before, that i s , the values decrease i n the order of Method 1, the B a s i c Method, and Method 4. However, when 2 these values are compared with those of the D a weight of Table 4 no c o n s i s t e n t trends are observed. With Method 1 values a c t u a l l y i n c r e a s e or remain the same, going from a D 2a to a D 2Ha weight. With Method 4 and the B a s i c Method values decrease s l i g h t l y . These r e s u l t s c e r t a i n l y do not encourage the measurement of height f o r weighting purposes, at l e a s t i n a volume sampling context; e s p e c i a l l y s i n c e height measurements are so time consuming. I t might be 2 worthwhile, however, to t e s t a D Ha weight fo r p a r t i t i o n i n g t r i a n g l e areas to form polygons which c o r r e l a t e w e l l with volume increment. These r e s u l t s c o nfirm F r a s e r ' s work and, i n a d d i t i o n , demonstrate that some improvement in s t a t i s t i c a l e f f i c i e n c y may be obtained with two of the methods proposed here. I t seems u n l i k e l y , though, that any of the four new methods o f f e r s any improvements i n sampling c o s t s . Another c o n c l u s i o n to be drawn from, t h i s i s that sampling methods which use volumes of p a r t s of t r e e s , rather than of the whole t r e e , introduce more v a r i a t i o n and hence r e q u i r e more PAGE 47 samples to obtain equal precision. Therefore, costs of making such samples would have to be reduced in order to make them p r a c t i c a l l y applicable. Thus i t appears that those methods which use whole tree volumes might provide the greatest potential for any future sampling work. PAGE 48 Table 7 C o e f f i c i e n t s of v a r i a t i o n (C) fo r each method using the D 2Ha weight Stand Type Method : HC HCB FPy B a s i c 59.3 78.6 113.5 1 : 132.1 139.6 213. 8 4 : 49.8 69.0 96.7 PAGE 49 CHAPTER FOUR i SUGGESTIONS FOR FURTHER WORK Obvi o u s l y , the most important need i s to o b t a i n f i e l d experience i n order to assess the c o s t of sampling t r i a n g l e based p r o b a b i l i t y polygons r e l a t i v e to c o n v e n t i o n a l methods. Such t r i a l s need to be r i g o r o u s l y t e s t e d with experienced f i e l d crews i n order to o b t a i n the most r e a l i s t i c r e s u l t s . Somewhat r e l a t e d to t h i s i s a need to t e s t f o r s e n s i t i v i t y of the volume estimate to measurement e r r o r s i n data c o l l e c t i o n . S t o c h a s t i c s i m u l a t i o n would probably be the best approach to accomplish t h i s . In the B a s i c Method and Methods 1 and 4 the volume estimates make no d i r e c t use of tree volumes which might be de r i v e d from the a d d i t i o n a l weighting measures taken f o r the purpose of c a l c u l a t i n g polygon areas. Estimates of volume may be improved by using these measures and i n a d d i t i o n , diameter d i s t r i b u t i o n s may be de r i v e d . (The assumption here i s that diameters are pa r t of the weighting measurements taken.) T h i s might r e s u l t i n Methods 1 and 4 becoming more PAGE 50 p a l a t a b l e sampling a l t e r n a t i v e s . C u r r e n t l y , the Research Branch of the B r i t i s h Columbia F o r e s t S e r v i c e i s t e s t i n g the i d e n t i f i c a t i o n of p l u s trees using c o m p e t i t i o n i n d i c e s as a b a s i s f o r the assessment of s t r e s s . Trees e x h i b i t i n g d e s i r a b l e t r a i t s may be doing so simply due to a s o c i a l p o s i t i o n which i s r e l a t i v e l y f r e e of s t r e s s , whereas trees e x h i b i t i n g the same t r a i t s while being subjected to high s t r e s s may be of e x c e p t i o n a l g e n e t i c stock. Thus competition i s an important q u a n t i t y to be determined. The advantage of using t r i a n g l e p r o b a b i l i t y polygons to estimate competition i s the ease of f i e l d measurement. Awkward and expensive stem mapping procedures are not necessary, nor i s there the p o s s i b i l i t y of measuring too few or too many tr e e s . The i n d e n t i f i c a t i o n of LDN p a i r s always r e s u l t s i n the optimum number of t r e e s . In a d d i t i o n , there i s the p o t e n t i a l to t e s t v a r i o u s p a r t i t i o n i n g and weighting a l t e r n a t i v e s to o b t a i n those which best s u i t the needs of t h i s work. Measures of d e n s i t y and p a t t e r n are important f o r t h i n n i n g and spacing work. R e l i a b l e assessment of stands p r i o r to entry f o r t h i n n i n g or spacing can be u s e f u l f o r determining the n e c e s s i t y of such treatments and f o r PAGE 51 d e f i n i n g c o n t r a c t s p e c i f i c a t i o n s . Likewise, s i m i l a r assessment a f t e r entry can be used to check s a t i s f a c t o r y completion f o r approval of f o r e s t r y c o s t s or p r i o r to c o n t r a c t p a y-off. Simple stems per hectare estimates do not s u f f i c e ; they g i v e no i n f o r m a t i o n as to degree of clumping. Using the s t a t i s t i c s of average t r i a n g l e area and v a r i a n c e of t h i s estimate proposed by F r a s e r and van den D r i e s s c h e (1971), one has i n d i c a t o r s of d e n s i t y and s p a t i a l d i s t r i b u t i o n . T h i s i s achieved simply through the sampling of i n d i v i d u a l LDN t r i a n g l e s and measuring s i d e d i s t a n c e s . S t a u f f e r (1979) has advanced t h i s aspect of F r a s e r and van den D r i e s s c h e ' s work and has estimated the d i s t r i b u t i o n s of these s p a t i a l i n d i c a t o r s so that confidence i n t e r v a l s may be c a l c u l a t e d f o r them. With these i n d i c a t o r s and the a b i l i t y to t e s t t h e i r s t a t i s t i c a l s i g n i f i c a n c e the next step i s to gain experience through s i m u l a t i o n and from the f i e l d i n order to develop an i n t e r p r e t a t i o n of what t h e i r magnitudes mean i n p r a c t i c a l terms. Development of f i e l d technique i s a l s o r e q u i r e d . Note that an o f f s h o o t of t h i s a p p l i c a t i o n would be to sample for volume info r m a t i o n (with l i t t l e e x t r a measurement required) f o r the purposes of e s t i m a t i n g volume of wood removed or f o r t e s t i n g growth response through p e r i o d i c measurements. Information about diameter d i s t r i b u t i o n s i s a l s o u s e f u l from the p o i n t of view of PAGE 52 spacing c o n t r a c t s ; t h e r e f o r e , i n v e s t i g a t i o n of the p r o d u c t i o n of diameter d i s t r i b u t i o n s from simple diameter measures of t r i a n g l e vertex trees would be u s e f u l . Regeneration surveys would a l s o enjoy a s i m i l a r a p p l i c a t i o n of t r i a n g l e s p a t i a l i n d i c a t o r s ; however, i t i s d i f f i c u l t to v i s u a l i s e a p r a c t i c a l f i e l d technique to apply to small s e e d l i n g s . R e s u l t s , though, could be much more r e l i a b l e than the use of stocked quadrats. Another area r e q u i r i n g more study i s the problem of i d e n t i f y i n g LDN p a i r s from stem map data. Shamos and Hoey (1975) provide a good summary of a l g o r i t h i m s f o r j o i n i n g p o i n t s a c cording to v a r i o u s c r i t e r i a . Included are algorithms f o r c o n s t r u c t i o n of t r i a n g l e s and Voronoi polygons. They p o i n t out that using a l i n e a r programming approach ( i . e . , the Simplex Method) i n two v a r i a b l e s with N c o n s t r a i n t s (where N represents number of p o i n t s ) r e s u l t s i n computation time i n c r e a s i n g as N , while polygons can be c o n s t r u c t e d , i n two dimensions, using geometric techniques which r e s u l t i n a computation time i n c r e a s i n g as N l o g N. Developing these algorithms to produce LDN t r i a n g l e networks, or t h e i r r e s u l t a n t polygons d i r e c t l y would g r e a t l y improve computational e f f i c i e n c y and f a c i l i t a t e s i m u l a t i o n PAGE 53 s t u d i e s i n v o l v i n g s p a t i a l p a t t e r n p rob lems. PAGE 54 LITERATURE CITED A d l a r d , P. G. 1974. Development of an e m p i r i c a l model f o r i n d i v i d u a l trees w i t h i n a stand. In Growth Models f o r t r e e and stand s i m u l a t i o n ( J . F r i e s , ed.),pp 22-37. Royal C o l l e g e of F o r e s t r y , Stockholm, Sweden, Res Note 30, 379 P. Brown, G. S. 1965. P o i n t d e n s i t y i n stems per acre. New Zealand For. Research Notes No. 38. F r a s e r , A. R. and P. van den D r i e s s c h e . 1972. T r i a n g l e s , d e n s i t y , and p a t t e r n i n p o i n t p o p u l a t i o n s . P. 277-286 i n Proc. 3rd Conf. Ad v i s o r y Group of F o r e s t S t a t i s t i c i a n s , Int. Union For. Res. Organ, I n s t . Nat. Rech. A g r i c , Jouy-en-Jonas, France. 332 p. F r a s e r , A. R. 1977. T r i a n g l e based p r o b a b i l i t y polygons f o r f o r e s t sampling. F o r e s t S c i . 23:111-121. Grosenbaugh, L. R. 1967. The gains from sample-tree s e l e c -t i o n with unequal p r o b a b i l i t i e s . J . For 65:203-206. Jack, W. H. 1967. S i n g l e - t r e e sampling i n even aged p l a n t a -t i o n s f o r survey and experimentation. P. 379-403 i n Proc. 14th Congr. Int. Union For. Res. Organ, Munich, 1967, Pt VI, Sect 25. Rogers, C. A. 1964. Packing and c o v e r i n g . Math T r a c t No. 54. Cambridge U n i v e r s i t y Press, London. 165 p. Shamos, M i c h a e l Ian, and Dan Hoey. 1975. C l o s e s t p o i n t problems. IEEF Trans. Comp. pp 151-162. S t a u f f e r , Howard B. 1979. D i s t r i b u t i o n s f o r F r a s e r ' s s p a t i a l i n d i c a t o r s . D r a f t . 

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