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Leaf area distribution and alternative sampling designs for hybrid spruce tree crowns Hailemariam, Temesgen 1998

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L E A F A R E A DISTRIBUTION A N D A L T E R N A T I V E SAMPLING DESIGNS F O R H Y B R I D S P R U C E T R E E C R O W N S By Temesgen Hailemariam B. Sc.(Agri.), Alemaya University of Agriculture M . Sc. (For.), Lakehead University '  A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T OF T H E REQUIREMENTS FOR T H E D E G R E E OF D O C T O R OF PHILOSOPHY  in T H E FACULTY OF G R A D U A T E STUDIES FOREST RESOURCES M A N A G E M E N T  T H E UNIVERSITY OF BRITISH COLUMBIA  Feb. 1998 © Temesgen Hailemariam, 1998  In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study.  I further agree that permission for extensive copying of this  thesis for scholarly purposes may be granted by the head of my department or by his or her representatives.  It is understood that copying or publication of this thesis for  financial gain shall not be allowed without my written permission.  Forest Resource Management The University of British Columbia 2357 Main Mall Vancouver, Canada V 6 T 1Z4  Date:  Abstract  Three-dimensional populations such as tree crowns present unusual sampling problems owing to the variability of morphological attributes at different heights, relative horizontal positions, and cardinal directions of the crown. High within- and among-tree crown variation have also contributed to the difficulty of tree crown sampling and single tree leaf area (TLA - area available for photosynthesis) estimation. The two objectives of this research were to: 1) describe the distribution of leaf area in hybrid spruce (Picea engelmannii * glauca * sitchensis) crowns grown in the northwestern interior of British Columbia, and 2) to assess and evaluate sampling alternatives that consider variations in the radial and vertical distribution of foliage in estimating TLA of hybrid spruce crowns. To meet these objectives, data were collected and used to: 1) examine the spatial distribution of leaf area, 2) create independent test populations of trees, and 3) evaluate five sampling alternatives for obtaining TLA estimates. Three-dimensional graphs of leaf area per centimetre (APCM) at different heights and relative horizontal widths of tree crowns illustrated that APCM increases towards the top of the tree and the tip of first-order branches. Trees between 50 to 60 years of age showed the highest APCM, whereas trees older than 120 years showed the least APCM. To compare selected sampling designs, test populations were created using several non-linear equations and a bivariate Weibull distribution function. The precision of these models was substantially greater when 1) the seemingly unrelated regression (SUR) fitting method was used instead of the separate multiple linear regression (MLR) or composite fitting method, and 2) branch position from the tip of the tree relative to tree height (zi\) was used rather than relative to crown length (zi<z). Also, bivariate Weibull and ii  beta distribution functions resulted in more precise representations of A P C M within tree crowns than the univariate Weibull and beta distribution functions. The bivariate distribution functions, unlike the univariate distribution functions, provided insights into positional (vertical and horizontal) variability of leaf area distribution in tree crowns. Five sampling designs were compared for mean square error, bias, sampling variance, distribution of the estimates, and cost. A l l sampling designs showed nearly zero bias, with the exception of the linear systematic sample selection procedure owing to the unequal inclusion probability of first-order branches and twigs. Stratified random sampling by tree height classes resulted in the lowest M S E values for the three sample sizes, followed by ellipsoidal, two-stage systematic, simple random, and two-stage unequal probability sampling.  Two-stage unequal probability sampling consistently resulted in the least  precise T L A estimates under different sampling intensities. Since the probabilities were based on branch and twig length, this reflects a poor relationship between twig leaf area and branch/twig length.  iii  T a b l e of C o n t e n t s  Abstract  ii  L i s t of T a b l e s  ix  L i s t of F i g u r e s  xiii  Acknowledgements  1  2  xviii  Introduction  1  1.1  Problem Statement  1  1.2  Objectives  2  1.3  Approach to the Problem  3  Literature Review  5  2.1  Tree Crowns  5  2.1.1  Leaf Area  7  2.1.2  Factors affecting Leaf Area  8  2.1.3  2.1.2.1  Site Quality  2.1.2.2  Disturbances  2.1.2.3  Tree Attributes  2.1.2.4  Stand Attributes  :  8 9 10 .  10  Spatial Variation Within Tree Crowns  11  2.1.3.1  Vertical Variation  11  2.1.3.2  Radial Variation  13 iv  2.1.3.3 2.1.4 2.2  Cardinal Directions  Distribution of Leaf Area and Leaf Weight  14 15  Methods Used for Estimating Canopy Leaf Area  16  2.2.1  Theoretical Basis for Estimating Crown Leaf Area  16  2.2.1.1  Pipe Model Theory  16  2.2.1.2  Hydraulic Model Theory  17  2.2.2  Direct Methods  18  2.2.3  Indirect Methods  19  2.2.3.1  Gap Fraction Analysis  19  2.2.3.2  Mathematical Relationships  19  2.2.3.3  Regression or Allometric Relationships  20  2.3  Sources of Error in Quantifying Crown Leaf Area and Leaf Weight . . . .  22  2.4  Modelling in Tree Crowns  24  2.5  Sampling Methods  27  2.5.1  Sampling Methods Used in Quantifying Attributes of Tree Crowns  27  2.5.1.1  Systematic Sampling  27  2.5.1.2  Ratio and Regression Sampling  28  2.5.1.3  Two-phase Sampling  28  2.5.1.4  Two-stage Sampling  29  2.5.1.5  Stratified Random Sampling  31  2.5.1.6  Ellipsoidal Sampling  32  2.5.1.7  Two-stage Unequal Probability Sampling  32  2.5.1.8  Randomized Branch Sampling  33  2.5.1.9  Importance Sampling  35  2.5.2  Cost Functions Used in Tree Crowns Sampling  37  2.5.3  Evaluation of Sampling Designs  38  v  2.5.4 3  Comparison of Sampling Designs  39  Methods  42  3.1  Collection and Preparation of Data  42  3.1.1  Study Area  42  3.1.2  Data Collection  43  3.1.3  Compilation of Data  46  3.1.4  Verification of Data  47  3.2  3.3  3.4  Examination and modelling of Leaf Area Distributions  47  3.2.1  Graphical Examination of Leaf Area  47  3.2.2  Leaf Area Distribution Models  48  3.2.3  Relationship of Leaf Area Distribution Models to Tree Attributes  49  Creation of Test Populations  51  3.3.1  Number and Distribution of First-Order Branches  52  3.3.2  Length of First-Order Branches  52  3.3.3  Number and Distribution of Second-Order Branches  55  3.3.4  Length of Second-Order Branches  55  3.3.5  Number and Distribution of Third-Order Branches  57  3.3.6  Length of Third-Order Branches  57  3.3.7  Fourth-Order Branches  58  3.3.8  Leaf Area Prediction  59  Evaluation of Sampling Designs  61  3.4.1  Selected Sampling Designs  61  3.4.2  Simulation  67  3.4.3  Measures Used in Assessing the Selected Sampling Designs . . . .  68  vi  4  Results and Discussion  72  4.1  Preliminary Data Analysis  72  4.2  Examining and Modelling Leaf Area  73  4.2.1  Graphical Examination of Leaf Area  75  4.2.2  Leaf Area Distribution Models  83  4.2.3  Relationship of Leaf Area Distribution Models to Tree Attributes  89  4.3  4.4  Generated Test Populations  91  4.3.1  First-Order Branch Length Predictions  91  4.3.2  Second-Order Branch Length and Number Predictions  96  4.3.3  Third-Order Branch Length and Number Predictions  103  4.3.4  Fourth-Order Branches  104  4.3.5  Description of the Test Populations Created  107  Evaluation of Sampling Designs  Ill  4.4.1  Bias  Ill  4.4.2  Root Mean Square Error and Sampling Variance  113  4.4.3  Change in Variance with Tree Characteristics  120  4.4.4  Distribution of the Estimates  121  4.4.5  Cost Efficiency and Relative Ease  131  4.4.6  Overall Ranking of Designs and Summary  131  5  Conclusions and Recommendations  137  6  Literature C i t e d  142  Appendices  158  A  158  Leaf area per centimetre at different relative horizontal lengths  vii  B  Leaf area per centimetre at different relative heights  162  C  Leaf area per centimetre at various heights and horizontal lengths  166  viii  List of Tables  3.1  Attributes, devices, and precision required in data collection  44  4.2  Tree height (HT), diameter outside bark at breast height (DBH), height to the base of live crown ( H T B L C ) , crown width (CW), and average leaf area per cm ( A P C M ) of the 12 hybrid spruce trees measured.  4.3  73  Summary of average leaf area per twig (cm ) and average leaf area per cm 2  ( A P C M in cm /cm) by three age class  74  2  4.4  Estimated parameters for the bivariate Weibull (ci to c ) and beta (c$ to 5  c$) leaf area per cm prediction models using branch position from the top of the tree relative to total tree height as the predictor variable (zii). 4.5  . .  84  Root mean square error ( R M S E in cm /cm) and coefficient of determina2  tion (R ) for the bivariate leaf area per cm prediction models using two 2  definitions of branch position 4.6  85  Root mean square error ( R M S E in cm /cm), coefficient of determination 2  (R ) values for estimating leaf area per cm ( A P C M ) as a function of rela2  tive vertical position (zii) 4.7  87  Estimated root mean square error (ERMSE) in c m / c m and I-square (I ) 2  2  values for two leaf area per cm prediction models using three fitting techniques 4.8  90  Pearson correlation coefficients between the estimated parameters of leaf area per cm prediction models  92  ix  4.9  Root mean square error (RMSE in m) and coefficient of determination (i? ) for five branch length prediction models using branch position from 2  the top of the tree relative to total tree height as the predictor variable  (zii).  n is the number of sampled first-order branches  93  4.10 Root mean square error (RMSE in m) and coefficient of determination  (R ) for five branch length prediction models using branch position from 2  the top of the tree relative to crown length as the predictor variable (zi2).  n is the number of sampled first-order branches  94  4.11 Estimated parameters for five branch length prediction models for each tree using branch position from the top of the tree relative to total tree height as the predictor variable (zi\)  95  4.12 Root mean square error (RMSE in m), coefficient of determination (R ) 2  and (I ) and estimated root mean square (ERMSE) values for estimating 2  the number of second-order branches (TOTSEC) as a function of branch length prediction (BLi).  n is the number of sampled first-order branches.  101  4.13 Root mean square error (RMSE in m) and coefficient of determination  (R ) values for estimating length of second-order branches (BL2) in hybrid 2  spruce crowns, n is the number of sampled second-order branches 4.14 Root mean square error (RMSE in m) and coefficient of determination (R  102 2  or I ) and estimated root mean square (ERMSE) values for estimating the 2  number of third-order branches (NOTRD) as a function of second-order length (BL2). n is the number of sampled third-order branches  104  4.15 Root mean square error (RMSE in m) and coefficient of determination  (R ) values for estimating the length of third-order branches (BL3) as a 2  function of BL\, DFT\, and DFT^. n is the number of sampled third-order branches  105  x  4.16 Branching ratios for the 12 hybrid spruce trees measured in this study.  . 106  4.17 Summary of average twig leaf area ( T W L A in cm ) and tree leaf area 2  ( T L A in m ) using the created test populations and the systematically 2  collected data (bold). N is the number of twigs on each tree crown, n is the number of sampled twigs  108  4.18 Bias (m ) by sample size, sampling design, and sample tree. Bias percents 2  are given in bold. Tree leaf areas (m ) of test populations are given in 2  brackets  112  4.19 Root mean square error (m ) by sample size, sampling design, and sample 2  tree. Tree leaf areas (m ) of test populations are given in brackets. The 2  average numbers of sampled first-order branches and twigs for two-stage systematic sampling, respectively, are given in bold  114  4.20 Efficiencies' of stratified random, ellipsoidal, two-stage systematic, and 1  two-stage unequal probability sampling design relative to SRS. A sample size of 36 twigs was used for comparison  116  4.21 Standard deviations of twig leaf area (cm ) by stratum  117  2  4.22 Summary of average R M S E (m ) values by age classes and sampling de2  signs. A sample size of 36 twigs was used for comparison  120  4.23 Minimum/maximum for the 1000 estimates of total leaf area using simple random, stratified random, ellipsoidal, two-stage unequal probability and two-stage systematic sampling. Tree leaf areas (m ) ( T L A ) of test 2  populations are given in brackets  122  4.24 Average skewness/kurtosis values for the 1000 estimates of total leaf area using simple random, stratified random, ellipsoidal, two-stage unequal probability, and two-stage systematic sampling. Tree leaf areas (m ) (TLA) 2  of test populations are given in brackets xi  123  4.25 Summary of the Shapiro-Wilk (1965) W test statistics of normality and the corresponding probability levels  125  4.26 Summary of sampling and leaf area measurement time (minutes) by sample tree  132  4.27 Summary of total costs (dollars) for sampling and leaf area measurement by sample tree and sampling design. A sample size of 36 twigs was used for comparison  133  4.28 Ranks of five sampling designs evaluated for estimating total tree leaf area. 134  xii  List of Figures  4.1  Leaf area per centimetre (cm /cm) at different relative horizontal lengths 2  by the four cardinal directions for Trees 3, 7, and 10. A relative horizontal length of one indicates values at the tip of the first-order branch 4.2  76  Leaf area per centimetre (cm /cm) at different relative heights by the four 2  cardinal directions for Trees 3, 6, and 10. A relative height of zero indicates values at the tip of the tree 4.3  78  Leaf area per centimetre (cm /cm) at different relative heights and rela2  tive horizontal lengths for Tree 1 (young tree). A relative height of zero indicates values at the tip of the tree. A relative horizontal length of one indicates values at the tip of the first-order branch 4.4  80  Leaf area per centimetre (cm /cm) at different relative heights and relative 2  horizontal positions for Tree 5 (middle age class). A relative height of zero indicates values at the tip of the tree. A relative horizontal length of one indicates values at the tip of the first-order branch 4.5  81  Leaf area per centimetre (cm /cm) at different relative heights and relative 2  horizontal lengths for Tree 10 (old tree). A relative height of zero indicates values at the tip of the tree. A relative horizontal length of one indicates values at the tip of the first-order branch 4.6  82  First-order branch lengths (estimated and measured) at different heights above ground using Mitchell's 1975 approach for each cardinal direction.  xiii  97  4.7  First-order branch lengths (estimated and measured) at different heights above ground using quadratic interpolation for each cardinal direction.  4.8  .  First-order branch lengths (estimated and measured) at different heights above ground using linear interpolation for each cardinal direction  4.9  98  99  Estimated tree leaf area (model versus sample estimates) of the 12 sample trees  110  4.10 Root mean square error (m ) and tree number for simple random sampling 2  (SRS), stratified random (STRS), ellipsoidal (ELLIP), two-stage systematic (TSYS), and two-stage unequal probability sampling (TSUEP) designs. A sample size of 36 twigs was used for comparison  119  4.11 Distribution of the 1000 estimates of tree leaf area using simple random sampling for Tree 5 for sample sizes of 12, 36, and 48  126  4.12 Distribution of the 1000 estimates of tree leaf area using stratified random sampling for Tree 5 for sample sizes of 12, 36, and 48  127  4.13 Distribution of the 1000 estimates of tree leaf area using ellipsoidal sampling for Tree 5 for sample sizes of 12, 36, and 48  128  4.14 Distribution of the 1000 estimates of tree leaf area using two-stage unequal probability sampling for Tree 5 for sample sizes of 12, 36, and 48  129  4.15 Distribution of the 1000 estimates of tree leaf area using two-stage systematic sampling for Tree 5 for sampling intensities of I, II, III and IV. . 130 A. 16 Leaf area per centimetre (cm /cm) at different relative horizontal lengths 2  by the four cardinal directions for Tree 1, 2, and 4. A relative horizontal length of one indicates values at the tip of the first-order branch  xiv  159  A. 17 Leaf area per centimetre (cm /cm) at different relative horizontal lengths 2  by the four cardinal directions for Tree 5, 6, and 8. A relative horizontal length of one indicates values at the tip of the first-order branch  160  A. 18 Leaf area per centimetre (cm /cm) at different relative horizontal lengths 2  by the four cardinal directions for Tree 9, 11, and 12. A relative horizontal length of one indicates values at the tip of the first-order branch  161  B. 19 Leaf area per centimetre (cm /cm) at different relative heights by the four 2  cardinal directions for Tree 1,2, and 4. A relative height of one indicates values at the tip of the first-order branch  163  B.20 Leaf area per centimetre (cm /cm) at different relative heights by the four 2  cardinal directions for Tree 5, 7, and 8. A relative height of one indicates values at the tip of the first-order branch  164  B. 21 Leaf area per centimetre (cm /cm) at different relative heights by the 2  four cardinal directions for Tree 10, 11, and 12. A relative height of one indicates values at the tip of the first-order branch  165  C. 22 Leaf area per centimetre (cm /cm) at different relative heights and rela2  tive horizontal lengths for Tree 2 (young tree). A relative height of zero indicates values at the tip of the tree. A relative horizontal length of one indicates values at the tip of the first-order branch  167  C.23 Leaf area per centimetre (cm /cm) at different relative heights and rela2  tive horizontal lengths for Tree 3 (young tree). A relative height of zero indicates values at the tip of the tree. A relative horizontal length of one indicates values at the tip of the first-order branch  xv  168  C.24 Leaf area per centimetre (cm /cm) at different relative heights and rela2  tive horizontal lengths for Tree 4 (young tree). A relative height of zero indicates values at the tip of the tree. A relative horizontal length of one indicates values at the tip of the first-order branch  169  C.25 Leaf area per centimetre (cm /cm) at different relative heights and relative 2  horizontal lengths for Tree 6 (middle age tree). A relative height of zero indicates values at the tip of the tree. A relative horizontal length of one indicates values at the tip of the first-order branch  170  C.26 Leaf area per centimetre (cm /cm) at different relative heights and relative 2  horizontal lengths for Tree 7 (middle age tree). A relative height of zero indicates values at the tip of the tree. A relative horizontal length of one indicates values at the tip of the first-order branch  171  C.27 Leaf area per centimetre (cm /cm) at different relative heights and relative 2  horizontal lengths for Tree 8 (middle age tree). A relative height of zero indicates values at the tip of the tree. A relative horizontal length of one indicates values at the tip of the first-order branch  172  C.28 Leaf area per centimetre (cm /cm) at different relative heights and relative 2  horizontal lengths for Tree 9 (old tree). A relative height of zero indicates values at the tip of the tree. A relative horizontal length of one indicates values at the tip of the first-order branch  173  C.29 Leaf area per centimetre (cm /cm) at different relative heights and relative 2  horizontal lengths for Tree 11 (old tree). A relative height of zero indicates values at the tip of the tree. A relative horizontal length of one indicates values at the tip of the first-order branch  174  C.30 Leaf area per centimetre (cm /cm) at different relative heights and relative 2  horizontal lengths for Tree 12 (old tree). A relative height of zero indicates values at the tip of the tree. A relative horizontal length of one indicates values at the tip of the first-order branch  xvii  Acknowledgements  I would like to extend special thanks to Dr. Valerie LeMay, my thesis advisor, for her guidance, constant support, and thought-provoking discussion throughout all phases of this research. I am also grateful to other members of my committee Drs. Philip Burton, Peter Jolliffe, Antal Kozak, Peter Marshall, and Mr. Ian Cameron for their confidence and direction since the inception of this project. Their encouragement is greatly appreciated. Thanks are also due to: Dr. Peter Marshall for his thorough editorial work, Dr. Peter Jolliffe for the fruitful discussion on fractal geometry, Dr. Julien Demaerschalk for his involvement at the start of this project, Ian Cameron for his keen interest in this project, and Christ Hope for his assistance in the data collection phase. The financial support of the National Science and Engineering Research Council of Canada (NSERC), the British Columbia Ministry of Forests, the Forest Renewal British Columbia ( F R B C ) , and the Faculty of Forestry at the University of British Columbia for this research is gratefully acknowledged. I dedicate this dissertation to the happy memory of my late father, Ato Hailemariam Abawana, in appreciation for his quintessence and wisdom that he has provided me over the years. May his soul rest in peace.  T. H / M .  xviii  Chapter 1  Introduction  1.1  P r o b l e m Statement  Efficient estimates of morphological tree crown attributes such as total crown biomass, leaf area, and leaf area index (i.e., leaf area per unit of ground area) are needed to investigate the productivity of a tree or stand. Total leaf area and leaf weight have been used extensively as indicators of the primary productivity of trees. Tree crowns form a complex three-dimensional surface and are the center of various physiological processes. They are important factors in determining the amount of light captured for photosynthesis (Massman 1982), tree form, and taper (Larson 1963). Attempts to model the impact of crown performance on tree growth are hampered by the difficulty in accurately estimating leaf area and leaf weight available for photosynthesis on individual trees. Variations in morphological attributes within tree crowns at different heights, relative horizontal positions, and cardinal directions of the crown contribute to this difficulty. Knowledge of within tree crown variation (i.e., in the vertical, radial, and cardinal directions) and the spatial distribution of foliage will contribute to the understanding of the complex nature of tree crowns and the ecophysiological processes. Several researchers have used probability densities to describe the vertical distribution of leaf area and leaf weight within tree crowns. For instance, Gillespie et al. (1994) used a Weibull probability density function on loblolly pine (Pinus taeda L), and Kellokami et al. (1980) used a beta  1  Chapter 1.  Introduction  2  distribution in young Scot's pine (Pinus sylvestris L). Three-dimensional populations, such as leaves within tree crowns, present unusual sampling problems. The wide variation of tree crowns in their size, shape, density, and distribution of branches has contributed to the difficulty of tree crown sampling. These variations have rendered some conventional sampling designs relatively inadequate. Crown biomass and total leaf area monitoring and evaluation involve detailed examination of samples from tree crowns. Measurement of foliage distribution in an entire forest stand requires a thorough understanding of within-crown variation and an efficient sampling scheme. Previous research reported that various morphological attributes appear to vary not only with species, but also with the methods and sampling schemes used for their estimation. Cognizant of the above facts, it is important to evaluate efficient sampling alternatives that account for the observed spatial and temporal differences within tree crowns.  1.2  Objectives  The principal objectives of this research were: 1. To describe the distribution of leaf area in hybrid spruce (Picea engelmannii * glauca * sitchensis) crowns grown in the northwestern interior of British Columbia (BC); and 2. To assess and evaluate sampling alternatives that consider variations in the radial and vertical distribution of foliage in estimating morphological attributes of hybrid spruce crowns.  Chapter 1.  1.3  Introduction  3  Approach to the Problem  Leaf area and leaf weight data were collected for hybrid spruce trees at the Ken Fire and Date Creek study areas within the Interior Cedar Hemlock (ICH) biogeoclimatic zone of the Kispiox Forest District, Prince Rupert Forest Region of British Columbia. In this study, hybrid spruce trees were selected based on having well defined whorl branches and deep green needles. The sample trees were selected from three age classes to represent a range of forest conditions. These data were used: (1) to examine the spatial distribution of leaf area, (2) to create independent test populations of trees, and (3) to evaluate five sampling alternatives for obtaining tree leaf area estimates. This dissertation is organized in five chapters. Chapter 1 gives a brief introduction and states the objectives of the study. Chapter 2 provides a literature review on tree crowns and sampling designs. The literature review focuses on single-tree leaf area estimates, rather than on stand-level leaf area estimates. This is done to match this review with the objectives of this study and to draw attention to forest management applications. Stand-level leaf area estimates are deliberately omitted in this dissertation. Chapter 3 embodies the four major parts of the methodology used in this study: 1. Evaluation of leaf area variability - graphical methods are used to select models for leaf area distribution and to analyze variability of leaf area in tree crowns; 2. Creation of the test population - the data collected on 12 sample trees are used to model the distribution of leaf area and then to create 12 independent test populations. 3. Simulation experiments - for each test population, simulation experiments are performed using three sample sizes (n=12, 36, and 48 sample twigs per tree); and  Chapter 1.  Introduction  4  4. Evaluation of sampling designs - the estimated leaf area for each simulation is compared to observed total leaf areas for the simulated trees (population parameters). The sampling strategies were evaluated for bias, sampling variance, the distribution of the estimator, and relative ease of use. Chapter 4 describes and discusses the results of the study in four sections, while Chapter 5 contains the conclusions and recommendations for further research. The sampling designs that are evaluated in this study provide a suitable framework for plant physiologists and ecologists who are faced with the task of estimating other morphological attributes and physiological processes of trees. Accurate estimates of tree leaf area and weight are needed: (1) to assess responses to silvicultural treatments, (2) to refine and/or extend existing tree growth models, and (3) as input to light extinction models. This research makes a number of contributions. 1. The study increases understanding of the variations of leaf area in hybrid spruce crowns via graphical examinations. 2. A bivariate distribution function was used to model leaf area per unit twig length. Previous studies have provided insights into the positional variability of leaf distribution in tree crowns, but have not produced any method that combined these variations (i.e., vertical and horizontal distribution) at specified points in tree crowns. 3. B y comparing the efficiency of selected sampling designs, this study improves the estimation of tree leaf area and, in turn, contributes to tree growth modelling.  Chapter 2  Literature Review  2.1  Tree Crowns  Tree crowns have been used to: (1) estimate competition via crown length (Assmann 1970); (2) describe the densities of forest stands in terms of crown closure (Clutter et al. 1983, Husch et al. 1982); (3) estimate tree growth development (Sprinz and Burkhart 1987), mortality (Arney 1972), and wood quality (Kershaw et al. 1990); (4) improve the accuracy of individual tree taper and volume equations (Hann et al 1987); (5) improve distance independent growth models via height to the base of live crown or crown ratio (Wykoff et al. 1982); and (6) indicate growth dynamics among individual trees via influence-zone overlap (Bella 1971), tree vigor via crown length (Kershaw et al. 1990), tree productivity via leaf area index (LAI) or via tree leaf area and leaf weight (Wang et al 1990). According to Ziede (1992), future forest growth and yield research will be focused on the structure and function of tree crowns. Ziede stated challenges will be associated with: 1) crown shape, by stating "Is crown form a cone, a paraboloid or a spheroid? It fits none of these, nor any other form known in Euclidean geometry"; and 2) the aggregate nature of a tree crown, by asserting that "Tree crown is neither a three-dimensional solid comprised of branches, twigs and leaves nor a two-dimensional photosynthetic surface. It can be viewed as a collection of holes that serve to conduct sunlight and gases or as a multi-leveled hierarchy of clustered dots (pigment molecules and chloroplasts)". One  5  Chapter 2. Literature Review  6  can extend Ziede's argument by stating that the tree crown is a quasi four-dimensional attribute, extending in space through three dimensions and changing in time, the fourth dimension. This feature is described as a space-time continuum in many physics textbooks. From the literature, there appears to be a separation between mensurational research (focusing on volume, tree height, tree taper, tree diameter) and ecophysiological research (focusing on light penetration, photosynthetic capability, shade tolerance).  This gap,  combined with the complex and non-merchantable nature of tree crowns, has contributed to the lack of knowledge of tree crowns (Ziede 1992). To interweave ecophysiological variables with mensuration variables, Schulze et al. (1977) suggested that whole-tree photosynthesis (a component of carbon gain or growth) be used as a surrogate variable for conventional mensuration variables such as tree diameter and height. Spurr (1952) noted that the erratic nature of tree crowns and the interlacing of branches in closed stands contribute to the inaccuracies of crown measurements such as crown diameter.  Despite the complex, erratic, and hard-to-measure nature of tree  crowns (Ziede 1992), estimates of morphological attributes and total crown biomass are useful in investigating many processes such as photosynthesis, light transmission, transpiration, shoot growth, cambial growth, and competition among trees in a stand (Kinerson and Fritschen 1971, Husch et al. 1982). In order to have a firm understanding of tree crowns, knowledge of leaf area, shape of tree crowns, and the spatial distribution of leaf area and weight are essential. Leaf area and weight are basic morphological attributes that are essential for examining and modeling of tree crowns because the amount of dry matter production by an individual tree depends on the photosynthetic surface area (leaf area) and the rate of photosynthesis per unit leaf area (Kramer and Kozlowski 1979).  7  Chapter 2. Literature Review  Leaf A r e a  2.1.1  Generally, leaf area denotes one half of the total surface area of leaves, which assume that a leaf receives light mainly from one side. Most leaf area estimates reported in the literature refer to this "projected leaf area". Indices that combine leaf area and leaf weight have been used to describe foliage morphology by Del Rio and Berg (1979), Kinerson and Fritschen (1971), Oren et al. (1986), Kimmins (1987), and others. Such indices include specific leaf area (SLA, projected leaf area per unit of dry leaf weight (cm /g)), and 2  its reciprocal, specific leaf weight (SLW, dry leaf weight per unit of projected leaf area (g/cm )). 2  Leaf area (assimilatory surface area) and leaf weight are widely used morphological attributes to describe photosynthetic capacity (PC) of a tree or a stand (Oren et al. 1986 and Del Rio and Berg 1979). Kinerson and Fritschen (1971) recommended that leaf weight be used in nutrient cycling and total biomass production studies, while leaf area be used in gas and energy exchange studies. Oren et al. (1986) and Drew and Ferell (1977) argued that SLW and mesophyll area per unit leaf area reflect environmental conditions and vary with P C . High SLW at high light intensity results primarily from increasing leaf thickness due to additional layers of mesophyll and high concentrations of soluble carbohydrate per unit leaf area (Oren et al. 1986). Tree leaf area can be used as a response variable in analyzing the impacts of silvicultural treatments such as fertilization and thinning. Estimates of tree leaf area can be either at the single-tree level or at the stand-level as L A I (leaf area per unit of ground area). Single-tree leaf area estimates are more useful than stand level estimates (Neumann 1990) when: 1) a study involves a small number of trees; 2) the individual tree response (e.g., respiration or productivity per tree) is of interest; or 3) the response of different crown classes within a stand is required. Despite these reasons, single-tree leaf  Chapter 2. Literature Review  8  area estimates are not widely used due to lack of efficient sampling designs that provide accurate estimates of tree leaf area (Neumann 1990). 2.1.2  Factors affecting Leaf A r e a  The spatial arrangement of leaf area affects light interception capability within individual crowns of a tree species. The probability of light interception is proportional to crown length, leaf surface area density, and leaf orientation (Martens et al. 1993). The longer the crown length, the higher the probability of light interception. Leaves that are facing the beam of radiation will have a higher probability of light interception than those that are parallel to the beam. However, the effect of leaf orientation on the amount of light interception is more highly pronounced in broad leaves than in conifers. L A I at a stand level has been reported to vary from 5 to 52 m per m of ground 2  2  in some western United States conifer forests (Kimmins 1987). Generally, factors that influence leaf area can be classed into four categories which are neither distinct nor independent.  These are site quality, disturbance, individual tree attributes, and stand  attributes. The first two of these factors influence both single-tree and stand-level foliage production. Individual tree and stand attributes and their influence on single-tree and stand-level leaf area estimates are discussed separately. The following subsections provide a brief overview of these factors.  2.1.2.1  Site Quality  Site quality is a result of the interaction of climatic factors (e.g., air temperature, precipitation, and wind), soil factors (e.g., physical and chemical characteristics, soil moisture, soil biota), and topographic attributes (e.g., slope, elevation, and aspect). Stand growth per unit land area is a function of site quality among other things. There are positive correlations among tree height growth, site quality, total leaf area and leaf weight of a tree  Chapter 2. Literature Review  9  (Spurr 1952). Trees on poor sites (i.e., trees grown on soils with poor nutrient content or on xeric soils) show lower L A I development than those grown on rich sites (Kimmins 1987). Significant influence of site fertility on L A I development in a Douglas-fir stand was reported by Brix and Mitchell (1983). Specifically, leaf area is known to vary with: 1) water balance of the site (the drier the site, the lower the leaf area; Kimmins 1987); and 2) nutritional deficiency (the poorer the site, the lower the leaf area; Waring et al. 1981). Stand level leaf area varies in different biogeoclimatic zones due to both site characteristics and species composition. For example, coastal Douglas-fir (Pseudotsuga menziesii var. Menziesii Franco), has higher L A I and total biomass per unit area and per tree than the interior Douglas-fir (Pseudotsuga menseizii (Mirb.) Franco var. glauca (Beissn.)) and true firs (Abies spp.) (Kimmins 1987). Waring et al. (1982) point out that climate plays a major role in determining the growth rate and the amount of leaf area that can be attained by a particular tree or stand. Extreme climatic conditions such as extreme temperatures limit the growing season, and thereby, constrain the foliage production. Kimmins (1987) provides an overview of the interaction of climatic, soil, topographic and biological factors and their influence on the amount of leaf area and leaf weight attained by a single tree or a stand.  2.1.2.2  Disturbances  Disturbances such as fire, wind, flooding and insect outbreaks usually result in the reduction or elimination of leaf area and leaf weight of a tree or a stand, depending on the extent of the disturbance. The physiological influence of drought is discussed in detail by Kramer and Kozlowski (1979). The authors assert that the cells shrink and the cell walls relax, characterized by lower hydrostatic (turgor) pressure and high solute concentration as the water content of the tree decreases. This finally results in the rupture of  Chapter 2. Literature Review  10  the membrane and in the senescence of the leaf. Water stress not only limits the size of individual leaves, but it also limits the number of leaves and decreases the growth rate of branches. The extent of, and mechanisms for leaf defoliation as a result of insects and diseases are reported by various authors in standard plant pathology (e.g., Manion 1990) and entomology textbooks (e.g., Coulson and Witter 1984). Defoliation due to wind and flooding are reported by Kramer and Kozlowski (1979) and Kimmins (1987). 2.1.2.3  Tree Attributes  In young trees, the crown length is not fully developed; young trees will have lower total leaf area and leaf weight than mature trees. When two trees with the same diameter and height but different crown lengths are compared, the one with the longer crown length will have higher total leaf area and leaf biomass. Dominant trees are taller and bigger in diameter and, usually, have wider crown width and longer crown length than intermediate and overtopped trees. Thus, dominant trees will have higher total leaf area and leaf weight per tree than the latter two. 2.1.2.4  Stand Attributes  In assessing total leaf area and leaf weight, stand density can be categorized based on average crown closure (% of crown cover), basal area per hectare, and the number of stems per hectare. However, the later two indicators are meaningful only if tree age or some measure of average tree size is known. To integrate these indicators into a measure of stand density, several indices have been developed (e.g., Curtis' relative density index, Reineke's stand density index, and the crown competition factor) (Clutter et al. 1983). In general, leaf area and leaf weight increase with density up to a certain point and then decrease. Kimmins (1987) argue that stand density affects the amount of total leaf  Chapter 2. Literature Review  11  area attained on a particular site; very open stands have low L A I due to fewer trees per unit of area. In a high density stand, trees have narrow growing space which reduces crown expansion and branch growth; as a result, the production of photosynthate will be reduced and so will the total leaf area and leaf weight per tree (Mitchell 1975). Trees that grow in open stands have wide crowns that cover most of the stem (high leaf area per tree) whereas trees that grow in dense stands have narrow crowns that are restricted to the upper part of the stem as the crowns of trees close more rapidly and the lower branches die. Generally, as the stand density increases, the ratio of crown length to tree height (live-crown ratio) decreases. Due to their rapid establishment and optimal spacing, planted monospecific stands will achieve higher L A I at a given period of time than naturally regenerated stands. Higher estimates of L A I in northwestern U.S. conifer forests were found in "mediumaged" and "well-stocked stands" (Kimmins 1987). 2.1.3  Spatial Variation Within Tree Crowns  Knowledge of within-tree crown variation (i.e., vertical, radial, and cardinal directions) and spatial distribution of foliage will contribute to understanding the complex nature of tree crowns and their ecophysiological processes.  Examination of the spatial and  temporal distribution of foliage in tree crowns is important, as leaf area and leaf weight vary with leaf age, tree vigor, the total quantity of foliage, and time. 2.1.3.1  Vertical Variation  Most leaves are produced in the upper one-half to two-thirds of the crown, with better lit portions producing more foliage than shaded portions (Flemming et al. 1990). The vertical distributions of needle area within the crown have been studied for: red pine (Pinus resinosa Ait.) (Stephen 1969); Scots pine (Whitehead 1978); Douglas-fir (Maguire and  12  Chapter 2. Literature Review  Bennet 1996, Woodman 1971); jack pine (Pinus banksiana Lamb.); lodgepole pine (Pinus contorta Dougl.) (Gary 1978); and eastern larch (Larix laricina (Du Rio) K . Koch) (Strong and Zavitkovski 1978). In Douglas-fir crowns, foliage is distributed asymmetrically with the maximum amount often located at a height approximately equal to 80% of the tree height (Woodman 1971). Needle area and biomass were distributed in the ratio 1.0:3.5:2.0 (top: middle: bottom) for open-grown Douglas-fir and 1: 2: 1 for silver fir (Abies concolor (Gord.) Engelm.) (Schmid and Morton 1981). Kinerson and Fritschen (1971) recognized significant vertical variation in their data set and developed three separate equations for the top-, middle-, and bottom-third of tree crowns for estimating leaf area from leaf weight. Using limited samples (one separate sample from the top, middle, and bottom) of three Juniper virginiana L. crowns, Gregg (1992) found that S L A was highest at the bottom of the crown and lowest at the top. This trend is consistent with the pattern of S L A variation with crown position shown for other conifers such as loblolly pine (Shelton and Swizer 1984), black spruce (Picea mariana Mill. B.S.P.) and eastern larch (Strong and Zavitkovski 1978). Webb and Ungs (1993) also observed an increase in needle area density (NAD) (i.e., needle surface area per crown volume m m ) with tree height for 2  - 3  an open grown Douglas-fir tree. Kershaw and Maguire (1995) reported "branch-level leaf area" values at various heights of Douglas-fir, western hemlock (Tsuga heterophylla (Raf.) Sarg.), and grand fir (Abies gradis (Dougl.) Lindl.) tree crowns. The existence of vertical variation of leaves in tree crowns is reported by various researchers at several scales (e.g., needles, shoots, N A D ) . Strong and Zavitkovski (1978) found that needles from the lower third of eastern larch crowns were shorter with smaller surface areas and dry weight than those from the upper third. Kramer and Kozlowski (1979) note that pine shoots in the upper crown generally show more annual growth flushes than those in the lower crown. In Larix, the lower 40% of the canopy represents  Chapter 2. Literature Review  13  about 5% of the total foliage biomass and 3% of the canopy photosynthesis index (Schulze et al. 1977). Wang et al. (1990) found that the maximum N A D within the crown was located approximately at one-third of the tree-crown length from the crown base. After recognizing the vertical complexity found within tree crowns, Binkley and Merrit (1977) divided pine crowns into seven strata to assess the extinction of light energy in each stratum. Similarly, Kinerson and Fritschen (1971) sampled one branch per whorl as a foliage sample in estimating needle surface area of Douglas-fir. 2.1.3.2  Radial Variation  Radial variation has not been as widely investigated as vertical variation of foliage in tree crowns. Studies of horizontal distributions have been done for: western hemlock, and grand fir by Kershaw and Maguire (1996); Douglas-fir by Jensen and Long (1983) and by Kershaw and Maguire (1996); and radiata pine (Pinus radiata D. Don) by Whitehead et al. (1990). In their intensive study on Norway spruce (Picea abies (L.) Karst.), Schulze et al. (1977) found significant differences in annual growth of needles and twigs, not only at different heights of the tree, but also along different radial positions of single branches at a particular tree height. They also found that the S L A of needles of identical age decreased from the trunk to the tip of the same branch. Schulze et al. (1977) found that two- to three-year-old needles were about 10% heavier per unit surface area (SLW) than one to two-year-old needles.  The authors argued that this might be caused by  changes in the epidermis or by an increased mineral content.  In studying the spatial  distribution of foliage in radiata pine, Siemon et al. (1980) found that the percentage of one-year-old foliage increased from 52% to 75%, whereas that of two-and three-year-old needles decreased from 28% to 22% and 20% to 3%, respectively, in the upper third of the crown. In the middle and lower crown, the spatial distribution of foliage by crown  Chapter 2. Literature Review  14  position was constant at 37% (one-year-old), 27% (two-year-old) and 36% (three-year old or more). Leave longevity varies from a couple of months (broad leaves) to fifteen years (e.g., Pacific silver fir (Abies amab'ilis (Dougl.) Forbes)) (Wilson 1989). The effects of fertilization and other silvicultural treatments on vertical distribution of foliage have been documented (Kellokami et al. 1980, Wang et al. 1990, Mori and Hagihara 1991), but no study has been done on their effect on radial distribution of foliage.  2.1.3.3  Cardinal Directions  In order to attain a good understanding of within-tree crown variation, it is important to assess not only the vertical and radial variations, but also the variations that occur among the cardinal directions. In their analysis of cross sections of sugar maple leaves (Acer saccharum Marsh.) from different crown positions, Kramer and Kozlowski (1979) noted that sun leaves from the south side of the tree had thicker cuticles over the upper epidermis and longer palisade parenchyma cells than shade leaves from the center of the crown and shade leaves from the base of the crown. Schulze et al. (1977) found that sample branches from the southern side of Engelmann spruce (Picea engelmannii Parry) had about 60% more one-year-old needles than sample branches from the north side, and that leaves from north aspects had higher S L A than those from south aspects. The photosynthetic capability of Douglas-fir varies significantly at different vertical and horizontal locations among the four aspects (Woodman 1971). The production of leaves also varies with aspect; the side facing the equator was usually the most productive in producing foliage and photosynthates (Schulze et al. 1977). Flemming et al. (1990) noted that more cones were produced at the south side of Jack pine crowns than at the other cardinal directions.  Chapter 2. Literature Review  2.1.4  15  Distribution of Leaf Area and Leaf Weight  The loose usage of the word "distribution" in ecological literature is a constant source of confusion, as it is often used both in its colloquial ("arrangement" or "pattern") and statistical senses without any mention of its context. To avoid this confusion, this thesis uses it in its statistical context as defined by Kogan and Herzog (1980) (i.e., "the way in which variate values are apportioned, with different frequencies, in a number of possible classes").  Knowledge of the density of leaf area and leaf weight at any given crown  position will indicate the zone of maximum and minimum amount within tree crowns. This, in turn, will enable selection of an efficient sampling design. Many researchers have fitted continuous probability distributions to describe the vertical distribution of foliage. For example, Gilmore et al. (1996) used the Weibull distribution for balsam fir (Abies balsamea (L.) Mill.). Among probability density functions used to describe the distribution of leaves at different heights in Douglas-fir, Massman (1982) found beta and chi-squared distribution functions to fit well. Wang et al. (1990) found that maximum N A D in different age classes was close to the center of the tree crown in the horizontal direction and the maximum N A D within the tree crown was up to four times larger than the average N A D . In their work, the vertical and horizontal distributions were standardized between 0 and 1. Kershaw and Maguire (1996) found no significant differences between Weibull, beta, normal and Johnson's SB probability distributions in modelling the horizontal distribution of foliage. Larson (1963) and Tadaki (1966) discussed how the vertical distribution of foliage strongly dictates the vertical distribution of bole increment using Pressler's law.  Chapter 2. Literature Review  2.2  16  M e t h o d s U s e d for E s t i m a t i n g C a n o p y Leaf A r e a  Various theories based on physiological and morphological reasoning have been used for estimating crown leaf area and leaf weight. Two of the most notable theories are the pipe model and the hydraulic model. Canopy leaf area has also been estimated directly using a leaf area meter, glass beads, plant canopy analyzer, planimetric methods or gravimetric methods, or indirectly by using gap fraction analysis, mathematical relationships, regression or allometric relationships.  2.2.1  T h e o r e t i c a l Basis for E s t i m a t i n g C r o w n Leaf A r e a  The pipe model and the hydraulic model theories are based on detailed morphological and physiological observations of trees. These theories have been used to estimate total leaf area and leaf weight of tree crowns by Shinozaki et al. (1964) and Whitehead et al. (1984). 2.2.1.1  Pipe M o d e l Theory  Shinozaki et al. (1964) presented the "pipe model theory" to explain the strong relationship between a tree's sapwood cross-sectional area at the base of the live crown and tree leaf area above this point. The theory contends that a given transpiring leaf mass needs a proportional amount of conducting stemwood. This theory is widely used to develop allometric relationships between the cross-sectional sapwood area at the base of the live crown and total leaf area of several species. Research results obtained by Maguire and Hann (1987), and Neumann (1990) support this theory. Several researchers have examined the validity of the pipe model theory by investigating the relationship of various stem measurements and by considering their ease of measurement.  For example, Snell and Brown (1978) asserted that the improvement in  17  Chapter 2. Literature Review  precision in biomass estimation from sapwood basal area does not justify its use considering the extra work required to measure sapwood area. Valentine et al. (1994) argued that the pipe model theory is not universally applicable because this theory does not hold when there is butt swelling, forking, and other circumstances such as top damage of a tree crown. The ratio of sapwood cross-sectional area to foliage quantity varies with the growth rate, the size of tree, and distance of the measuring point from the crown. The pipe model theory was also used by Valentine (1985) to derive growth rates of tree basal area and height from the growth rate of the cross-sections of the "pipes" (tracheids or vessel elements which contribute to the xylem) and from the growth rates of pipes that extend to the tip of the tree, respectively. Using this theory, Valentine et al. (1994) assessed the spatial distribution of dry leaf weight per unit cross-section of branches at different heights on the tree. Branches in the lower third stratum had less dry weight per unit of cross-sectional area than branches in the upper two strata. To facilitate the use of the pipe model theory for estimating canopy leaf area, Valentine et al. (1994) developed a function to estimate cross-sectional basal area at the base of live crown (A) from the cross-sectional basal area at breast height (B), crown length (CL), and height of a loblolly pine tree (H).  A = B x R  (2.1)  R = CL/(H - 1.3)  (2.2)  where R is the modified live-crown ratio.  2.2.1.2  Hydraulic M o d e l Theory  Zimmermann (1983) demonstrated that the ratio between the amount of foliage and the water conducting wood varies at different positions in tree crowns. The author also  Chapter 2. Literature Review  18  stated that resistance to water flow in the transport path increases away from the stem. To compensate for the poorer water conductivity at a greater distance from the base of the stem, less leaf area would be supported by the same cross-sectional area that occurs closer to the stem base. Berninger and Nikinmaa (1994) found low needle area to sapwood area ratios in the canopy supporting Zimmermann's (1983) hypothesis which stated that the hydraulic structure of trees favors leaves on the main axis. The hydraulic model contends that the flow rate of water (6F) through a tree crown is positively correlated to: the cross-sectional sapwood area (As), the water potential difference(A), and the saturated permeability of the sapwood (K); and negatively correlated to the viscosity of the water(C) and the length of the stem (L) (Equation 2.3).  SF = KAsA/C  xL  (2.3)  Whitehead et al. (1984) used the hydraulic model to account for the influence of the hydraulic pathways between the stem and foliage in calculating tree leaf area. Hinckley and Ritchie (1970) observed the underlying water potential variation within tree crowns. Branches at the base of the crown had more negative water potential than those at the top of amabilis fir trees. The pipe model theory does not consider the characteristics of the hydraulic pathway.  2.2.2  Direct M e t h o d s  Since the early 1960's, several direct methods of leaf area measurement have been used for agricultural crops. Such methods include plant canopy analyzer, planimetric and gravimetric methods. Most plant canopy analyzers use fisheye lenses to project a hemispheric  Chapter 2. Literature Review  19  image of the canopy. In the planimetric method, leaf area is obtained by direct measurement using calibrated video or light transmission devices. In the gravimetric method, an outline of the leaf is traced, or a print is made on a paper which has a uniform area distribution. The leaf shape is cut from the paper and weighed. The leaf area is then calculated from the weight to area relation of the paper, established by weighing pieces of paper of known area. Accurate determinations of the area to weight ratio of leaves (from a subsample) then allows estimation of total crown leaf areas by weighing, which is generally easier than direct area measurement. However, it assumes a constant area to weight ratio within the crown.  2.2.3 2.2.3.1  Indirect M e t h o d s G a p Fraction Analysis  Gap fraction analysis determines the fraction of sky visible from within the canopy for a designated region in the sky hemisphere. This method is fast, less expensive than the direct methods, and is based on fisheye photographs. The accuracy of fisheye photographs depends on the sky conditions and meeting certain assumptions such as random leaf positioning and random azimuthal orientation of foliage. However, these assumptions are rarely met in nature, and thus, the technique should not be used in canopies with clumped foliage (Martens et al. 1993). This method uses the Beer-Lambert Law (Jarvis and Leverenz 1983, Marshall and Waring 1986) to calculate the L A I of a stand.  2.2.3.2  M a t h e m a t i c a l Relationships  Martens et al. (1993) developed a mathematical relationship to estimate L A I , using the measurement of solar transmittance through a tree crown:  Chapter 2. Literature Review  LAI  20  = - \n{9i/ej) x CosQ/k  (2.4)  where LAI is the leaf area index; #;= solar radiance below the canopy; 9j— solar radiance above the canopy; Cos0=the cosine of the solar incident angle; and k= light extinction coefficient. Martens et al. (1993) called the ratio of  9i/0j,  the gap fraction.  The Beer-Lambert logarithmic equation assumes that foliage is randomly distributed in space and leaf inclination angles are spherically distributed (Jarvis and Leverenz 1983). Chacko (1960) noted that if sampling units in a population are assumed to be arranged in space at random then a systematic sample is clearly equivalent to simple random sampling of the units.  2.2.3.3  Regression or Allometric Relationships  Canopy leaf area has been estimated on a few sample trees using regression or allometric relationships that have been established between leaf area and other attributes that are easy to measure and determine. Such attributes include: diameter outside bark at breast height (Baldwin 1989, Catchpole and Wheeler 1992), cross-sectional sapwood area at breast height (Waring et al. 1982); cross-sectional sapwood area at the base of the live crown (i.e., using the pipe model theory) (Maguire and Hann 1987); diameter at stump height (diameter outside bark at 0.3 m above ground) (Helgerson et al. 1988); the ratio of live crown length to total tree height (Loomis et al. 1966); the distance from breast height to the base of live crown point (BLC) (Dean and Long 1986); and the proportion of light that passes through the canopy (Martens et al. 1993). Regression models developed using hemispherical photos vary by stand age and sky condition. In their comparison of allometric relationships based on stem measurements and gap-fraction analysis methods, Bidlake and Black (1989) found that L A I estimates  Chapter 2. Literature Review  21  obtained by the two methods were not significantly different. Marshall and Waring (1986) compared methods of L A I estimation using litter fall, light interception, sapwood crosssectional area, and tree diameter at breast height. Among these variables, sapwood crosssectional area was the best predictor of L A I . A comprehensive review of numerous linear and nonlinear regression models for various North American tree species was provided by Neumann (1990). Regression models are developed from data collected from a specific forest population (e.g., a particular tree species at a specific location). Despite their predictive and analytical uses, regression models are limited in quantifying morphological attributes (Gregoire et al. 1995). First, extrapolation outside the range of data on which these models are built is likely to reduce their accuracy. Second, the predictive ability of the regression model will be uncertain when the model is applied to a subpopulation (e.g., a subregion, Long and Smith 1988), a new population (e.g., the same tree species in a different location or a similar species in the same location (Kaufmann and Troendle 1981)) or the same population with alternative features (e.g., the same species at the same location but with different stand structures and densities (Long and Smith 1988, Thompson 1989), or stand treatments (Brix and Mitchell 1983, Whitehead et al. 1984)). Several researchers have attempted to include other predictor variables such as mean annual ring width, sapwood permeability or crown length to overcome the site- or region-specific nature of the regression models (Whitehead et al. 1984, Maguire and Hann 1987). However, the limited scope of regression models still remains a problem. Third, even when different equations are used for a subpopulation or a different population, the accuracy and precision of a prediction is highly dependent on the validity of the underlying model which is, in turn, dependent on the quality and number of observations to which the model is fitted. Cochran (1977) noted that an invalid model results in biased and inconsistent regression coefficients and bias from this model does  22  Chapter 2. Literature Review  not decrease with increasing sample size. Gregoire et al. (1995) asserted that a strong relationship between the response variable and the predictor variable(s) should exist even if the underlying model is valid and the sample size is large. However, regression models can be used as a source of auxiliary information to guide sample selection (Cochran 1977, Gregoire et al. 1995). If a mathematical model is known to adequately describe a given relationship, a sampling strategy that would optimize data collection for estimating the parameters of that model can be obtained (Schreuder et al. 1993). Despite all the attempts to quantify total leaf area and leaf weight of the entire crown, accurate and efficient estimation continues to be a problem. To date, there have been few sampling designs for estimating crown attributes developed or assessed for efficiency.  2.3  Sources of E r r o r in Quantifying C r o w n Leaf A r e a and Leaf Weight  In quantifying morphological attributes of tree crowns, the use of an improper sampling design or estimator can result in biased and/or inefficient estimates.  This can  be attributed to the complex and erratic nature of tree crowns. Errors encountered in quantifying morphological attributes of tree crowns can be classed into non-sampling and sampling errors. Sources of non-sampling errors include: 1. Misuse of regression equations or research results. For example, extrapolation (scaling up) from single-leaf estimates to whole-tree estimates or extrapolation of regression equations outside the range of data on which these equations were built (Gregoire et al. 1995, Yang and Cunia 1990); 2. Measurement error as a result of faulty instruments and procedures (e.g., inaccurate leaf area meter, misalignment of leaves) (Cochran 1977);  Chapter 2. Literature Review  23  3. Negligence and/or accidental error (e.g., mistakes made at recording or entering data or at any stage of a research) (Cochran 1977); 4. Inappropriate statistical models and estimation procedures (Yang and Cunia 1990, Dean et al. 1988); 5. Researcher bias (Cochran 1977, Schreuder et al.  1993).  For example, falling  trees consistently in one cardinal direction and not accounting for broken pieces or branches results in measurement and sampling bias, or using green (fresh) weight which would vary with the moisture content of the tree at a given point in time; 6. Confounding of variables. For example, disregarding the strong intercorrelation among tree crown attributes or spatial autocorrelations in tree crowns (Newton and Bower 1990); and 7. Inclusion of epicormic branches or spur shoots in selecting paths for randomized branch sampling or in calculating total leaf area and leaf weight (Gregoire et al. 1995). Non-sampling errors decrease the accuracy and efficiency of associated statistical estimators (Cochran 1977). Bias of individual observations will not affect estimates of precision if additive and constant or if they are a linear function of sample size. Bias which is multiplicative will affect sampling variance estimates and, therefore, estimates of precision (Schreuder et al. 1993). Sampling errors measure the spread of an estimate from its expected value. The magnitude of sampling error depends on the size of the sample, the variability of the population, and the sampling method used (Cochran 1977). Yang and Cunia (1990) identified four sources of errors (i.e., sampling error, measurement error, statistical model error, and application error) in biomass estimation. The last three types of error can be  Chapter 2. Literature Review  24  classed as non-sampling error. Williams (1989) discussed the following possible sources of error in defining branch diameter and length for selection probability in Randomized Branch Sampling (RBS) and Importance Sampling (IS): 1. Different points of measurement of branch diameter. For example, Ek (1979) and Loomis et al. (1966) measured branch diameter at 2.5 and 5.1 cm from the trunk of trees, respectively. 2. When selection probabilities do not accurately reflect the area or biomass contained on a branch as compared to the remainder of the tree. The selection probability can be influenced by differences in tree species, shade tolerance, and other factors; and 3. The use of randomized branch sampling (RBS) on first-order branches to estimate tree-level foliar dry matter. Valentine et al. (1994) discussed how subsampling using R B S resulted in imprecise tree level estimates. It is argued that ocular estimates of selection probabilities can be employed to avoid the third source of error. Williams (1989) also suggested that this source of error can be rectified by selecting more paths within each tree when RBS and IS sampling designs are used. However, more paths will result in additional cost, making these designs less efficient than with fewer paths.  2.4  M o d e l l i n g in Tree Crowns  The complex structure of a tree crown can be viewed as a population of twigs that are distributed over the tree crown from the base to the top of the crown, and from the main stem to the tips of first-order branches. These twigs bear leaves and are scattered erratically (Wilson 1989). In assessing total leaf area and leaf weight of a tree crown,  Chapter 2. Literature Review  25  it is difficult and unreasonable to analyze the leaf area and weight of each twig separately. However, by treating branches as populations of twigs (Wilson 1989), estimates of individual tree leaf area and weight can be obtained. Since variation in branch length strongly contributes to the differences in the number and spatial location of twigs in tree crowns (Wilson 1989), accurate prediction of branch length at different heights of a tree improves the estimates of leaf area and weight carried by each branch, and therefore improves the estimates of total tree leaf area and weight. Leopold (1971) used the "bifurcation ratio" and "length ratio" in analyzing and modelling tree branching patterns. He reported these ratios to be the same for the whole tree.  For a branch length prediction model, Mitchell (1975) suggested integrating a  branch growth function. One of the parameters in his model indicates the profile (curvature) of the tree crown. As the value of the parameter increases, the curvature of tree crown decreases toward a conical shape. On the other hand, smaller positive values of the parameter indicate greater curvature of tree crowns. Several researchers have used univariate probability density functions to describe the vertical distribution of foliage within tree crowns. For instance, Gillespie et al. (1994) used a Weibull probability density function to quantify the amount and vertical distribution of young loblolly pine foliage. Kellokami et al. (1980) used a beta distribution function to quantify needle biomass of Scots pine. Wang et al. (1990) used a beta distribution function to approximate the vertical needle area density distribution of radiata pine. Among five models compared by Massman (1982), the beta distribution function followed by the chi-square distribution function best described the foliage spatial distribution of old-growth Douglas-fir tree crowns. A modified chi-square distribution function performed the worst among the models compared. Flemming et al. (1990) used the following equation to estimate the amount of receptive seed cones produced by Jack pine:  Chapter 2. Literature Review  26  Y = aH (l - H);a > 0,6 > 0 b  (2.5)  where Y is the number of receptive seed cones produced at a given relative crown height; a and b are regression coefficients; and H is the relative height of the tree. Using Flemming's approach, the following equation could be developed for leaf area estimation of a tree.  Y  ijk  = a hf(l k  - hi) * c lf (1 - Ij); a > 0, b > 0, c > 0, d > 0 k  (2.6)  where Yijk is the amount of leaf area found at a given relative crown height and branch length in a particular cardinal direction; a, b, c, and d are regression coefficients for a particular tree; i is the ith crown length position; j is the jth relative horizontal position of a first-order branch; k the kth tree; h is relative height of the tree; and I is relative branch length. The total leaf area of a tree can be estimated as follows.  (2.7) where Yi is the total leaf area over the entire tree; H is total crown length; L is total k  branch length; dh is the change in h; and dl is the change in /. The literature on univariate probability distributions is voluminous and well documented, while the literature on bivariate probability distributions is rather limited. The Weibull distribution is one of the most widely used probability distributions in biological analysis due to its versatility, ease of interpretation, and flexibility. In addition to some biological application, bivariate Weibull probability distributions have been used in examining the times to first and second failures of a reparable device or the survival times in two-component systems (Lu and Bhattacharyya 1990). The use of bivariate distribution functions, such as the bivariate Weibull and beta  Chapter 2. Literature Review  27  distribution functions, has been limited in tree crown research. However, bivariate distribution functions provide flexibility in assessing the clumpy nature of foliage spatial distributions in tree crowns. While univariate density functions are easier to interpret than bivariate density functions, the latter can provide more information when the spatial distribution of foliage within a tree crown varies in both vertical and horizontal directions. When considering a univariate or bivariate probability distribution function for estimating a tree crown attribute, ease of parameter estimation and numerical computation should be considered. Generally, the fewer the number of parameters, the easier the parameter estimation and the numerical computations.  2.5  Sampling M e t h o d s  2.5.1  Sampling M e t h o d s U s e d in Quantifying Attributes of Tree Crowns  It is desirable to attain a sample from which an estimate of tree leaf area can be obtained. Representative samples are used to estimate the parameter via an unbiased estimator. However, to obtain these samples, an efficient sampling design is required.  2.5.1.1  Systematic Sampling  Murthy (1967) compared linear and circular systematic sampling procedures. He demonstrated that the circular systematic sampling procedure, unlike the linear, provides fixed sample size and an unbiased estimator for any sampling interval. These methods are identical when the population size is a multiple of the sample size and the difference between them becomes minimal when the sampling fraction is very small. The difference between the two systematic sampling procedures is rarely noted in tree crown research. Payandeh (1970) noted that systematic sampling from populations with linear trends,  Chapter 2. Literature Review  28  or with periodic variation results in statistical inefficiencies and inaccuracies. This problem can be avoided or minimized by adjusting the sampling interval and using multiple random starts (Iachan 1982). Payandeh (1970) compared the efficiency of two-dimensional systematic and simple random sampling for crown area estimation. He found that two-dimensional systematic sampling was less precise than simple random sampling when applied to uniformly spaced test populations. However, the converse was true when applied to the original forest types. 2.5.1.2  R a t i o a n d Regression S a m p l i n g  Ratio estimates are used to improve estimation efficiency using an assumed proportional relationship between the variable of interest and an auxiliary variable. Ratio-of-means estimation has been used to estimate total dry weight of leaves on a tree (Marchand 1984). The dry leaf weight to fresh leaf weight ratio from a sample was multiplied by the total fresh leaf weight on the tree (total of the auxiliary variable) to obtain the total dry weight of leaves. Tree biomass has been estimated using whole-tree biomass equations, with diameter and height as input (Baldwin 1989, Yang and Cunia 1990) and using density integral approaches (Parresol and Thomas 1989, Van Duesen and Baldwin 1993). The literature on whole-tree biomass equations is voluminous and is deliberately omitted in this review. 2.5.1.3  Two-phase S a m p l i n g  The use of two-phase sampling (double sampling) to estimate leaf area was first reported by Watson (1937). He used leaf weight as an auxiliary variable to estimate leaf area per plant. Two-phase sampling is widely used for estimating volume or biomass of forests. In the first phase, large samples are measured for some selected tree attributes (e.g., fresh  Chapter 2. Literature Review  29  leaf weight, diameter) and for some site attributes (e.g., forest type, age class, and site quality). In the second phase, fewer sample trees are measured for leaf area and other attributes measured in the first phase. The second phase samples can be a subset of or completely independent from the first phase samples. Measurements from the second phase are used to estimate a relationship between the variable(s) measured in the second phase and the first phase. In two-phase sampling, both the first and the second phase samples contribute to the error of estimates (Yang and Cunia 1990). Valentine and Hilton (1977) favored two-phase sampling in estimating tree leaf area, weight, and number in a tree crown when leaves are borne in terminal clusters as in oak (Quercus spp). When there is a relationship between a variable of interest and an auxiliary variable, a regression estimate from double sampling is always more efficient than estimation obtained from simple random sampling with no auxiliary variable (Sukhatme and Sukhatme 1970). However, the reduction in variance should be sizable to offset the cost of measuring the auxiliary variables. Two-phase sampling is advantageous when the gain in precision due to regression or ratio estimates is not as much as the precision lost due to a reduced level of sampling of the variable of interest (Cochran 1977). Numerous applications of two-phase sampling in forest inventories can be found in any standard forest inventory textbook (e.g., Freese 1962J Loetsch et al. 1973, Avery 1975, Husch et al. 1982, Schreuder et al. 1993).  |  i 2.5.1.4  Two-stage Sampling  Two-stage sampling has primary (first-stage) and secondary (second-stage) sampling units. These units, either separately or simultaneously, contribute to the variance of an estimate. In most two-stage applications,) the variance of an estimate is dominated by the variance among the primary sampling units (PSUs). The variance associated with the secondary sampling units (SSUs) is usually expected to be smaller in size than the  Chapter 2. Literature Review  30  variance associated with the PSUs (Cochran 1977, Valentine et al. 1984). This is due to more homogeneity within PSUs (among SSUs) than among PSUs in most applications. The mean square error among PSUs is a major component in calculating confidence intervals. Williams (1978) argued that the sample units should be spread out as much as possible over the primary sampling units if the primary variance component is the largest. Valentine et al. (1984) employed efficient procedures that increased the number of PSUs and thereby increased the precision of the design. Pike et al. (1977) used two-stage sampling to estimate total tree biomass and biomass of the tree crown epiphyte community (microorganisms, lichens, and bryophytes). In stage one, all the components of the tree crowns' weight were predicted and in the second stage, sample units were chosen with probability proportional to the predicted weight and were measured in detail. Biomass estimates from the sampled units were expanded to tree totals using the predicted values from the first sampling stage. Pike et al. (1977) selected second stage sampling units using probability proportional to size (PPS) (Equation 2.8). n  Pi = Y, Xi/X n  (2.8)  i=l  where Pi is the selection probability; Xi is the predicted weight for the ith unit; n is sample size; and X is the sum of Xi for the entire population. The total weight for all units of the population was calculated as: n  Yt = J2 /  Yi Pi  (2-9)  i=l  where Yi is the actual weight of each of the n units sampled and Yt is the total weight of all units in the population. Cochran (1977) argued that if first-stage sample units can be divided into smaller  Chapter 2. Literature Review  31  units without giving up significant precision, subsampling of first-stage units may lead to greater cost effectiveness. A major advantage of two-stage sampling over simple random sampling is the reduction in sampling cost.  2.5.1.5  Stratified R a n d o m Sampling  Stratification of the crown is an important variance reduction strategy (Gregoire et al. 1995). To account for the difference in morphological attributes, stratification of the crown has been performed by: layers or sections (Feller 1992, Schulze et al. 1977); segments and aspects (Grace et al. 1987); defining the boundary between sun and shade foliage (Beadle et al. 1985); branch whorl and aspects (Woodman 1971); length into thirds (Valentine et al. 1994); strata of equal lengths (Kershaw and Maguire 1996, Snell and Brown 1978); by whorl (Whitehead et al. 1984); and subjectively (Kaufman and Troendle 1981, Baldwin 1989). Schulze et al. (1977) estimated total biomass and its spatial distribution in Norway spruce by measuring sample branches taken from four different crown layers. Twenty-five percent of the total number of main branches were sampled at the "lower shade crown zone", 11% in the "upper shade crown zone", 8% in the "lower sun crown zone", and the 3% in the "upper sun crown zone". In an attempt to determine leaf area from leaf weight or volume, Gregg (1992) collected three samples of 10 to 20 grams (fresh weight) of leaves from the bottom, middle and upper third of each tree. After stratifying the crown by relative heights, Kershaw and Maguire (1996) took one sample branch from every group of 20 consecutive branches. Wolf et al. (1995) stratified leaves of poplar clones (Populus spp.) into four maturity classes (i.e., immature, recently mature, mature, and overmature) in estimating whole-tree photosynthesis. Stratification by maturity classes reduced the sampling error, as the production of photosynthate significantly differed among these classes. Likewise, the foliage nutrient status evaluation sampling guidelines  Chapter 2. Literature Review  32  developed by Ballard and Carter (1985) stipulate that samples be collected between one-fourth and one-half of the crown length from the top of the tree. 2.5.1.6  Ellipsoidal Sampling  One sampling design which is not widely referred to in the literature, but which may prove useful in the quantification of morphological attributes of tree crown, is ellipsoidal sampling. Point and variance estimators for this design should be developed and their performance be tested against existing techniques. In modelling canopy photosynthesis in radiata pine stands, Grace et al. (1987) assumed that the crown of each tree could be represented by an ellipsoid which was truncated at the base and the tree crown was divided into a maximum of 52 segments. The estimates of net photosynthesis from each segment were summed to estimate net photosynthesis for the tree crown. Ellipsoidal sampling can be considered as stratification by ellipsoids. However, in stratified sampling, the population (tree crowns) is divided into non-overlapping strata, while in ellipsoidal sampling the ellipsoids are nested within one another (the narrower ellipsoids nesting within the wider ones) which may result in overlapping of some sampling units such as first-, second-, and third-order branches. Ellipsoidal sampling can be considered as multi-stage sampling. However, the different shells are not stages of sampling as used in multi-stage sampling; they are rather a different set of sampling units that happen to be nested within other shells of sampling units. Ellipsoidal sampling is similar to the "foliage bands" used by Kershaw and Maguire (1996).  2.5.1.7  Two-stage U n e q u a l P r o b a b i l i t y S a m p l i n g  The main difference between SRS and two-stage sampling with unequal probability selection is that the probability of selecting a sampling unit remains constant at each selection in SRS, while in two-stage unequal probability sampling the selection probability varies  33  Chapter 2. Literature Review  at each successive selection. Statistical theory suggests that the unequal probability sampling results in accurate and precise estimates when the variable of interest is strongly and linearly related to probability of selection (Cochran 1977, Sukhatme and Sukhatme 1970). 2.5.1.8  R a n d o m i z e d B r a n c h Sampling  Gregoire et al. (1995) and Valentine et al. (1984) proposed randomized branch sampling (RBS) and importance sampling designs (IS) to estimate individual tree morphological attributes, including leaf area or weight. As stated by Gregoire et al. (1995), RBS has been used to estimate: the number of fruits on a tree (Jessen 1955); aboveground woody dry matter (Valentine et al. 1984); surface area of the stem (Valentine et al. 1994); leaf area, leaf weight, and leaf number (Valentine and Hilton 1977); insect egg or larvae population on a tree (Furness 1976); foliar dry matter; and average stem length (Gregoire et al. 1995). RBS is a special case of multistage sampling where samples are selected at successive stages with probability proportional to size. Gregoire et al. (1995) labeled the possible samples of each successive higher order branch as a "path" and the commencement point of a path is the butt of the first segment of the path. The probabilities of selection assigned to branches at each node must sum to one. R B S also provides flexibility to estimate leaf area and weight of either a first- or higher-order branch or the entire tree by using different "paths". These "paths" can be selected and truncated using different criteria. For example, in his use of RBS to estimate loblolly pine biomass, Williams (1989) truncated the selection of a path when a main stem diameter of 5 cm was reached. Gregoire et al. (1995) used probability rules to develop unbiased estimators of the mean and variance. The unconditional probability of selecting of the r  th  branch segment of a path is:  (r = 1, 2 , R )  Chapter 2. Literature Review  34  Qr=Y[q  (2.10)  k  k=l  where k is the selected branch segment and q is the probability of selecting a branch k  segment. This equation, evidently, assumes that each branch segment is independently chosen, as the probability of selecting r  th  branch segment is equal to the product of their  individual probabilities of selection q . k  If LA denotes the leaf area measured on the r  th  r  branch segment of a path, an unbiased  estimator of leaf area, Ai, of the entire branch emanating from the first node is given as: R  Ai = Y, rlQr  (2.11)  LA  r=l  If Ai represents the estimated leaf area calculated from equation 2.11 for the i  th  path  (i = 1, 2, ...,p), then an unbiased estimator of A and its variance are given as: A = l/pj^A t=i  (2.12)  i  Var(A) = l/(p(p - 1)) J2(At - A)  2  (2.13)  i=i  To estimate the variance of A unbiasedly, it is necessary to choose more than one path. This is also necessary when R B S is used in combination with stratified sampling (i.e., select more than one path per stratum). Some of the more important reasons given by Gregoire et al. (1995) as to why R B S is useful for sampling individual trees include: 1. It is cheaper than double sampling as there is no need for intensive weighing of branches at first phase sampling; 2. Its estimates are unbiased;  Chapter 2. Literature Review  35  3. Its estimates are more efficient than simple random sampling as branches can be selected with probability proportional to size (PPS); 4. It can be employed in different strata to increase precision; and 5. It can be used with importance sampling to estimate aboveground woody components of individual trees. The authors also presented two and three-stage sampling applications of RBS and IS for aggregate foliage estimates (per hectare values).  2.5.1.9  Importance Sampling  Gregoire et al. (1995) defined IS as "a continuous analog of sampling discrete units with probability proportional to size (pps)". Williams (1989) asserted that IS may be used for estimating the value of any definite integral. In IS, an auxiliary variable is used to determine the sample selection probabilities. Efficiency of an estimator is improved when the auxiliary variable is strongly and positively associated with the variable of interest. IS increases the precision of estimation by utilizing a proxy function, X (r), that p  is expected to mimic the shape of the unknown integrand, X(T), well and consistently (Gregoire et al. 1995). Fitted taper equations that contain stem diameter or crosssectional area as a function of height above ground, tree height and size of stem near the base can be used as a proxy function. The proxy total or integrand proxy function is:  (2.14) where r is tree height. Gregoire et al. (1995) gave an unbiased estimator for IS to estimate some continuous independent variables (e.g., bole biomass, cambial area, heartrot fungus, bole volume  36  Chapter 2. Literature Review  increment, and bark surface area) as: m  (2.15)  Y =l/mY X{T )lf{T ) i=l IS  J  l  l  where X(ji) is the integrand of a continuous independent variable (e.g., bole and branch circumference) and m is the number of sample locations. The values X(TJ), i = 1, ...,m are chosen at random according to a probability density function / ( r ) = X (T)/Y . p  P  An  unbiased estimator of the variance of YJS is: ra  Var(Y ) IS  = l/(m(m - 1))  £((*(^)//(0) -  Y)  2  IS  (2.16)  Gregoire et al. (1986) asserted that IS is faster and less expensive than direct measurement of tree attributes such as volume, increment, biomass, and nutrient content. The accuracy of IS increases with increasing sample size and its precision improves with the use of an auxiliary variable. IS also avoids bias and the use of imprecise volume equations. The authors suggested the use of IS: 1. When the accuracy of (a) volume equation(s) for a particular species and region is (are) not verified; 2. For continued validation of volume equations as tree growth is dynamic and variable from site to site; and 3. As an alternative to developing new volume equations. In his use of IS to estimate loblolly pine biomass, Williams (1989) selected sample branches using probability proportional to diameter squared and branch length. Gregoire et al. (1986) noted that in PPS' sampling, the variance of the estimate decreases as the selection probability becomes nearly proportional to the size of the characteristics of  37  Chapter 2. Literature Review  interest for each element of the population. Similarly, in IS, the variance of the estimate decreases as the probability density function becomes more closely related to the spatial distribution per unit length of the attribute of interest. Wiant et al. (1992) used IS to estimate the volume of the entire or part of a bole of a standing tree. Wood and Wiant (1990) developed centroid sampling as a variant of IS for estimating tree bole volume. The use of centroid sampling in estimating morphological attributes of tree crowns has not yet been investigated. This may be attributed to the difficulty of obtaining the center of gravity of tree crowns.  2.5.2  Cost Functions Used in Tree Crowns Sampling  Accurate cost estimates for sampling are required to compare sampling designs for a specific objective. Tree crowns are expensive to measure and study. To obtain an optimal sampling design, appropriate cost functions must be developed. Despite their succinct summary and intensive discussion of IS and RBS, Gregoire et al. (1995) did not discuss cost functions or the costs incurred by these designs. Sampling cost can be categorized into fixed and variable costs. Fixed costs are independent of the number of samples taken and can be ignored when determining sampling fractions for various strata. Fixed costs are not dependent of sample fraction; only variable costs determine the optimum level of sampling (Watson 1973). The following cost function can be used to calculate total cost in stratified sampling: fc  C = a + Y,C n k  (2.17)  k  t=i where a is the fixed cost, C is the cost per observation in the k  th  k  number of sample observations in the k  th  stratum, and n is the k  stratum.  Proportional allocation of sample units is not synonymous to optimum allocation, as  Chapter 2. Literature Review  38  the latter may have variable sampling fractions when the cost and variability of sampling units are varied among strata.  Sukhatme and Sukhatme (1970) argued that single-  stage sampling would be preferred to multistage sampling if sampling costs were not considered as the efficiency of the design increases when the last multi-stage units are selected independently from the whole population. Sukhatme and Sukhatme (1970) observed that the surveying cost for a multi-stage sampling depends on the size of each stage. For example, the cost of two-stage sampling is given as:  (2.18)  C = C\n + Conm-  where C is the cost of the survey, C\ and Ci are positive constants, n is number of PSUs, and m is the number of SSUs per P S U . From the above equation one can infer that the cost of k-stage sampling can be calculated as:  C = C\n + Conm + ... + C nm...p k  (2.19)  where C\, Co, and C are positive constants and p is the number of the k-th sampling k  units.  2.5.3  Evaluation of Sampling Designs  A number of empirical studies have been conducted to evaluate the efficiency of systematic, stratified, two-stage, and simple random sampling in the field of forestry (e.g., Hasel 1938, Palley and O'Regan 1961, Schreuder 1984, Thrower 1989). However, such comparisons in tree crown studies are limited. Sampling designs have been compared using bias, sampling variance, and relative ease  Chapter 2. Literature Review  39  (Schreuder et al. 1993, Thrower 1989). Clearly, the crux of sampling design studies comes down to a search for methods which have sampling distributions with smaller variance, and. a sampling plan with lower cost and unbiased estimators (Schreuder et al. 1993). Manly (1990) outlined three computer-intensive methods for hypothesis testing and confidence interval estimation in evaluating sampling designs (e.g., Monte Carlo methods, the Jackknife method, and the Bootstrap method). The Bootstrap and Jackknife methods are used in estimating variances of estimators in complex sampling designs (Green 1979). Monte Carlo Simulation (MCS) allows hypothesis testing and the estimation of confidence intervals for any pre-specified pattern analysis statistics (Manly 1990). MCS tests the significance of an observed test statistic by comparing it with a sample test statistic obtained by generating random samples using some assumed model. These random samples are generated using some form of random number generator. Rubinstien (1981) stated that the random numbers must be uniformly distributed, statistically independent, reproducible and that the method used to generate them should be fast and require minimum memory capacity. M C S is a cost effective method in comparing sampling methods; it gives an indication of the relative efficiency of sampling methods. The Jackknife is a general approach for testing and calculating confidence intervals in situations where no better methods are easily available (Manly 1990). This method has been used as a means of reducing bias. The Bootstrap method is a convenient technique for estimating the standard error of an estimator that has complicated analytic properties. An intensive review of Bootstrap and Jackknifing methods is given by Manly (1990).  2.5.4  Comparison of Sampling Designs  There are several sampling designs available for estimating single-tree leaf area values. However, no single sampling design will be efficiently applicable in all situations (Catchpole and Wheeler 1992, Schreuder et al. 1993). In their review of techniques used to  Chapter 2.  Literature  Review  40  estimate plant biomass, Catchpole and Wheeler (1992) identified the size of the sampling frame, accuracy, sampling cost, structure, and components of vegetation as major attributes in selecting a sampling design. The adequacy of various sampling designs are assessed by their precision and cost. A design with the smallest variance is the most efficient statistically (Cochran 1977). Although some authors have used only precision as the basis for comparing sampling designs, this measure by itself is inadequate. So long as minimum precision needs are met, the best design is determined by the amount of precision per unit cost (Cochran 1977, Schreuder et al. 1993). Monte Carlo methods are widely used to investigate the performance of sampling designs (e.g., Schreuder 1984, Thrower 1989, Schreuder et al. 1993, Gregoire et al. 1995). Generally, the cost efficiency of a sampling design can be examined on: 1. Actual population data or numerous actual sample units; 2. Artificial population or sample units; or 3. Actual units in a population or sample units combined with Monte Carlo simulation. All of these approaches have both merits and shortcomings. The merits of using actual data are (1) the data will represent actual crown conditions and (2) the data may be collected with accuracy and high precision by the researcher. However, this approach is expensive, time consuming, and sometimes destructive (Cochran 1977, Sukhatme and Sukhatme 1970). Tests on an artificial population are cheaper due to a reduced cost for data collection. However, the results may not reflect reality or may not be consistent (Schreuder 1984). Tests made on data obtained from the combination of actual and Monte Carlo simulated data show repeatability (i.e., are consistent) and are fairly inexpensive (Schreuder et al. 1993).  Chapter 2. Literature Review  41  Tree crowns are comprised of hard-to-measure attributes. The preceding literature review explicitly conveys the existence of spatial variation within tree crowns and a need to quantify morphological attributes of tree crowns. Devising and/or evaluating sampling methods that account for the spatial and temporal variations of leaf area and leaf weight is important. In selecting a sampling design, knowledge or quantification of the vertical and radial distribution of leaf area and leaf weight should improve the efficiency of the sampling design, as the nature of the frequency distribution of the variable being sampled will influence the sampling design efficiency. Evaluation of potentially useful sampling designs minimizes the difficulties that forest canopy scientists face in sampling from three-dimensional complex tree crowns for various canopy studies. The complexities of the unpredictable three-dimensional structure warrants the use of spatial statistics so that one can extrapolate up and down spatial scales (Ziede 1992).  Chapter 3  Methods  This chapter is divided into four sections and describes the details of the methods used to achieve the two objectives of this study. Section 3.1 outlines the methods used in collecting and processing the field data for further analysis. The second section describes the examination and modelling of leaf area (objective one of this study). Section 3.3 outlines the methods used to create test populations that were used to evaluate selected sampling designs (objective two of this study). Section 3.4 describes how the selected sampling designs were evaluated.  3.1 3.1.1  Collection and Preparation of D a t a Study A r e a  Twelve trees were sampled at the Ken Fire and Date Creek study areas within the Interior Cedar Hemlock (ICH) biogeoclimatic zone, Kispiox Forest District, Prince Rupert Forest Region of British Columbia (BC). The Ken Fire study area is located 71 km north of Hazelton and east of the Kispiox River and is composed of young stands of fire origin. The Date Creek study area is about 400 ha and is located 21 km north of Hazelton and west of the Kispiox River (Clement and Banner 1992). The I C H moist-cold subzone, Hazelton Variant (ICHmc2) is the predominant biogeoclimatic unit in the I C H zone in the region followed by the I C H moist-cold subzone, Nass Variant (ICHmcl) (Meidinger and Pojar 1991). The elevation ranges from 350 to 1,100 m. These study areas were  42  Chapter 3.  Methods  43  selected because they contain a wide range of age classes and forest structures, and also for the purpose of coordinating this project with on-going research projects of the B C Ministry of Forests.  3.1.2  Data  Collection  Based on available forest cover maps, two stands from Date Creek and one from the Ken Fire study area were selected. These stands represented three age classes: a) younger than 20 years, from the Ken Fire study area; b) 50 to 60 years, from the Date Creek study area; and c) older than 120 years, from the Date Creek study area. Selected stands were composed of the following conifers and deciduous trees: western hemlock; hybrid spruce; subalpine fir (Abies lasiocarpa (Hook.) Nutt.); western red cedar; lodgepole pine; paper birch (Betula papyrifera Marsh); trembling aspen (Populus tremuloides Michx.); and black cottonwood (Populus trichocarpa Torr. & Gray.). Stands were located on an undulating morainal landscape ranging in soil texture from loamy sand to clay loam, with Eluviated Dystric Brunisols being the most common soil type (Clement and Banner 1992). Four of the tallest hybrid spruce trees from each stand were selected and sampled. These trees had uniform crowns, straight boles, and were free of evidence of past crown damage. The four cardinal directions were marked at breast height on each stem with different colours of paint. Each sample tree was then carefully felled between 0.0 and 0.3 m above ground. After felling, the following measurements were taken for each tree: 1) diameter outside bark at breast height (DBH); 2) total tree height (HT); 3) height to the base of live crown ( H T B L C ) ; 4) age at breast height(BHAGE); 5) crown width (CW); and 6) stem diameter outside bark at the base of live crown (DOBLC) (Table 3.1). A l l sample trees were felled between July 10 and August 22, 1994. Felling was carried out during this time period based on the assumption that the expansion of the current year's  C h a p t e r  3.  44  M e t h o d s  Table 3.1: Attributes, devices, and precision required in data collection.  Attribute  Device  DBH Tree height (HT) HTBLC DOBLC Age at breast height Internodal distance Branch weight Branch angle Branch length Leaf area Leaf weight  Diameter tape Eslon tape Eslon tape Diameter tape  Precision  ±2 % ±2% ±2% ±2 % ±2 years  Eslon tape Hanging scale Protractor Eslon tape Leaf area meter Electronic balance  ±2cm ±0.1kg  ±2° ±2 cm ±2% ±0.1  gm  Note: D B H = diameter outside bark at breast height; H T B L C = height to the base of live crown; and D O B L C = diameter outside bark at the base of live crown.  needles terminated in the first half of July. Each stem was cut at breast height, a disk was removed and annual rings were counted ( B H A G E ) . On each tree, each first-order branch in the crown was classed into one of the four cardinal directions, and the total number of first-order branches in each cardinal direction was counted. For example, branches that were within 45 degrees of north were classed as North. Systematic sampling was used to choose every first, followed by every fourth firstorder branch by cardinal direction. Only whorled first-order branches were included. On each selected first-order branch, the branch angle, length, fresh weight, and internodal distances between second-order branches were measured. If sample branches were broken off during felling, their origin was determined using the broken pieces. Subsequently, from each selected first-order branch, every fourth second-order branch (starting from  Chapter 3.  Methods  45  the branch tip, inwards) was systematically sampled. On the selected second-order branch, branch length and distance from the tip of the first-order branch were measured. In events where a selected second-order branch had more than one third-order branch, the number of third-order branches was counted and one of them was randomly selected using a random number (generated using a calculator). Also, the length of the selected third-order branch and its distance from the tip of the second-order branch were measured. The same selection procedure was used when a selected third-order branch had more than one fourth-order branch. On each selected second-, third-, or fourth-order branch, all needles less than five years old and of the same age were clipped and placed in a paper bag. It was assumed that leaves older than five years did not significantly contribute to the total leaf area of a tree (Kramer and Kozlowski 1979). In this study, a twig is defined as an axis of a sampling unit arising from any order of branch and comprised of the first five years of foliage. Samples were taken each morning and transported to a laboratory for green leaf area and dry leaf weight measurements. Projected leaf area (the vertical projection of needles) was measured in the afternoon to prevent any leaf shrinkage that would occur due to drying. In a laboratory, needles of the same age were removed from each twig and were vertically aligned to scan their projected area using a leaf area meter. The leaves were dried for 24 hours at 80°C and dry weights were measured using an electronic balance at the University of British Columbia (UBC), Forest Pathology Laboratory. Leaf weights were measured for future studies and were not used in this study. On selected sampling days, a stop watch was initiated as sampling of leaves began, and then it was stopped as sampling for the day was completed. Sampling time was recorded for all first-order branches sampled on that day. These data were used to calculate the amount of time required to sample a first-order branch, and then an entire tree crown.  Chapter 3.  46  Methods  Leaf area measurement time was also recorded for all the samples collected on each day. Time was to be used as a surrogate to develop a cost function and to estimate the cost of sampling for a given sampling design. This, in turn, was used to compare the efficiency of sampling designs selected in this study. 3.1.3  Compilation of Data  To carry out further analysis of leaf area the following variables were calculated: 1. Crown lengths (CL) were obtained by subtracting the height to the base of live crown ( H T B L C ) from tree height (i.e., C L = H T - H T B L C ) ; 2. Crown ratio (CR) = C L / H T ; 3. Height above ground ( H T A G R ) values represent different vertical positions on the tree. For each branch, two relative heights above ground were calculated. These were 1) the position of a branch from the the top of the tree (DFTi) relative to the total tree height (i.e., zii — DFTi/HT)  and 2) the position of the branch  from the top of the tree relative to the crown length (i.e., zi = DFTi/CL). 2  A zii  or zii value close to zero indicates that the sample point was taken close to the tip of the sample tree. A zi value of one indicates that the sample point was at the 2  base of the crown; 4. Relative horizontal length (li) values were calculated as the ratio of the horizontal distance (m) of the sampling point along the first-order branch from the trunk of the tree, to the respective branch length (m) of the first-order whorl branch. A n li of one indicates that the sample point was at the tip of the first-order branch whereas li equal to zero occurs at the main stem; and  Chapter 3.  Methods  47  5. Leaf area per centimetre ( A P C M ) values were obtained by dividing the leaf area of a sampled twig branch by the twig length.  3.1.4  Verification of D a t a  A l l data were checked for anomalies before creating three-dimensional distributional analyses, displays, and creating test populations. The data were further checked for outliers by plotting pairs of variables. These diagrams were also used to examine the dispersion and irregularities, such as groupings and extreme points in the data. Data recording and measurement errors in twig length and leaf area measures were found and ten twigs were removed from the sample because of these errors.  3.2  E x a m i n a t i o n and modelling of Leaf A r e a Distributions  To characterize the spatial distribution of leaf area in hybrid spruce crowns (objective one), three dimensional A P C M graphs (SYSTAT Inc. 1990) were generated to (1) observe the variability and/or trend of leaf area distribution in tree crowns and (2) to select models for leaf area spatial distribution. Also, the predicted leaf area distribution model parameters were related to tree size variables to: (1) assess trends and variability of these parameters over the tree size variables and (2) examine relationships between the model parameters and tree size variables.  3.2.1  G r a p h i c a l E x a m i n a t i o n of Leaf A r e a  Two-dimensional scatter diagrams of A P C M by horizontal position and A P C M by vertical position over the four cardinal directions were graphed (SYSTAT Inc. 1990). These graphs were used to: 1) determine what kind of probability distributions could adequately describe the vertical and horizontal distributions of leaf area, and 2) evaluate trends in  Chapter 3. Methods  48  leaf area over horizontal, vertical, and cardinal directions. These were used as aids in selecting candidate models for leaf area distribution. Three dimensional graphs of leaf A P C M as a function of horizontal and vertical positions were also prepared. 3.2.2  Leaf Area Distribution Models  The bivariate Weibull (Lu and Bhattacharyya 1990) and beta probability distribution functions were selected for predicting leaf A P C M using relative heights (vertical position) and relative branch lengths (horizontal position). The bivariate leaf A P C M prediction models were fitted for each of the two definitions of relative height (zii and zi-i).  1. Bivariate Weibull distribution function  APCM  =  [ x c /c x c ] x Cl  2  [(zi/ ) C 3  (l/c  5  3  C l / c 6  4  [ ( ^ / C 3 )  + (Zz/c )  C 2 / c 5  4  C  1  /  C  5  _  1  ( ^ / C 4 )  2  /  p- x {[(^/c ) 2  C  5  _  1  C l / c 5  3  - 1)} x (exp -[( i/ )<*/<* + ( z  C  ]  x + (^/c )  fc/c^TH  C3  C2/c5  4  ]  C5  + (3.20)  Gi  where zi is either zii or z%2, c\ to C5 are model parameters to be estimated, 0 < C5 < 1, i to c > 0, and € 1 is the prediction error. c  4  This model was constructed based on survival functions by Marshall and Olkin (1967) and later updated by L u and Bhattacharyya (1990). The cumulative function of Equation 3.20, F(zi,li),  F(zi,  where F(zi,li)  can be calculated as:  li) = exp{-[(zi/a ) / ai  3  a5  (3.21)  + (li/a ) ^} } a  a5  4  is cumulative leaf area, zi is either zii  o  r  ^2, i  parameters to be estimated, 0 < 05 < 1, and ai to a > 0. 4  a  t°5 a  a  r  e  model  Chapter 3. Methods  49  2. B i v a r i a t e B e t a d i s t r i b u t i o n f u n c t i o n  APCM  =  r(c + c + c ) 7  6  r(c ) x 6  r(  C 7  )  zi^-Hi - {\  8  x  r(  Cl  C 8  x  )  -zi-  li) *- -{- e c  l  2  (3.22)  where zi is either zi\ or zi2, 0 < (zi + li)/2 < 1, zi > 0,li > 0, CQ to c§ > 0 are model parameters to be estimated, and G2 is the prediction error.  T(t) is  the gamma distribution with parameter t. A-Grivas and Asaoka (1982) used the bivariate beta distribution function to assess the joint distribution of cohesive and friction coefficients in a slope stability analysis. Each bivariate distribution function was fitted to each of the 12 sample trees using a derivative-free, nonlinear least squares, optimization technique P R O C N L I N (SAS Institute Inc. 1989). The starting value for each parameter was varied and several runs were performed in order to find the global minimum. After each bivariate model was fitted to each tree, models were compared using the coefficient of determination (R ) and root 2  mean square error (RMSE) values. For comparison with other literature, univariate Weibull and beta distribution models were also fitted to assess the improvements obtained using a bivariate model. These univariate models were also compared using R and R M S E values. 2  3.2.3  R e l a t i o n s h i p of L e a f A r e a D i s t r i b u t i o n M o d e l s to Tree A t t r i b u t e s  The relationship of model parameters to tree attributes was used to detect any trend that existed and to assess the possibility of using tree size variables as a proxy to estimating model parameters. Each estimated parameter of the two bivariate models was plotted against bole size (HT, D B H , and B H A G E ) and crown size (CL, C W and CR). Multiple linear regression  Chapter 3.  Methods  50  (MLR) was used to predict each estimated parameter as a function of all the variables, followed by backward elimination (a — 0.05) to select predictor variables. The assumptions of M L R were checked using residual plots, Shapiro-Wilks (Shapiro and Wilk 1965) normality test, and an F-test for homogeneity of variances (Goldfeld and Quandt 1965). Transformations of predictor variables or the dependent variables were carried out when the assumptions of M L R were not met. The regressions of each estimated parameter with the selected predictor variable(s) were again checked for meeting the assumptions of M L R . Further transformations were carried out until all assumptions were met. Once a prediction equation was selected for each estimated parameter, the errors in predicting A P C M were calculated by replacing the estimated parameters by the fitted parameter prediction equations (separate M L R fits). These errors in predicting A P C M were summarized into two fit statistics. 1. The I-squared value  I = 12  SSE/SSY  (3.23)  where SSE is sum squares of error and SSY is sum squares of total. The I  2  is  similar to the R value calculated for M L R . 2  2. Estimated root mean squared error (ERMSE)  (3.24) where n is the number of observations used to calculate SSE and p is the number of coefficients from the model used to predict the dependent variable.  51  Chapter 3. Methods  As an alternative, each leaf area prediction equation was refit by replacing the parameters by the appropriate parameter prediction functions identified using M L R . The leaf area prediction model was then refit (composite fitting method, LeMay 1988). The errors in leaf area prediction using these fitted models were also summarized into I and R M S E 2  fit statistics by replacing the estimated parameters by the fitted parameter prediction equation as with the M L R fit. The parameters of each leaf area prediction model were also estimated using seemingly unrelated regression (SUR procedure, SAS Institute Inc. 1984) on a system of linear functions which were identified using M L R . Each leaf area prediction equation was refit by replacing the estimated parameters with those obtained by using the joint functions of the parameters. The errors in leaf area prediction using these fitted models were also summarized into I and R M S E fit statistics. 2  3.3  C r e a t i o n of Test Populations  Since data were collected by systematic sampling, the total leaf area per tree was not available. Test populations were created by using measured leaf area of twigs taken from detailed sampling points, and estimates of leaf area for non-sampled points on the 12 sample trees. The process used to reconstruct each tree crown was to create and position first-order branches, second-order branches, third-order branches, and then fourth-order branches hierarchically. Only first- and second-order branches were created for trees younger than 20 years, as the sampling points mainly consisted of leaf area measurements taken from the tip of first-order branch and second-order branches. First-, second-, and third-order branches were created for trees 40 to 60 years old. A l l four orders of branches were created for trees older than 120 years. The steps used to create the test populations are outlined in Sections 3.3.1 to 3.3.8. The created test populations were used to assess  Chapter  3.  52  Methods  the efficiency of selected sampling designs to meet the second objective of this study.  3.3.1  N u m b e r a n d D i s t r i b u t i o n of F i r s t - O r d e r B r a n c h e s  The sampled first-order branches of each tree were sorted by cardinal directions and height above ground. For each cardinal direction, the internodal distance between two consecutive sampled first-order branches was divided into four equal segments and the three non-sampled first-order branches were placed evenly over the distance between the bordering sampled first-order branches from the base of the crown to the tip of the tree. For the tree tip, the remaining non-sampled first-order branches were positioned between the last sample first-order branch and the tip of the tree for each respective cardinal direction. Following this, the estimated distance of each non-sampled first-order branch from the tip of the tree (DFT\  = HT - HTAGR )  e  3.3.2  e  was calculated.  L e n g t h of F i r s t - O r d e r B r a n c h e s  The length of non-sampled first-order branches (BL\) was estimated as a function of relative vertical position (zi\ or zi ) on a stem. Six approaches were selected and compared 2  based on the interpret ability of the coefficients, the ease of parameter estimation, the accuracy of prediction when zi\ or zi was used, and known biological trends. The six 2  models were:  1. W e i b u l l d i s t r i b u t i o n  BLi  function  = [fo/P [(zi  -  2  ft)//?/ ] 3-1  exp [-[(zi  -  /?i)//? ] 2  /33  ]+ G  3  (3.25)  where zi is either zi\ or zi , j3\ < zi < oo, f3\, (3 , and (3$ are, the location, scale and 2  2  shape parameters, respectively (Clutter et al. 1983), exp is the Naperian constant  Chapter  3.  53  Methods  (i.e., exp « 2.718), and €3 is the prediction error. 2. Chi-square distribution function  BLi  where zi is either zii  o r  = [d (l  exp (8 zi)+  - zif ] 5  4  6  G  (3.26)  4  <^2, (1 — zi) indicates the relative position of the branch  from the base of the tree  (zii)  or  from the base of the crown  (2^2), /?4 to B§ are  model parameters to be estimated, and 64 is the prediction error. 3. Beta distribution function  \z%f'-\\  BU  -  zif*'  1  x  T(p  7  +P) S  T(3 ) x F(8s)_  + e  5  (3.27)  7  where zi is either zii or zi , 0 < zi < 1, 87 and 8% are model parameters to 2  be estimated, G is the prediction error, and F(t) is the gamma distribution with 5  parameter t. 4. Modified Chi-square distribution function  BLi  where 21 is either zii  = o  r  \8g(l - zi)(zi  -  8 )} W  e x p O M l " **))+ e  6  (3.28)  ^ 2 , /?9 to 8n are model parameters to be estimated, and  €6 is the prediction error. 5. Mitchell's (1975) model  BLi  = /?i m[£>FTi/(/? + 1)]+ G 2  13  7  (3.29)  Chapter 3.  Methods  where DFTi  54  is distance from the tip of a tree, f3\ and /3 are regression coeffi2  i3  cients, and € 7 is the prediction error. The latter coefficient, /?i3, governs the profile (curvature) of the tree crown. 6. I n t e r p o l a t i o n  Branch lengths at different heights in a tree were estimated through linear and modified quadratic interpolation between consecutive sampled first-order branches. In linear interpolation, the difference in branch length between two consecutive firstorder branches was divided into four parts and added to the three non-sampled firstorder branches cumulatively. In modified quadratic interpolation, the difference of the square of the two consecutive branch length was divided into four equal parts and the square root added to the three non-sampled branches cumulatively. Since quadratic interpolation uses at least three points to interpolate along a curve, the technique used here is referred to as modified quadratic interpolation. The first five models were fitted to each of the 12 trees using a nonlinear, least squares, Marquardt optimization technique P R O C N L I N (SAS Institute Inc.  1989).  Both definitions of relative height (zi\ and zi ) were used in fitting Equations 3.25 to 2  3.28. The starting value for each parameter was varied and several runs were obtained in order to find the global minimum. Initial approximations for each parameter were obtained from linear transformation of the models where possible. The first five models were compared using R and R M S E (SAS Institute Inc. 1989). 2  The estimated first-order branch lengths for all six models were then plotted against the height above ground for each tree. These graphs were used to assess the accuracy and the biological rationality of the six approaches.  Chapter 3. Methods  3.3.3  55  Number and Distribution of Second-Order Branches  On the sampled first-order branches, the total number of second-order branches (TOTSEC) had been counted.  The three non-sampled second-order branches between the  sample branches were positioned by dividing the internodal distance between two sampled second-order branches into four equal parts and the estimated distances from the branch tip (DFT ) 2e  were calculated.  For non-sampled first-order branches, the number of second-order branches was estimated as a function of the estimated first-order branch length (BL\ ) e  as outlined in  Section 3.3.2. Two models were assessed using measured first-order branches:  TOT SEC = B  14  \og(TOTSEC)  + 0is x BLx+  8  = B + B x log(£Li)+ G 16  ir  (3.30)  €  9  (3.31)  where du to 3n are regression coefficients and 68 and Eg are the prediction errors. After fitting using M L R (SAS Institute Inc. 1989), models were compared using R or I , and R M S E or E R M S E . The estimated number of second-order branches was 2  2  calculated using BL\ in the selected equation. These were then positioned equally over e  the estimated first-order branch length and then their respective distances from the tip of a first-order branch (DFT2 ) were calculated. e  3.3.4  Length of Second-Order Branches  The lengths of the second-order branches were not estimated using interpolation (as was used for estimating BL\), as there were no BLi measurements on the non-sampled first-order branches to carry out an interpolation.  56  Chapter 3. Methods  For the sampled first-order branches, the lengths of the non-sampled second-order branches were estimated by obtaining a function of measured second-order branch length and BL\ or distance from the tip of a first-order branch [DFT^)-  The following six  models were assessed using measured second-order branches:  BL  = 0 i + 0 i x BL\ + (3 x DFT +  2  BL  2  BL  2  8  = 0  2x  9  + 022 x DFTi  2  BL  2  BL  2  = exp^ xBLf 28  = exp  = exp?  34  ftl  xBL^  26  xDFTi™  35  x  Gn  2  2  + 027 x DFT + 2  x DFT^x  e  x DFT^x  DFT?  36  (3.32)  w  + 023 x DFT +  = 024 + 025 x BL\ + 0 6 x DFTi  BL  E  2  20  x DFT^x  (3.33)  E  12  (3.34)  (3.35)  x3  E  (3.36)  14  e  15  (3.37)  where 0is to 0 7 are coefficients to be estimated and Gio to €15 are the prediction errors. 3  After fitting using M L R or nonlinear regression (SAS Institute Inc. 1989), models were compared using B? and R M S E . On the non-sampled first-order branches, the length of each non-sampled second-order branch (£?L ) was estimated as a function of the estimated distance from the tip of a 2e  tree (DFTi ),  the estimated length of the first-order branch BL\ , and the estimated  e  e  distance from the tip of a first-order branch (DFT2 ) e  as outlined in Sections 3.3.1, 3.3.2.,  and 3.3.3, respectively. The selected model from Equations 3.32 to 3.37 was used to estimate BL . 2  Chapter 3.  3.3.5  Methods  57  Number and Distribution of Third-Order Branches  Using data from sampled first-order and second-order branches, two models were assessed to estimate the number of third-order branches on a second-order branch.  = 8  NOTRD  log(NOTRD)  38  =B + 3 42  43  + /? x BLi + PAO x BL + 3 39  41  2  x log(BLi) + 3  44  x DFTi+  x log(BL ) + 3 2  £  x DFT +  45  (3.38)  16  X  e  (3.39)  17  where 3 8 to #45 are regression coefficients to be estimated, and G i 6 and 617 are the 3  prediction errors. After each model was fitted, models were compared using R  2  or I  2  and R M S E or  E R M S E values (SAS Institute Inc. 1989). For sampled first- and second-order branches, the estimated number of third-order branches (NOTRD) BL  was obtained by inputting  using the selected model. These third-order branches were then positioned at equal  2e  intervals over the estimated second-order branch length and their respective distances from the tip of a second-order branch (DFT ) 3e  were calculated. For the non-sampled  first- and second-order branches, the selected model was also used to estimate the number of third-order branches (NOTRD),  3.3.6  but BL  2  was replaced by BL  2e  and BL\ by BL\  e  .  Length of Third-Order Branches  The lengths of third-order branches were not estimated using interpolation (as was used for estimating BL\) as there were no BL  3  measurements on the non-sampled first-order  branches to carry out an interpolation. Using data from sampled first- and second-order branches, models to estimate the lengths of non-sampled third-order branches as a function of DFTi,  a n  d DFT  2e  were  C h a p t e r  3.  58  M e t h o d s  assessed. Initially, DFT  was also included in models to estimate BL . However, DFT  3  3  3  did not improve the fit of the models and was later discarded.  BL  = 046 + 047 x DFTi + 048 x DFT +  3  BL  = 0  3  4 9  + 050 x BLi + 05i x DFTi + 052 X DFT + 2  BL  Z  = exp^ x D F T f 3  3  BL  Gis  2  = exp'  356  xBLi  57  5 4  x DFT x  <E  65  2  x DFTi  58  x  59 2  Gig  (3.41)  ( 3.42)  20  DFT  (3.40)  x € i 2  (3.43)  where 046 to 0sg are coefficients to be estimated, and €is to €21 are the prediction errors. After each model was fitted using M L R or nonlinear regression (SAS Institute Inc. 1989), models were compared using R and R M S E values. For sampled and non-sampled 2  second-order branches, the selected equation was used to estimate third-order branches by replacing DFT  2  by DFT . 2e  Similarly, for non-sampled first- and second-order branches,  lengths of non-sampled third-order branches (BL ) 3e  DFT  and  3.3.7  Fourth-Order Branches  U  were estimated as a function of  DFT . 2e  Steps given in Sections 3.2.5 to 3.2.6 were repeated to create simulated fourth-order branches for trees older than 120 years. During data collection, the numbers and lengths of fourth-order branches were not measured. As a result, reconstructing the non-sampled fourth-order branches was difficult and a bifurcation ratio (the ratio of first- to secondto third- order branch numbers) had to be used. Using the collected data set, average bifurcation and length ratios were developed for each tree and later used for the creation of  Chapter 3. Methods  59  fourth-order branches in the test populations. This was needed for four test populations (i.e., trees older than 120 years). The following assumptions were made to create the fourth-order branches: 1. Proportional relationships (bifurcation ratio) exist between the numbers of 1° and 2°, and 2° and 3° branches. The average of the two proportional ratios was calculated for each tree. This average proportion was used to estimate the number of fourth-order branches by dividing the number of third-order branches for each second-order branch by this ratio. 2. Proportional relationships (length ratio) exist between the lengths of 1° and 2°, and 2° and 3° branches. The average of the two length ratios was calculated for each tree. This average proportion was used to estimate the length of fourth-order branches by dividing the estimated length of each third-order branch by the average of this ratio. The estimated fourth-order branches  (NOFOR)  were positioned at equal intervals  over the measured or estimated third-order branch lengths. 3.3.8  Leaf Area Prediction  The steps in Sections 3.3.1 to 3.3.7 resulted in a list of twig positions and their respective lengths in each tree crown. This list of positions was used to select sample twigs for evaluating selected sampling designs. The leaf A P G M values were estimated as a function of relative height in the crown and relative branch length using the selected leaf area models described in Section 3.2. The estimated A P C M values were then multiplied by the estimated twig length to obtain the leaf area of each twig. A l l of the measured and the estimated twig leaf area values were combined and then summed to obtain the total leaf area of each tree.  Chapter 3. Methods  60  The total tree leaf area values for the created populations were compared to the estimated total leaf area values obtained using the measured sample data. The estimated total leaf area values using the systematic sampling data were computed as follows. The leaf area of each sampled first-order branch (LAPB ) k  n = (£ LA /n )  was estimated by:  fc  LAPB  k  jk  x NTB  k  (3.44)  k  3=1  where j is a sample point on the kth first-order branch, k is the kth first-order branch, LAjk is the leaf area of a sampled twig from the kth first-order branch, n is the number k  of sample points from the fcth first-order branch, Yl]=i LAj /n k  all sample points on the fcth first-order branch, and NTB  k  k  is the average leaf area of  is the number of twigs on the  sampled kth. first-order branch (i.e., the total number of second- or third-order branches on the sampled first-order branch). Next, the total leaf area of the sample tree was estimated as: 6  TL.4 =  LAPB Jb fc=i k  x N  (3.45)  T  where b is the total number of sampled first-order branches, Y,k=i LAPB /b k  is the esti-  mated average leaf area per first-order branch, and NT is the total number of first-order branches per tree. The estimated leaf area values for the test populations (model estimates) were compared to the values estimated using the systematic sampling data (sample estimates) by graphing the former against the latter and by using the Chi-square test. The graphs were also used to detect any trend and anomalies that may exist in the model estimates. The Chi-square value was calculated as:  Xl = E(Oi i=l  Erf/El  (3.46)  Chapter  3.  where x  2  61  Methods  is the chi-square distribution with v (i.e., 11) degrees of freedom, Oi is the  estimated tree leaf area value for the recreated tree i, and Ei is the estimated tree leaf area value for tree i using systematic sampling. If the calculated  w a  s significant at  a = 0.05, it indicated that the models used to create the test populations created values different from those obtained using the sample data. The crowns of each tree were divided into three equal positions (strata) and the variances of twig leaf area were computed for each stratum and were compared with the variances observed in the sample estimates. The crowns of each tree were also classified into various ellipses and the variance of twig leaf area in each ellipse was calculated and was also compared with the variances observed in the sample estimates.  3.4  E v a l u a t i o n of S a m p l i n g  3.4.1  Selected Sampling  Designs  Designs  Based on literature, statistical, and practical (i.e., ease of sampling) considerations, the five sampling designs that were considered for further evaluation were (1) simple random sampling without replacement, (2) stratified random sampling (stratified by relative height) without replacement, (3) ellipsoidal sampling without replacement, (4) two-stage systematic sampling, and (5) two-stage sampling with unequal probability of selection with replacement. The five sampling designs selected are possible for use in the operations described in the literature review. Other sampling designs could have been included, but this study focused on these sampling designs to restrict the scope of this research. Three sample sizes (i.e., n=12, 36, and 48 twigs) were selected and allocated for the first three sampling designs. The sample sizes of 12 and 48 twigs were used to assess the effect of small and large sample sizes, respectively, in estimating single tree leaf area estimates. The sample size of 36 twigs was chosen to detect nonlinear responses to sample  Chapter 3. Methods  62  size. These sample sizes are selected for ease of distributing samples into various strata and ellipses proportionally or equally within each stratum or ellipse, as they are multiples of three. For the two-stage systematic sampling design, four different intensities of firstand second-order branches were assessed (i.e., 10 and 20%, 10 and 40%, 20 and 20%, and 20 and 40%). These sampling intensities, instead of fixed sample sizes, were used to (1) avoid the need of a detailed sampling frame (i.e., to maintain one of the advantages of systematic sampling) and (2) mimic operational systematic sampling of tree crowns. For two-stage sampling with unequal probabilities of selection, the three sample sizes (i.e., 12, 36, and 48) were selected from three sets of primary (3, 9, and 12 first-order branches) and four sets of secondary (four sample twigs) sampling units. These sampling combinations were used to mimic operational two-stage sampling of tree crowns. Samples were selected with replacement and with probability proportional to first-order branch and twig lengths.  Simple R a n d o m  Sampling  SRS is a basic sampling design and was included in this study as a basis for comparing the other sampling designs. The observations for each of the three sample sizes (n=12, 36, and 48) were randomly selected from the entire set of twigs in each test population. Random numbers were generated using a F O R T R A N algorithm. The average twig leaf area, total leaf area and associated variances were calculated from the selected samples.  Stratified R a n d o m  Sampling  Stratification of tree crowns by relative height or ellipsoids (ellipsoidal sampling) could be an important variance reduction method for tree crown sampling. Stratification of tree crown by relative height accounts for the vertical variation of leaf area while  Chapter 3. Methods  63  stratification of tree crowns by ellipsoidal sampling accounts for horizontal and vertical variations of leaf area in tree crowns. Therefore, both stratified random and ellipsoidal sampling designs were selected for subsequent evaluation in this study. For stratified random sampling, the crowns of each tree were divided into three equal positions or strata, as it is easier and faster to locate these positions in the tree crowns. Although division of tree crowns into thirds has been widely used in tree crown research, the variances of twig leaf area were computed for each stratum to assess whether stratification by relative height in the crown would likely be useful or not and also to detect any trend that may exist. The number of twigs in each stratum was counted and proportional allocation (by the number of twigs in each stratum) was used to distribute the three selected sample sizes (i.e., n=12, 36, and 48) to each stratum. Subsequently, random numbers were generated and sample twigs were selected from each stratum. Following this, relative proportions (weights), the average twig leaf area, and variance of leaf area in each stratum were calculated. These values were later used to calculate the stratified mean twig leaf area, total leaf area, and associated variances.  Ellipsoidal  Sampling  The twigs in each tree crown were classed into three ellipses based on their relative horizontal length (i.e., hi 0.0 to 0.33 into ellipse one, 0.33 to 0.65 into ellipse two, 0.66 to 1.0 into ellipse three). The number of twigs in these ellipses varied based on the branching frequency, age of the tree, and the distribution of twigs in the crown. The variance of twig leaf area in each ellipse was calculated to assess whether stratification by ellipses is likely to be more efficient than SRS and also to detect any trend that may exist. The number of twigs in each ellipse was counted and proportional allocation (by the number of twigs in each ellipse) was used to distribute the three selected sample sizes to each  Chapter  3.  Methods  64  ellipse. Subsequently, random numbers were generated and sample twigs were selected from each ellipse. The relative proportion (weights), average twig leaf area, and variance of each ellipse were calculated. These values were used to calculate the stratified mean twig leaf area, total leaf area, and associated variances.  Two-stage Systematic Sampling  Systematic sampling ensures complete coverage of populations of interest. Thus, twostage systematic sampling with two random starts was selected for further evaluation in this study. First-order branches were systematically selected at the first stage and sample twigs were, systematically, selected at the second stage. The number of sampled first-order branches was calculated by multiplying the selected sampling intensity by the total number of first-order branches (B) on the tree crowns. The sampling interval (/) between the sampled first-order branches was calculated by dividing the total number of first-order branches by the number of sampled first-order branches. Linear and fractional interval systematic sample selection procedures (Murthy 1967) were used to select the first sample first-order branch and twig. In a linear systematic sample selection procedure, a random number was generated (between 1 and /) to select the first sample first-order branch randomly. On the selected first-order branch, the number of sample twigs was calculated by multiplying a given sampling intensity by the total number of twigs on the selected first-order branch. The sampling interval between sample twigs was calculated by dividing the number of twigs on the selected first-order branches by the number of sample twigs. Then, a random number was generated (between one and the total number of twigs on the selected firstorder branch) to select the first sample twig randomly. Subsequent sample first-order branches and twigs were selected systematically.  Chapter 3.  Methods  65  In fractional interval systematic sample selection procedure, a random number was generated between 1 and B. The selected random number was divided by / and the remainder of the division was multiplied by I to select the first sample first-order branch. The same selection procedure was used to select the first sample twig on selected firstorder branches. Subsequent sample first-order branches and twigs were selected systematically. The linear systematic sample selection procedure does not ensure equal probability of inclusion in the sample for every element in the population if B is not an integral multiple of sample size and / . The fractional interval systematic sample selection method ensures equal probability of inclusion in the sample for every first-order branch in the crown and every twig on the selected first-order branches. Following this, the average twig leaf area, total leaf area, and associated variances were calculated from the selected samples.  Two-stage Sampling  with Unequal Probability  of S a m p l e Selection  This method involves selecting branches with probability proportional to size (PPS) and is expected to provide more efficient leaf area estimates than simple random sampling (SRS) providing the probability is positively correlated with the variable of interest (Cochran 1977, Schreuder et al. 1993). Sampled first-order branches in the tree crown were selected at random with probability proportional to the length of first-order branch (q\ = BLu/ Ylf=\ BLu where B is the number of first-order branches in the tree crown). Sample twigs were also selected with probability proportional to the length of twigs on selected first-order branches samples. Selection probabilities for each sample twig were determined by dividing the length of a twig (TLij) by the sum of the lengths of all twigs on the selected first-order branch (<7 = TLij/ Jlf=i TL^ where BTi is the number of twigs on first-order branch i). 2  Chapter 3.  66  Methods  To select first-order branches and twigs, two random numbers were selected between: (1) 0 and the cumulative totals of the first-order branch lengths (J2f=i BLu), and (2) 0 and the cumulative totals of the twig lengths on the selected first-order branches (Jlf=i  TLij).  When the selected random number fell between two successive cumulative totals, the latter first-order branch was selected. Once the first-order branch was selected, a similar selection procedure was used to select the secondary sampling units (i.e., four sample twigs). This selection procedure was repeated until the desired sample size was achieved. Samples were selected with replacement as it is difficult and sometimes impossible to keep the probabilities proportional to size as the sample size increases (Cochran 1977 p 251). The first-order branch as well as the twig selection probabilities summed to one. The probabilities of twigs selection are:  Qn  BTi  B  x  = [BLu  TLyl/E  x £ TL ]  BL  U  i3  i=l  To calculate the total leaf area of the tree (Yri),  (3.47)  j=l  the selected sample twig leaf area  was multiplied by the inverse of its probability of occurring in the sample.  Y  TI  = LAij/Qij  (3.48)  where LA{ - is the leaf area measured on selected twig. 3  The within-tree standard errors can be determined by taking more than two sample twigs. The sum of the tree leaf area is an unbiased estimate of the true leaf area (YT) n  Y  T  = 1/UJ2YT  (3.49)  i=i  where n is the number of sample twigs (i.e., 12 or 36 or 48). The Horvitz-Thompson (1952) estimator that is coupled with probability proportional to size is given by:  Chapter 3. Methods  67  Y  T  where  P i  = Y LA /p j  ij  i  (3.50)  = n x LA^j Yli=\ LA  i:j  In randomized branch sampling (RBS), the sampling frame is developed as the population is sampled (Valentine and Hilton 1977). However, this was not done during data collection for this study the length of all the parent shoots (i.e., second-, third-, and fourth-order branches). Nor were the internodal distances between all the second-, third, and fourth-order branches measured). Thus, a test population with complete paths in the tree crowns could not be created and R B S was not included in this study. 3.4.2  Simulation  The twelve generated trees were used as populations in the simulation experiments. Each simulation was run 1000 times to obtain stable estimates of population parameters. A total of 192,000 runs (sample sets) were made (i.e., 3 sample sizes x 4 sampling designs x 12 trees x 1000 repetitions plus 4 sampling intensities x 1 sampling design x 12 trees x 1000 repetitions) The following steps were used to estimate total leaf area in the simulation experiments: 1. Select a tree from the test population; 2. Select a sampling design; 3. Select one of the three sample sizes (number of twigs to be sampled); 4. Select n sample twigs using the sampling design specified in 2; 5. Calculate the sample mean and sample variance of leaf area for the n selected sample twigs using the appropriate estimator for the sampling design under consideration (Cochran 1977);  Chapter 3. Methods  68  6. Calculate the estimated total tree leaf area using the respective estimator for the population total for the sampling design specified in step 2; 7. Repeat steps 4 to 6 1000 times; 8. Repeat steps 4 to 7 with another sample size; 9. Repeat steps 3 to 8 for each sampling design; and 10. Repeat steps 2 to 9 for each tree in the test population. The Monte Carlo sampling algorithms were written in F O R T R A N 90 (Brooks 1997) and compiled on the mainframe computer at the University of British Columbia under a Unix operating system.  3.4.3  M e a s u r e s U s e d in A s s e s s i n g the Selected S a m p l i n g  Designs  The estimated leaf area for each simulation was compared to true total leaf area for the simulated trees obtained by summing all the possible sample points (twigs) in the test population. The sampling strategies were evaluated for bias, sampling variance, the distribution of the estimates, cost efficiency, and relative ease of use. 1. Mean Square Error (MSE) M S E was estimated as the squared difference between the mean of the Monte Carlo estimates and the population parameter. The M S E over the 1000 samples for each tree sampling design was computed as: 1000  MSE = J2(Yk - V) /i000 2  fc=i  (3.51)  where Y is the estimated total leaf area per tree for the fcth replication; and Y is k  the known total leaf area of the tree.  Chapter  3.  69  Methods  2. Bias Bias was estimated as the difference between the mean of the Monte Carlo estimates and the population parameter. The bias over the 1000 samples for each tree sampling design, was computed as:  1000 Bias  =  53(y  fc=i  - y)/1000  fc  (3.52)  Percent bias was calculated as a percentage of Y . 3. Sampling Variance The variance of the sampling designs was estimated from the sampling distribution obtained using the Monte Carlo simulations. The variance of the estimates for the 1000 samples of each simulation was computed as:  1000  _  4 = E(^- )V999 . fe=i y  where  (3.53)  is sampling variance of the j t h tree, and Y is the average of the 1000  total leaf area estimates for the zth tree. _  1000  Y = £ y /1000 fc=i fc  (3.54)  4. Distribution of the Estimates The frequency distributions of the Monte Carlo estimates of the total leaf area per tree were graphed to determine if estimates followed a known distribution. Distributions were compared to normal, Weibull, and beta distributions using graphical examination and estimates of skewness and kurtosis (Rice 1988).  Chapter 3. Methods  70  5. Cost Efficiency For each sampling design, the total sampling cost was estimated from the sampling and measurement time data. The estimated sampling cost was based on a twoperson crew at a cost of $400.00/day for an eight-hour day. The crew cost was multiplied by the estimated sampling and measurement time for a given sampling design. The sampling cost (TC) by sampling design was estimated as:  TC = 0.83[T + T ] S  m  (3.55)  where the multiplier 0.83 is crew cost (dollars/minute based on an eight hour working day), T is the total sampling time (minutes) for sampling a tree crown using s  a specified design. T  m  is the measurement time required (minutes) for stripping  needles, aligning, scanning leaf area, and recording. Only variable costs were considered in this study, since fixed costs are not a function of sample size and sampling design (Watson 1973). T included the time spent for (1) marking the boundaries for some sampling des  signs, (2) labelling and identifying the sample units, (3) clipping the sample twigs, (4) fresh leaf weighing, and (5) bagging and recording. The last three attributes were not affected by the type of sampling design used. Only time spent on (1) and (2) were used to compare the sampling designs. Travel time was assumed to be the same for all sampling designs. Stratified random sampling by relative height strata involves dividing the population of twigs into non-overlapping sections and independently using SRS in each stratum. The expected cost of sampling in any stratum is the cost of selecting samples plus the cost of positioning the necessary boundary markers. Labelling and identifying the stratum and sample twigs in each stratum would be faster than  Chapter 3.  Methods  71  in SRS. In ellipsoidal sampling, each sample belongs to a shell and random samples are taken from each shell of foliage. Besides the time spent in selecting sample twigs, the cost of ellipsoidal sampling includes time spent in setting up markers for identifying the shell boundaries. Identification and sampling within each ellipse would be slower than in stratified sampling. Based on these assertions, the following assumptions were made. (a) The cost of measuring each sample twig is identical regardless of its position in tree crowns; (b) SRS would take 1.75 times more time than for two-stage systematic sampling to identify the population and then to select sample twigs; (c) Stratified sampling with three strata would require 1.25 times more time than required by two-stage systematic sampling; (d) In ellipsoidal sampling, the cost of identifying and positioning markers for each ellipse would require 1.5 times more than the cost of two-stage systematic sampling; and (e) For two-stage unequal probability sampling, measuring and identifying sample first-order branches and twigs were assumed to take twice as much time as systematic sampling. 6. Relative Ease The relative ease of using sampling designs in terms of selecting and measuring sample points, identifying first-, second-, third-and/or fourth-order branches, identifying strata or ellipses, and overall operational ease was assessed subjectively, and used in ranking the designs.  Chapter 4  Results and Discussion  This chapter is divided into four sections. Section 4.1 describes the preliminary analysis and summaries of the data. Section 4.2 presents the examination and modelling of leaf area in three parts: (1) graphical examination of leaf area, (2) leaf area distribution models, and (3) the relationship of leaf area distribution model parameters to tree attributes (the first objective of this study). Section 4.3 presents the generated test populations that were used to achieve the second objective of this study. Section 4.4 provides the evaluation of results, and a discussion of the performance of the five sampling designs (the second objective).  4.1  Preliminary Data  Analysis  The sample trees ranged from: 4.08 to 36.4 m in height; 5.9 to 44.4 cm in D B H ; 0.16 to 21.10 m in H T B L C ; 1.80 to 4.68 m in C W ; 15 to 133 years in B H A G E ; 0.38 to 0.96 in crown ratio; 3.43 to 17.22 m in crown lengths; and 1.42 to 3.68 (cm /cm) in A P C M 2  (Table 4.2). Crown widths and lengths were smaller for smaller D B H (younger) trees. The average leaf area per twig for the sampled first-, second-, third- and fourth-order branches did not show a linear trend with age classes (i.e., < 20 years, 50 to 60 years, and > 120 years) (Table 4.3). Trees between 50 to 60 years of age showed the highest A P C M , whereas trees older than 120 years showed the least A P C M , although these differences were not great.  72  4.  Chapter  Results  and  73  Discussion  Table 4.2: Tree height (HT), diameter outside bark at breast height (DBH), height to the base of live crown ( H T B L C ) , crown width (CW), and average leaf area per cm ( A P C M ) of the 12 hybrid spruce trees measured.  4.2  Tree No.  Tree HT (m)  DBH HTBLC (cm) (m)  CW (m)  BHAGE (years)  No. of sampled 1° branches  No. of sampled twigs  APCM  1 2 3 4  4.70 4.08 4.85 6.55  5.90 8.00 7.30 11.00  0.16 0.65 1.35 0.26  1.80 2.60 1.95 3.50  15 15 15 15  17 13 13 14  109 120 101 141  3.01 2.27 3.68 2.86  5 6 7 8  15.0 14.0 16.9 15.4  16.70 16.50 38.50 20.10  5.20 4.60 7.65 6.60  4.10 4.20 4.30 4.45  52 50 51 55  17 16 17 22  237 269 237 245  2.83 2.98 3.32 3.11  9 10 11 12  32.5 34.2 36.4 30.4  44.40 42.20 41.00 40.00  15.28 21.10 19.48 17.75  4.68 4.10 4.10 3.70  130 132 133 132  29 27 18 23  428 378 217 266  2.59 3.27 1.42 2.96  Examining and Modelling Leaf A r e a  Three dimensional graphs were generated to observe the variability and/or the trend of leaf area spatial distribution in tree crowns and to select models for leaf area distribution. Also, the selected leaf area estimated model parameters were related to tree attributes. These results are presented in Sections 4.2.1, 4.2.2, and 4.2.3.  Chapter 4. Results and Discussion  74  Table 4.3: Summary of average leaf area per twig (cm ) and average leaf area per cm ( A P C M in cm /cm) by three age class. 2  2  Age class (years)  Variable  Mean  Standard deviation  Minimum  Maximum  < 20  Leaf area APCM  120.24 2.92  67.47 0.92  19.71 0.14  268.69 5.94  50 to 60  Leaf area APCM  121.34 3.06  92.33 1.21  17.24 0.16  383.22 8.15  > 120  Leaf area APCM  44.47 2.67  20.03 1.28  7.81 0.01  175.78 12.22  ALL  Leaf area APCM  95.34 2.85  62.94 1.21  7.81 0.01  383.22 12.22  Chapter  4.2.1  4.  Results  and  Discussion  75  G r a p h i c a l E x a m i n a t i o n of L e a f A r e a  Two-dimensional scatter diagrams of A P C M by horizontal position and A P C M by vertical position within the four quadrants of one tree from each of the three age classes are presented in Figure 4.1. Graphs for the remaining trees are included in Appendix A . A l l the graphs indicate that A P C M did not vary among the cardinal directions of the first-order branches. Therefore, cardinal directions were not included in further analysis. Nine of the 12 sample trees showed no significant trend in A P C M over relative horizontal length of first-order branches (Figures A. 16 to A. 18 in Appendix A ) , in that deviations of A P C M from the mean A P C M values were not high at different horizontal lengths on these trees. Three trees showed some trend in A P C M along the relative horizontal length. The highest A P C M values for these trees were obtained between relative horizontal lengths of: 0.4 to 0.6 for tree 3, 0.3 and 0.6 for tree 7, and 0.5 to 0.7 for tree 10 (Figure 4.1). Despite the low number of trees, these results were consistent with ranges reported in other studies. For example, maximum foliage was found for radiata pine at 0.5 horizontal length (Wang et al. 1990) and between 0.5 to 0.75 (Whitehead et al. 1990). For Douglas-fir, peak foliage density was found between 0.5 and 0.73 relative horizontal length (Kershaw and Maguire 1996), at 0.25 relative horizontal length near the top and at 0.75 relative horizontal length near the base of crown (Jensen and Long 1983). Webb and Ungs (1993) found no significant differences in leaf area distribution along first-order branches of a Douglas-fir trees. The highest A P C M was found between the relative heights of: 0.35 to 0.5 for tree 3 (Figure 4.2), 0.25 to 0.5 for tree 6, and 0.1 to 0.3 for tree 9; 0.05 to 0.4 for trees 7 and 8 (Figure B.20 in Appendix B); 0.4 to 0.6 for tree 5 (Figure B.20 in Appendix B); 0.05 to 0.3 for trees 6, 9, and 10 (Figure 4.2 and Figure B.21). No noticeable differences  Chapter  4.  Results  and  Discussion  76  Direction  0.0  0.2 0.4 0.6 0.8 Relative horizontal length  1.0  *  West South North East  Direction  0  J_  0.2 0.4 0.6 0.8 Relative horizontal length  ^ + x o  1.0  ^ + x o  West South North East  Direction A + x o Relative horizontal length  west South North East  1.0  Figure 4.1: Leaf area per centimetre (cm /cm) at different relative horizontal lengths by the four cardinal directions for Trees 3, 7, and 10. A relative horizontal length of one indicates values at the tip of the first-order branch. 2  Chapter  4.  Results  and  Discussion  77  in A P C M over relative height were detected within three of the 12 sample trees (Figure B.19). These results were consistent with the relative height positions reported to carry the maximum leaf area: (1) between 0.1 and 0.3 (Maguire and Bennett 1996, Massman 1982) and at 0.2 (Woodman 1971) for Douglas-fir crowns; (2) between 0.2 to 0.4 for loblolly pine crowns; (3) between 0.3 and 0.7 for western larch (Bidlake and Black 1989); and (4) at one-third of the tree crown length from the crown base for radiata pine (Wang et al. 1990). Based on one sample tree, Webb and Ungs (1993) reported an increase in N A D from the base of the crown to the tip of the tree for open grown Douglas-fir. The relative horizontal and vertical (height) positions where maximum A P C M values were obtained indicate the zone of maximum photosynthetic capacity in the crown (Maguire and Bennett 1996, Schulze et al. 1977). Results of this (Figures 4.1 and 4.2) and other studies indicate that horizontal and vertical distribution of foliage is highly variable throughout the crown (Maguire and Bennett 1996, Whitehead et al. 1990, Woodman 1971). This could be attributed to differences in the spatial distribution of trees in a stand, stand development history, and other ecophysical factors. The significant variation of foliage distribution at various relative heights of tree crowns prompted Binkley and Merrit (1977) to divide pine crowns into seven strata to assess the distribution of light in tree crowns.  C h a p t e r  4.  R e s u l t s  a n d  78  D i s c u s s i o n  Direction  0.2  0.4 0.6 Relative height  0.8  1.0  A + x o  West South North East  Direction  0.2  0.4 0.6 Relative height  0.8  1.0  ^ + x o  West South North East  Direction  0.2  0.4 0.6 Relative height  0.8  1.0  A + x o  West South North East  Figure 4.2: Leaf area per centimetre (cm /cm) at different relative heights by the four cardinal directions for Trees 3, 6, and 10. A relative height of zero indicates values at the tip of the tree. 2  Chapter  4.  Results  and  Discussion  79  Three dimensional graphs of A P C M of one tree from each of the three age classes are presented in Figures 4.3 to 4.5. The remaining graphs are included in Appendix C (Figures C.22 to C.30). The three-dimensional graphs indicate the dispersion of leaf area in tree crowns and an increase in A P C M towards the tip of the tree and towards the tips of the branches. A simple explanation is that younger leaves (higher leaf area) are concentrated at the tip of the first-order branches and the tree (Schultze et al. 1977). However, no noticeable trend in A P C M was detected in tree 10 (Figure B.21). Webb and Ungs (1993) found significant differences in leaf area distribution in Douglas-fir crowns vertically, but not horizontally. Within the tree crowns, A P C M was found to be highly variable, as is typical for spruce leaf attributes (Schultze et al. 1977). This portrait of A P C M ignores variations in branch lengths, number of second-, third-, and fourth-order branches at different heights of the tree displaying the distribution of leaf area within tree crowns. The maximum A P C M was four times as large as the average A P C M (Table 4.3). Wang et al (1990) also found non-uniform leaf area distribution within radiata pine crowns. The maximum N A D they found was four times larger than the average N A D . This variation will have a large influence on the amount of light intercepted (i.e., light extinction coefficients), crown transparency (Gertner and Kohl 1995), radiation transfer models (Webb and Ungs 1993), photosynthesis, and respiration by the foliage.  Chapter  4.  Results  and  Discussion  80  F i g u r e 4.3: L e a f a r e a p e r c e n t i m e t r e ( c m / c m ) at different r e l a t i v e h e i g h t s a n d r e l a t i v e 2  h o r i z o n t a l l e n g t h s for T r e e 1 ( y o u n g tree).  A r e l a t i v e h e i g h t of zero i n d i c a t e s values at  t h e t i p of t h e tree. A r e l a t i v e h o r i z o n t a l l e n g t h of one i n d i c a t e s values at the t i p of t h e first-order  branch.  C h a p t e r  4.  R e s u l t s  a n d  81  D i s c u s s i o n  Figure 4.4: Leaf area per centimetre (cm /cm) at different relative heights and relative horizontal positions for Tree 5 (middle age class). A relative height of zero indicates values at the tip of the tree. A relative horizontal length of one indicates values at the tip of the first-order branch. 2  Chapter  4.  Results  and  Discussion  82  F i g u r e 4.5: L e a f a r e a p e r c e n t i m e t r e ( c m / c m ) at different r e l a t i v e h e i g h t s a n d r e l a t i v e 2  h o r i z o n t a l l e n g t h s for T r e e 10 ( o l d tree).  A r e l a t i v e h e i g h t of zero i n d i c a t e s values at  t h e t i p of t h e t r e e . A r e l a t i v e h o r i z o n t a l l e n g t h of one i n d i c a t e s v a l u e s at t h e t i p of the first-order b r a n c h .  Chapter  4.2.2  4.  Results  and  83  Discussion  Leaf Area Distribution Models  In this study, the variation of A P C M among tree crowns was large owing to vertical and horizontal variation, and differences in crown size and shape. Assessed individually, the bivariate Weibull model parameter estimates varied widely. The range of each parameter estimate was from: 0.0008 to 0.313 for cu 0.00003 to 0.0418 for c ; 0.000006 to 1.813 for 2  c ; 0.00003 to 2.035 for c ; and 0.0018 to 0.930 for c (Table 4.4). On the other hand, 3  4  5  the bivariate beta model parameter estimates did not vary as widely (i.e., CQ ranged from 0.742 to 1.515, c ranged from 1.010 to 1.877, and c ranged from 0.883 to 1.975). 7  8  Parameter estimates c\ and c (of the bivariate Weibull distribution function) varied 2  less in terms of absolute values among the sample trees than the parameter estimates of the bivariate beta distribution function, while parameter estimates c to C5 varied more 3  (Table 4.4). Prediction of A P C M using branch position from the top of the tree relative to total tree height (zi{) was more precise (higher R and lower R M S E ) than using branch po2  sition from the top of the tree relative to crown length (zi ) except for trees 10 and 11 2  (Table 4.5). Trees 10 and 11 were the two tallest trees and had the two lowest crown ratios. This precision is more pronounced in the bivariate Weibull distribution function than in the bivariate beta distribution function. When zi\ instead of zi was used, R  2  2  increased by 35% and 1% on the average and R M S E decreased by 15.9% and 3% in the bivariate Weibull and the beta distribution functions, respectively (Table 4.5). However, the differences in the fit statistics for the two bivariate distribution functions were minimal.  4.  Chapter  Results  and  84  Discussion  Table 4.4: Estimated parameters for the bivariate Weibull (ci to C5) and beta (CQ to c$) leaf area per cm prediction models using branch position from the top of the tree relative to total tree height as the predictor variable (zii).  Tree No.  Weibull Cl  C2  C3  1 2 3 4  0 3130 0 2799 0 0259 0 7560  0.0013 0.0113 0.0130 0.0002  0.6530 1.8130 0.0289 0.9680  5 6 7 8  0 0220 0 0199 0 0398 0 0105  0.0002 0.0195 0.0060 0.0007  9 10 11 12  0 0192 0 0008 0 0534 0 0016  0.0127 0.0068 0.00003 0.0418  Beta c  C5  C6  C7  eg  0.0095 0.0019 0.0016 0.0182  0.0123 0.1633 0.2720 0.0018  0.742 0.989 0.979 1.373  1 010 1 293 1 198 1 709  1.538 1.758 1.821 1.691  1.0120 0.0017 0.0017 0.0031  0.0002 0.0273 0.0765 0.00005  0.0042 0.2800 0.5810 0.9300  0.939 0.899 0.904 0.904  1 270 1 032 1 290 1 084  1.310 1.210 1.630 1.476  0.0232 0.0647 0.8940 0.000006  0.00047 0.00003 0.00008 2.0350  0.8630 0.0986 0.0053 0.0075  1.149 1.290 1.515 1.428  1 136 0.883 1 175 1.910 1 877 1.975 1 581 1.437  4  85  Chapter 4. Results and Discussion  Table 4.5: Root mean square error (RMSE in cm /cm) and coefficient of determination (R ) for the bivariate leaf area per cm prediction models using two definitions of branch position. 2  2  zi! = DFTj/ET Tree No.  Weibull  zi =DFT /CL 2  Beta  Weibull  1  Beta  RMSE  R  RMSE  R  RMSE  R  RMSE  R  1 2 3 4  0.92 1.31 1.31 0.88  0.91 0.85 0.86 0.92  0.94 1.33 1.27 0.76  0.90 0.84 0.88 0.94  1.18 1.41 1.40 0.90  0.85 0.82 0.85 0.92  0.99 1.35 1.34 0.76  0.89 0.83 0.86 0.94  5 6 7 8  0.60 0.82 0.88 1.03  0.94 0.91 0.93 0.91  0.64 0.78 0.88 1.02  0.92 0.92 0.93 0.91  0.72 1.60 1.00 1.09  0.91 0.66 0.91 0.91  0.65 0.79 0.95 1.14  0.93 0.92 0.92 0.89  9 10 11 12  1.11 1.63 0.87 0.70  0.61 0.77 0.96 0.78  0.97 1.57 0.99 0.73  0.69 0.78 0.94 0.95  1.16 1.56 0.75 1.57  0.60 0.79 0.95 0.78  0.96 1.59 1.06 0.74  0.71 0.78 0.92 0.94  2  2  2  2  86  Chapter 4. Results and Discussion  The findings in this study are consistent with the biological attributes of tree crowns. Crown length is a less consistent attribute than tree height, as C L is easily influenced by density of a stand and by factors that lead to abscission of branches. The length of a first-order branch is highly influenced by DFTi  (Mitchell 1975) because of strong apical  dominance of the stem. The reference of a branch to the top of a tree (DFTi)  allows the  relative height (zii) to be zero at the top of the tree and increase towards the base of the live crown. Thus, zii will be correlated with first-order branch length and higher-order branch length. Prediction of A P C M using the univariate beta distribution function was more precise than using the univariate Weibull distribution function. The R and R M S E values ranged 2  from 0.34 to 0.77 and 1.20 to 2.32 for the univariate Weibull distribution and from 0.62 to 0.84 and 1.03 to 2.30 for the univariate beta distribution function. On average, the use of the univariate beta distribution instead of the univariate Weibull distribution function increased R values by 12.3% and reduced R M S E values by 11.5% (Table 4.6). 2  The 87 and 83 parameter estimates of the univariate beta distribution function (Equation 3.27) ranged from 2.52 to 5.76 and 1.84 to 7.63, respectively, in estimating A P C M at various heights on the trees in this study. These values ranged from 2.4 to 4.6 and 1.0 to 2.8, respectively, in Maguire and Bennett's (1996) study describing the vertical distribution of foliage within individual tree crowns. M y results are consistent with theirs and confirms the use of a univariate beta distribution function for describing the vertical distribution of foliage in conifer tree crowns. The 81, 82, and 8% (i.e., the location, scale, and shape) parameter estimates of the univariate Weibull distribution function varied widely from -5.083 to -0.41, 0.2 to 5.66, and 0.84 to 43.45, respectively. The shape parameter estimate had the highest variability of all the parameters. The bivariate Weibull and beta distribution functions resulted in better fits (higher R and lower R M S E ) than the univariate Weibull and beta distribution functions. On 2  87  Chapter 4. Results and Discussion  Table 4.6: Root mean square error (RMSE in cm /cm), coefficient of determination (R ) values for estimating leaf area per cm ( A P C M ) as a function of relative vertical position (zii). 2  Tree  n  no.  Weibull  2  Beta  RMSE  R  RMSE  R  2  2  1 2 3 4  109 120 101 141  2.01 1.23 1.98 2.26  0.60 0.73 0.73 0.34  1.88 1.14 1.90 1.23  0.65 0.77 0.75 0.83  5 6 7 8  237 269 237 245  1.64 1.82 2.32 1.70  0.71 0.69 0.62 0.73  1.46 1.72 2.30 1.67  0.77 0.72 0.64 0.74  9 10 11 12  428 378 217 266  1.20 1.66 1.24 1.82  0.58 0.77 0.50 0.71  1.08 1.39 1.03 1.71  0.62 0.84 0.65 0.74  Chapter  4.  Results  and  Discussion  88  average, R values increased by 15% and R M S E values decreased by 39% with the use of 2  the bivariate distribution functions (comparing Tables 4.5 vs 4.6). Maguire and Bennett (1996) strongly advocated the use of bivariate beta distribution function (Wang et al. 1990) to describe the vertical and horizontal distribution of foliage within individual tree crowns owing to its closed interval form. However, the bivariate Weibull distribution function (Lu and Bhattacharyya 1990) also provides a closed interval form and avoids the need for truncation. This argument is also supported by the better fit statistics (higher R  2  and lower R M S E ) of the bivariate Weibull distribution function than the bivariate  beta distribution function for most sample trees (Table 4.5). Maguire and Bennett (1996) also used the same argument for the use of a univariate beta distribution function instead of the more widely used univariate Weibull distribution function for describing the vertical distribution of foliage within individual tree crowns. However, Schreuder and Swank (1974) reasoned that the Weibull distribution, unlike the beta distribution, is simple and does not require numerical estimation of beta and gamma functions (see Equation 3.27). The results found in this study confirm the approach used by Maguire and Bennett (1996) (Table 4.6). Beside the difficulty of fitting (i.e., weak convergence) the univariate Weibull distribution model for A P C M at various relative heights, the univariate beta distribution function showed better fit statistics (higher R  2  and lower R M S E ) than the univariate Weibull distribution function (Table 4.6). This could be credited to the truncation of the univariate Weibull distribution function at the base of the crown (Maguire and Bennett 1996). Some studies on the vertical and horizontal distribution of foliage (e.g., Kershaw and Maguire 1996, Maguire and Bennett 1996) have provided insights into the positional variability of leaf area in tree crowns, but have not produced any method that combined these variations (vertical and horizontal) at specified points in tree crowns. The approach used in this study examined the combined variability of leaf area within tree crowns as  Chapter 4. Results and Discussion  89  indicated in Figures 4.2 to 4.4 and Appendix C (Figures C.22 to C.30).  4.2.3  R e l a t i o n s h i p of L e a f A r e a D i s t r i b u t i o n M o d e l s to T r e e A t t r i b u t e s  Estimated parameters of the two bivariate leaf area distribution models were related to tree and crown size variables. There was a direct proportional relationship between the bivariate Weibull parameters parameter c% and CR, and inverse proportional relationships between: c% and D B H ; c% and C W ; t% and H T , and Sj and C W . Parameters c\ and £4 did not have significant correlations with any of the tree size variables at a = 0.05. For the bivariate Weibull and beta distribution functions, the S U R fitting method resulted in more precise A P C M estimates than separate M L R fits or the composite fitting method (Table 4.7). For the bivariate Weibull distribution function, the equations to predict the estimated parameters using the S U R fitting method were:  logfo) = -0.126 + 0.057 x log(DBH) + 0.096 x CR  c% = 1.921 - 1.138 x \n(DBH)  eg = 0.877- 0.309 x (l/log(v ffT)) A  (4.56)  (4.57)  (4.58)  The ln(x) transformation, unlike the base 10 logarithm transformation of variable x (log(x)), met the assumptions of M L R in Equation 4.57. For the bivariate beta distribution function, the equations to predict the estimated parameters using the S U R fitting method were:  log(c ) = 0.272 - 0.067 x CW 6  (4.59)  Chapter  4.  Results  and  90  Discussion  Table 4.7: Estimated root mean square error (ERMSE) in c m / c m and I-square (I ) values for two leaf area per cm prediction models using three fitting techniques. 2  Model  Separate M L R fit ERMSE I  2  Weibull Beta  0.99 1.14  0.47 0.12  SUR Fit ERMSE  I  0.77 1.13  0.60 0.14  2  Composite Fit ERMSE I  2  1.10 1.14  c = 2.13 — 0.223 x CW  8  HT  +  0.984 x  0.17 0.11  (4.60)  7  c = 0.77 - 0.05 x  2  (1/CR)  (4.61)  CR, D B H , H T , C W , and their transformations were selected as predictor variables for estimating the above parameters on the basis of their significance at a = 0.05, compliance with the assumptions of M L R and simple linear regression (SLR), and their predictive ability. Stands with lower bivariate c (Weibull) and c (beta) model parameter estimates 2  7  were denser than those with higher values, if other conditions were equal. This is due to the inverse relationship between these estimated model parameters and C R and C W . A P C M and average twig leaf area were poorly related to most of the tree size variables. This is also shown by the poor relationships between the leaf area model parameter estimates and the tree size variables. This result is in accordance with Wang tt a/.'s (1990) findings of poor relationship between N A D and D B H , tree height, and total leaf area within the crown.  Chapter  4.  Results  and  91  Discussion  Since the correlations among the estimated parameters (Table 4.8) were not considered in the single parameter prediction equations, the separate M L R fits did not result in efficient estimates of A P C M . Because the bivariate model parameter estimates and the significant tree attributes are fitted simultaneously in the S U R fitting method their effects are confounded (Judge et al. 1985).  4.3  G e n e r a t e d Test Populations  To achieve the second objective of this study, test populations were created using the steps described in Section 3.3.  4.3.1  First-Order Branch Length  When zii instead of zi  Predictions  was used on Equations 3.25 to 3.28, the R values on average 2  2  increased from (1) 0.70 to 0.95 (26.3%) for the Weibull function, (2) 0.89 to 0.95 (6.3%) for the beta function, and (3) from 0.88 to 0.98 (10.10%) for the modified Chi-square model. R M S E values decreased from (1) 0.70 to 0.53 (24.3%) for the Weibull function, (2) 0.57 to 0.35 (38.6%) for the beta function, and (3) from 0.45 to 0.26 (42.2%) for the modified Chi-square model. The Chi-square distribution model was not influenced by the position of branches in tree crowns (zii °r zi ), as the fit statistics did not differ 2  (compare Table 4.9 vs 4.10). Therefore, zii was used in further analysis. The use of DFTi  also enabled a comparison of the four distribution functions tested for branch  length prediction with Mitchell's (1975) model at the same reference points. The R M S E values for the five models were ranked from one (best) to five for each tree. The ranks for each model were summed over the 12 trees. The Chi-square and modified Chi-square distribution functions had the lowest rank sum followed by Mitchell's (1975) model. However, the modified Chi-square distribution had the widest variability  Chapter  4.  Results  and  92  Discussion  Table 4.8: Pearson correlation coefficients between the estimated parameters of leaf area per cm prediction models.  Parameter  Weibull: ci c c c c Beta: c c c 2  3  4  5  6  7  8  c\  c  c  C4  C5  1.00 -0.321 0.545 -0.176 -0.357  1.00 -0.36 0.847 -0.078  1.00 -0.278 -0.494  1.00 -0.235  1.00  0.116 0.305 0.208  0.259 -0.020 -0.274  0.048 0.307 0.307  0.431 0.302 -0.115  -0.345 -0.339 -0.475  2  3  c^  1.00 0.81 0.31  C-J  1.00 0.44  eg  1.00  Chapter  4.  Results  and  93  Discussion  Table 4.9: Root mean square error (RMSE in m) and coefficient of determination (R ) for five branch length prediction models using branch position from the top of the tree relative to total tree height as the predictor variable (zii). n is the number of sampled first-order branches. 2  Model type Tree No.  Weibull  Chi-square  Beta  n RMSE  R  RMSE  R  RMSE  R  2  2  2  Modified Chi-square RMSE R 2  Mitchell (1975) RMSE R  2  1 2 3 4  17 13 13 14  0.90 0.18 1.01 0.28  0.13 0.98 0.27 0.97  0 14 0 17 0 23 0 09  0.99 0.98 0.99 0.99  0.27 0.19 0.07 0.14  0.93 0.97 0.99 0.99  0.28 0.19 0.18 0.44  0.99 0.98 0.99 0.96  0.16 0.17 0.07 0.09  0.95 0.98 0.99 0.99  5 6 7 8  17 16 17 22  0.61 0.35 0.69 0.34  0.90 0.97 0.90 0.97  0 0 0 0  26 25 21 27  0.98 0.98 0.99 0.98  0.46 0.29 0.80 0.36  0.94 0.97 0.86 0.96  0.21 0.23 0.23 0.24  0.99 0.99 0.99 0.98  0.28 0.23 0.27 0.32  0.97 0.98 0.98 0.98  9 10 11 12  29 27 18 23  0.75 0.38 0.45 0.28  0.90 0.94 0.94 0.97  0 23 0 25 0 46 0 32  0.99 0.99 0.94 0.97  0.70 0.22 0.46 0.27  0.91 0.98 0.93 0.97  0.16 0.16 0.47 0.27  0.97 0.99 0.93 0.97  0.13 0.35 0.48 0.35  0.99 0.94 0.93 0.96  Chapter  4.  Results  and  94  Discussion  Table 4.10: Root mean square error (RMSE in m) and coefficient of determination (R ) for five branch length prediction models using branch position from the top of the tree relative to crown length as the predictor variable (z^) • n is the number of sampled first-order branches. 2  Model type Tree No.  Weibull  Chi-square  Beta  n RMSE  R  RMSE  R  RMSE  R  2  2  2  Modified Chi-square RMSE R  2  1 2 3 4  17 13 13 14  0.94 0.20 1.04 0.31  0.14 0.98 0.24 0.97  0.15 0.18 0.07 0.07  0.98 0.98 0.99 0.99  0.28 0.21 0.20 0.18  0.92 0.97 0.97 0.99  0.89 0.29 0.30 0.69  0.23 0.95 0.93 0.86  5 6 7 8  17 16 17 22  1.04 0.63 1.30 0.89  0.72 0.89 0.68 0.79  0.28 0.28 0.22 0.38  0.98 0.98 0.99 0.96  0.75 0.42 1.09 0.70  0.84 0.95 0.77 0.87  0.35 0.67 0.23 0.53  0.97 0.88 0.99 0.92  9 10 11 12  29 27 18 23  1.40 0.67 0.88 0.86  0.68 0.81 0.78 0.77  0.27 0.37 0.04 0.28  0.99 0.94 0.99 0.98  1.16 0.53 0.70 0.62  0.78 0.88 0.86 0.88  0.44 0.38 0.41 0.26  0.97 0.94 0.95 0.98  Chapter  4.  Results  and  95  Discussion  Table 4.11: Estimated parameters for five branch length prediction models for each tree using branch position from the top of the tree relative to total tree height as the predictor variable (zi\).  Tree No  Weibull  b  Chi-square  4  Beta  6  0.56 1.22 1.81 1.98  0.54 0.86 0.89 1.11  11.22 -0.63 0.01 15.14  -1.53 21.25 -1.63 -0.99  6.57 3.79 3.75 5.20  2.65 0.61 4.62 0.51  9.56 0.72 10.50 0.28  8.73 5.08 8.11 15.37  2.45 2.76 72.80 3.27  3.35 2.83 4.45 4.64  6.81 9.17 6.96 7.62  0.01 0.01 -0.12 0.02  -0.51 0.42 -0.14 -0.39  0.67 0.90 0.36 1.23  0.36 1.13 0.01 1.45  9.29 43.42 24.98 31.78  3.09 2.02 2.02 2.28  5.75 3.64 3.89 4.51  13.25 0.83 2.69 2.16  -0.04 0.02 -0.01 0.02  0.24 -3.44 -1.83 -2.32  1.53 0.67 0.64 0.98  2.18 0.75 0.51 1.13  5  -5.14 -6.25 -0.10 -4.40  12.09 6.99 10.10 5.23  24.44 22.79 33.10 23.81  0.19 0.42 0.27 1.08  0.11 0.52 0.75 -0.01  2.43 2.38 3.66 0.42  5 6 7 8  -2.95 -2.97 -7.03 -3.08  3.38 3.50 7.42 3.52  17.85 17.71 44.79 23.12  0.69 0.64 1.09 0.39  4.53 2.13 4.47 8.51  9 10 11 12  -2.89 -0.78 -0.13 -0.03  3.25 1.14 0.50 0.40  22.85 6.14 2.81 2.24  0.92 0.42 0.59 0.51  4.89 32.03 17.22 22.35  7  b  9  b  10  Mitchell (1975)  S  1 2 3 4  6  Modified Chi-square b  b  i>  2  3  b  1  fc  b  ll  12  b  b  13  b  of parameter estimates (69 ranged from -0.63 to 15.14, 610 ranged from -1738.63 to 21.25, and fen ranged from -3.44 to 6.57) (Table 4.11). The Chi-square distribution function showed better fit statistics (higher R and lower RMSE) than Mitchell's (1975) model 2  for most sample trees. However, the differences in R values between them were often 2  small (Table 4.9). The Chi-square distribution and Mitchell's (1975) branch length prediction models provided reasonable branch length estimates at various heights. The fit statistics showed that these models agree well with the hybrid spruce branch length data. Approximately 98% and 97% of the branch length variation were explained by the Chi-square distribution and Mitchell's (1975) branch length prediction models respectively (Table 4.9). However, when the estimated branch lengths for each tree were plotted against height above ground, they showed under- or over-estimation of non-sampled branch lengths compared to the consecutive sample branches and non-curvature trends (Figures 4.6 to 4.8). This likely  96  Chapter 4. Results and Discussion  occurred because the parameter estimates for these models were estimated using least squares regression which minimized the sum of squared errors over the entire tree height. After discovering this problem, interpolation techniques were evaluated for estimating the length of non-sampled first-order branches. The modified quadratic interpolation estimates showed smoother and better curvature than the linear interpolation estimates, and better predictions than the Chi-square distribution or Mitchell's (1975) models (Figures 4.6 to 4.8). Thus, modified quadratic interpolation was used to estimate first-order branch length for the non-sampled branches for the population of 12 trees. Mitchell (1975) asserted that the profile of a tree crown is governed by the value of &13 in his model. Large 613 values decrease the degree of convex curvature towards a conical shape. Smaller positive values lead to greater curvature. Based on this assertion, the sample trees in this study had high curvature, except for trees 1 and 3 which had a conical shape (Table 4.11). Both 6  1 2  and 6  1 3  of the Mitchell's (1975) model were not  related to tree size variables.  4.3.2  Second-Order  B r a n c h Length and N u m b e r  Predictions  Equations 3.30 and 3.31 indicated a strong relationship (i.e., high R  2  and low R M S E  values) between the number of second-order branches and the length of the first-order branches (Table 4.12). Both regression equations were significant (a = 0.05). Based on I , E R M S E , and residual plots, Equation 3.31 was selected for estimating T O T S E C in 2  creating test populations. This model explained 26 to 89% of the variation in the number of second-order branches. The R M S E ranged from 1.21 to 2.81 in the estimated number of second-order branches. The wide range of I could be attributed to high variation in 2  the number of second-order branches among the sample trees. Equations 3.34 and 3.37 had higher R and lower R M S E values for the sample trees 2  Figure 4.6: First-order branch lengths (estimated and measured) at different heights above ground using Mitchell's 1975 approach for each cardinal direction.  C h a p t e r  4.  R e s u l t s  a n d  D i s c u s s i o n  98  Figure 4.7: First-order branch lengths (estimated and measured) at different heights above ground using quadratic interpolation for each cardinal direction.  Chapter 4. Results and Discussion  99  33 !  °°  0.4  0.8  1.2  1.6  2.0  2.4  2.8  3.2  3.6  Branch length (m)  0 I R  Figure 4.8: First-order branch lengths (estimated and measured) at different heights above ground using linear interpolation for each cardinal direction.  Chapter 4. Results and Discussion  100  than Equations 3.32, 3.33, 3.35, and 3.36. This can be explained by the number of independent variables (i.e., BL\  y  DFT\, and DFT2) they included. The later equations  had only two of the three independent variables. Equation 3.37 had slightly higher R  2  and lower R M S E than Equation 3.34 for seven of the 12 trees. However, when Equation 3.37 was used to create the test populations, it resulted in unreasonable BL2 estimates. Thus, Equation 3.34 was used for estimating BL2 in creating the test population as the difference in R  2  and R M S E values between the two equations were negligible. The JR  2  values for Equation 3.34 ranged from 0.11 to 0.76 and the R M S E ranged from 0.08 to 0.13 m (Table 4.13). The wide range of R values indicates the importance of variables other 2  than BLi, DFT\, and DFT  2  and the influence of the sum squares total. A n extensive  search for other potential models through a review of the literature and personal communications with various scientists in the field failed to find a better model. Nonetheless, Equation 3.34 provided acceptable estimates of second-order branch lengths in creating all the test populations as evidenced by reasonable R M S E values. In this study, the number and lengths of second-order branches on a given first-order branch varied by position in the crowns. This positional variability was also observed by Maguire and Bennett (1996) and Karkkainen (1972). The use of the first-order branch lengths for estimating BL2 improved R  2  and reduced R M S E values (Equations 3.33  versus 3.34 in Table 4.13). The use of first-order branch length as the driving variable for estimating BL  2  also allowed for a proportional relationship between BL\ and BL2  as exhibited by most conifer trees (Leopold 1971, Schultze et al. 1977). The long firstand second-order branches were located close to the base of the tree crowns for most trees. Analyses of the second-order branch lengths indicated that BL2 and T O T S E C were positively related to branch size and distance from the tip of the tree. Similar patterns have been reported for eastern larch (Remphrey and Powell 1987), Douglas-fir (Mitchell 1975), and Norway spruce (Schulze et al. 1977).  Chapter  4.  Results  and  101  Discussion  Table 4.12: Root mean square error (RMSE in m), coefficient of determination (R ) and (I ) and estimated root mean square (ERMSE) values for estimating the number of second-order branches ( T O T S E C ) as a function of branch length prediction (BLi). n is the number of sampled first-order branches. 2  2  Equation number Tree no.  I  3.31 ERMSE  n  R  1 2 3 4  17 13 13 14  0.76 0.65 0.85 0.86  2.81 2.37 2.41 1.22  0.75 0.65 0.84 0.86  2.86 2.37 2.47 1.23  5 6 7 8  17 16 17 22  0.67 0.56 0.81 0.89  2.51 2.64 2.30 1.97  0.67 0.51 0.82 0.89  2.50 2.80 2.28 1.96  9 10 11 12  29 27 18 23  0.73 0.69 0.86 0.28  2.29 2.78 1.31 1.58  0.73 0.70 0.84 0.27  2.28 2.73 1.53 1.59  2  3.30 RMSE  2  Chapter 4. Results and Discussion  102  Table 4.13: Root mean square error (RMSE in m) and coefficient of determination (R ) values for estimating length of second-order branches (BL2) in hybrid spruce crowns, n is the number of sampled second-order branches. 2  Equation number Tree  n  3.32  3.33  3.34  3.35  3.36  3.37  Ft?  RMSE  R  RMSE  R?  RMSE  R  RMSE  R  RMSE  R  RMSE  2  2  2  2  1 2 3 4  30 33 25 34  0.10 0.17 0.57 0.19  0.13 0.12 0.09 0.17  0.19 0.26 0.63 0.70  0.12 0.12 0.08 0.10  0.34 0.27 0.63 0.71  0.12 0.12 0.09 0.10  0.10 0.17 0.55 0.17  0.13 0.13 0.09 0.17  0.19 0.27 0.66 0.67  0.11 0.12 0.08 0.11  0.36 0.28 0.66 0.69  0.12 0.12 0.08 0.11  5 6 7 8  23 39 41 27  0.10 0.25 0.24 0.10  0.09 0.10 0.11 0.19  0.10 0.30 0.05 0.05  0.09 0.10 0.13 0.19  0.11 0.32 0.26 0.17  0.09 0.10 0.11 0.19  0.10 0.23 0.24 0.12  0.09 0.11 0.11 0.19  0.09 0.32 0.18 0.10  0.09 0.10 0.12 0.19  0.10 0.33 0.25 0.18  0.09 0.10 0.11 0.19  9 10 11 12  15 41 19 24  0.40 0.11 0.35 0.40  0.11 0.10 0.17 0.12  0.51 0.21 0.58 0.73  0.10 0.10 0.58 0.73  0.66 0.21 0.63 0.76  0.09 0.10 0.13 0.08  0.39 0.11 0.31 0.40  0.11 0.10 0.17 0.12  0.44 0.22 0.54 0.71  0.11 0.09 0.14 0.08  0.59 0.22 0.58 0.78  0.10 0.10 0.14 0.08  Chapter 4. Results and Discussion  103  Leopold (1971) used the following equation to estimate length of higher-order branches as a function of branch order andfirst-orderbranch length.  BL  = BLi * exp  1360  2  + e2  (4.62)  2  In this study, 060 ranged from -0.93 to -0.36; Leopold reported -0.85 for 060- However, this equation assumes that the lengths of all second-order branches, on a first-order branch, are identical. Therefore, this model was not used in this study.  4.3.3  T h i r d - O r d e r Branch Length and N u m b e r Predictions  Equations 3.38 and 3.39 indicated a reasonably strong relationship between NOTRD and BLi, BL , and DFTi. 2  Based on the RMSE or ERMSE and R or I values (Table 4.14), 2  2  Equation 3.39 was selected for estimating NOTRD in creating test populations. Equations 3.41 and 3.43 had higher R and lower RMSE values than Equations 3.40 2  and 3.42 owing to the larger number of independent variables they included (Table 4.15). Based on higher R and lower RMSE values, Equation 3.41 was selected for estimating 2  BLz in creating test populations, despite a low R value for tree 9. This low R value can 2  2  be attributed to the high sum squares of total. An extensive search for other potential models through a review of the literature and personal communications with various scientists in the field failed to find a better model. Nonetheless, Equation 3.41 provided reasonable estimates of third-order branch lengths in creating all the test populations. The estimated lengths of third-order branches were positively related to distance from the tip of the tree and the tip offirst-orderbranches. This result is consistent with that reported by Remphrey and Davidson (1992), who documented a significant positive relationship between the number of lateral shoots and the length of associated parent shoots. Cannell (1974) also stated that the longer the branches, the higher the number  Chapter  4.  Results  and  104  Discussion  Table 4.14: Root mean square error (RMSE in m) and coefficient of determination (R or I ) and estimated root mean square (ERMSE) values for estimating the number of third-order branches (NOTRD) as a function of second-order length (BLi)is the number of sampled third-order branches.  2  2  n  Equation number Tree no.  n  5 6 7 8  32 23 17 29  0.45 0.61 0.70 0.25  1.28 1.12 0.63 1.28  0.42 0.57 0.70 0.23  1.32 1.17 0.63 1.32  9 10 11 12  46 41 32 36  0.46 0.16 0.86 0.80  3.51 1.61 0.76 1.01  0.38 0.20 0.85 0.78  4.06 1.56 0.77 1.09  R  2  3.38 RMSE  I  2  3.39 ERMSE  of the laterals produced.  4.3.4  F o u r t h - O r d e r Branches  A n average branching ratio (the ratio of first- to second- to third- order branch numbers) of 8.0 (1:9.68:60.8) and length ratio of 1.8 (142.7:52.8:34.4) was obtained. Frequency of branching decreased with branch order (Table 4.16). The average number of second-order branches per first-order branch increased with the age class of the trees. The estimated average number of fourth-order branches per third-order branch was more variable from tree to tree than the estimated average length of the fourth-order branches. There was no distinct trend between age class of the trees and the average length of second-order  Chapter  4.  Results  and  105  Discussion  Table 4.15: Root mean square error (RMSE in m) and coefficient of determination (R ) values for estimating the length of third-order branches ( B L 3 ) as a function of BLi, DFTi, and DFTiis the number of sampled third-order branches. 2  n  Equation number 3.42 RMSE  n  5 6 7 8  31 23 17 29  0.42 0.86 0.54 0.37  0.16 0.09 0.10 0.10  0.42 0.87 0.58 0.47  0.16 0.14 0.09 0.10  0.46 0.83 0.56 0.36  0.15 0.11 0.08 0.10  0.47 0.83 0.57 0.46  0.15 0.11 0.10 0.10  9 10 11 12  46 41 32 36  0.10 0.72 0.44 0.60  0.11 0.14 0.13 0.10  0.10 0.79 0.49 0.89  0.11 0.16 0.13 0.07  0.10 0.89 0.59 0.90  0.11 0.08 0.11 0.06  0.10 0.89 0.59 0.90  0.11 0.08 0.10 0.05  R  2  3.40 RMSE  3.41 RMSE  Tree no.  R  2  R  2  R  2  3.43 RMSE  Chapter 4. Results and Discussion  106  Table 4.16: Branching ratios for the 12 hybrid spruce trees measured in this study.  Tree no.  AVEBL1 (cm)  n  Ave. no 2nd/lst O B  AVEBL2 (cm)  Ave. no. 3rd/2nd O B  AVEBL3 (cm)  Ave. no. 4th/3rd O B  AVEBL4 (cm)  1 2 3 4  81 102 93 153  17 13 13 14  6.6 9.0 8.0 6.4  32.3 49.5 47.3 58.2  4.2 7.6 6.6 4.7  -  -  -  5 6 7 8  164 160 192 128  17 16 17 22  9.3 10.1 10.7 11.2  53.2 48.0 52.1 66.3  5.1 6.4 7.5 6.2  39.5 32.2 29.2 40.3  _  _  -  -  9 10 11 12  217 128 148 146  29 27 18 23  13.7 10.5 10.7 10.0  59.2 60.2 54.6 52.2  8.6 6.6 6.4 5.5  35.9 32.7 30.2 35.0  5.4* 4.2* 3.8* 3.0*  13.4* 16.4* 13.4* 16.4*  2nd/lst O B = average number of second-order branches per a first-order branch, 3rd/2nd OB=average number of third-order branches per a second-order branch, 4th/3rd OB=average number of fourth-order branches per a third-order branch AVEBLx = average length of x-order branches. * indicates estimated value using bifurcation and length ratios, n is the number of sampled first-order branches.  branches and the average length of third-order branches (Table 4.16). Branching and length ratios are known to vary with a variety of exogenous (e.g., stand density, spatial distribution of trees in a stand) and endogenous (i.e., genetics, tree species) factors. For example, Leopold (1971) found a branching ratio of 4.8 (1:4:21:88) and a length ratio of 2.7 (490:187:77:27) for white fir (Abies concolor ((.Gord. & Glend.) Lindl.).  Cannell (1974) found branching ratios of 3.5 and 4.1 for lodgepole pine and  Sitka spruce, respectively. Kempf and Pickett (1981) reported a significant difference in branching ratios between open- and forest-grown trees. In this study, twig length decreased with increasing order of branches as expected for  107  Chapter 4. Results and Discussion  most conifer trees that exhibit excurrent growth. This decrease was also documented by Remphrey and Davidson (1992) and Cochrane and Ford (1978). They found a strong quantitative relationship between the lengths of consecutive branches, with a gradual decrease in twig length with increasing branch order. For example, in green ash (Fraxinus pennsylvanica (Valhl) Fern.), 35% of the fourth- and fifth-order branches were greater than 30 mm compared to 98.1% of the first-order branches (Remphrey and Davidson 1992). Most of the published studies on tree architecture have reported on the size, number, and distribution of first-order branches (Maguire et al. 1994). Little has been discussed or reported on the size, number, and distribution of higher-order branches. Higher order branches have a temporal and a spatial distribution within tree crowns. These temporal and spatial distributions are important for the distribution of foliage within tree crowns.  4.3.5  Description of the Test Populations Created  The created test populations showed high variation in twig leaf area ( T W L A ) . The average varied from 39.57 to 153.30 cm with a standard deviation ranging from 21.13 2  to 105.36 c m (Table 4.17). Average T W L A decreased with an increase of age class, 2  tree height and D B H . The average T W L A for trees younger than 20, between 50 to 60 years, and older than 120 was 114.08, 88.55, and 47.41 cm , respectively. However, the 2  variability of T W L A among trees younger than 60 years and older than 120 years was minimal. Average A P C M decreased with an increase in age class (3.98, 3.06 and 2.61 cm , 2  respectively, for the three age classes).  Single-tree leaf area ( T L A ) estimates ranged  from 4.14 to 11.24 m (model estimates), and 3.01 to 8.52 m (sample estimates) (Table 2  2  4.17). T L A increased with age class (4.44, 8.07, and 9.29 m for the three age classes), 2  D B H , and height of the trees.  Chapter  4.  Results  and  108  Discussion  Table 4.17: Summary of average twig leaf area ( T W L A in cm ) and tree leaf area (TLA in m ) using the created test populations and the systematically collected data (bold). N is the number of twigs on each tree crown, n is the number of sampled twigs 2  2  Age (years)  Tree N or No. n  Average TWLA  Standard deviation  Minimum  Maximum  Average T L A APCM (cm /cm) (m ) 2  2  15 1 2 3 4  422 30 497 33 297 25 537 34  108.13 115.28 83.32 75.48 153.30 151.33 81.60 138.88  32.01 25.26 28.63 45.40 72.99 60.08 42.13 83.52  20.30 20.30 11.22 11.22 1.76 31.18 16.23 16.23  245.54 245.54 229.25 203.21 271.43 271.43 410.46 354.57  3.98 3.01 3.10 2.27 4.00 3.68 4.82 2.86  4.56 3.01 4.14 4.50 4.56 4.45 4.70 4.49  1100 54 922 62 727 58 856 56  76.28 123.55 107.41 97.36 82.04 124.27 78.73 140.17  69.11 87.82 105.36 98.76 87.67 128.51 49.25 92.75  6.97 12.71 12.27 12.27 5.98 6.25 12.00 37.70  435.63 435.63 433.27 358.85 433.14 433.14 305.27 305.27  3.39 2.83 2.69 2.98 2.87 3.32 3.29 3.11  8.39 4.78 9.90 4.64 5.96 6.34 6.74 8.52  2844 90 1791 83 2283 51 1946 60  39.51 50.12 52.76 48.06 39.57 28.41 52.97 51.18  18.01 32.21 25.41 35.59 21.13 51.10 48.04 66.28  1.15 9.14 5.02 10.34 3.20 0.90 1.02 10.85  127.83 127.83 129.23 132.36 144.64 137.64 253.63 186.37  2.38 2.59 2.61 3.27 2.66 1.42 2.79 2.96  11.24 7.03 9.45 6.05 9.03 5.95 10.31 6.95  50-60 5 6 7 8  120+ 9 10 11 12  Note: The standard deviation values for the systematically collected data were calculated using the variance formula derived for a two-stage sampling design (Cochran 1977 p. 277).  109  Chapter 4. Results and Discussion  Branch orders higher than second- and third-order were not common for trees younger than 20 and 60 years, respectively. Trees older than 120 years produced more twigs than the other trees because of their high number of first- and second-order branches (Table 4.17). This is consistent with Cannell (1974) finding a large number of second-order branches on each first-order branch that contributed the highest amount of twig length in older crowns. A Chi-square test comparing T L A values of the sample and model estimates showed no significant differences for trees younger than 60 years at a = 0.05. Differences between these estimates were detected for trees older than 120 years (Figure 4.9). This is due to including estimated leaf area of fourth-order branches in the model estimates that were not in the sample estimates.  Overall, it can be concluded that the created test  populations were reasonable. In this study, only whorl branches were considered to contribute to T L A in the test population. Therefore, these T L A values are not comparable to T L A values obtained in other studies that used both whorl and inter-whorl branches. The use of actual and estimated leaf area values for evaluation of sampling designs was fairly inexpensive compared to a completely measured tree and accurate, as few biologically reasonable data were collected with accuracy and high precision. Noticeable differences in T W L A occurred due to differences in the length of secondand third-order branches. This study demonstrated that differences in branch length, number of higher-order branches, and A P C M contributed to the variation of T L A . T L A is dependent on a multitude of factors, relationships, and environmental factors. Therefore, in estimating single-tree leaf area estimates, not only tree attributes but also stand attributes should be considered. The use of twig length to estimate T L A was also explored by Schulze et al. (1977).  Chapter  4.  Results  and  Discussion  110  Estimate  0  1 2  3 4  5 6  7  8 9 10 11 12 13  • Sample • Model  Tree number  Figure 4.9: E s t i m a t e d tree leaf area (model versus sample estimates) of the 12 sample trees.  Chapter 4. Results and Discussion  111  The dimensions (lengths and numbers) and the locations of twigs determine the structure of the crown system. In this study, the crown structure of hybrid spruce test populations can be regarded as modular, where the population of twigs form structural modules in tree crowns. Kellomaki et al. (1980) assumed that branches form structural modules in Scots pine crowns.  4.4  Evaluation of Sampling Designs  This section presents the results of evaluating the sampling designs (objective two of this study). The measures used in assessing the selected sampling designs are presented in Sections 4.4.1 and 4.4.2. The distribution of the estimates, cost efficiency and relative ease, and overall ranking of the sampling designs are presented in Sections 4.4.4, 4.4.5, and 4.4.6, respectively. 4.4.1  Bias  A l l sampling designs showed nearly zero bias as expected, with the exception of the linear systematic sample selection procedure (Table 4.18). This selection procedure resulted in biased estimates when the test populations were not integral multiples of sample sizes and sampling intervals as supported by Murthy (1967). This result is also in agreement with Payandeh (1970) who stated that a linear systematic selection procedure results in unbiased estimates if the elements in the population are distributed randomly. Since twig leaf area was not randomly distributed in tree crowns, the biased estimates are not surprising. Minor deviations from zero bias were also observed due to the use of pseudo random numbers and low sampling intensity from highly variable twig populations (e.g., 48 sample twigs (1.7%) were taken from 2844 twigs for tree 9).  Chapter 4. Results and Discussion  112  Table 4.18: Bias (m ) by sample size, sampling design, and sample tree. Bias percents are given in bold. Tree leaf areas (m ) of test populations are given in brackets. 2  2  Sample size  Sampling design  SRS  Tree  Number  1  2  3  4  5  6  7  8  9  10  11  12  (4.56)  (4.14)  (4.56)  (4.70)  (8.39)  (9.90)  (5.96)  (6.74)  (11.24)  (9.45)  (9.03)  (10.31)  12  -0.018  0.010  0.010  0.021  -0.001  -0.058  0.010  -0.028  -0.117  -0.063  -0.004  -0.154  -0.40  0.24  0.22  0.47  0.01  -0.59  0.17  -0.42  -1.05  -0.71  -0.04  -1.52  36  -0.022  0.009  0.009  0.014  0.013  -0.109  0.005  -0.026  -0.124  -0.069  -0.002  -0.092  -0.48  0.22  0.20  0.30  0.15  -1.11  0.08  -0.39  -1.12  -0.78  -0.02  -0.90  -0.028  0.007  0.011  0.026  0.014  -0.123  -0.001  -0.024  -0.122  -0.075  -0.003  -0.071  -0.62  0.24  0.24  0.55  -0.17  -1.26  -0.02  -0.36  -1.10  -0.85  -0.03  -0.69  0.008  0.002  -0.005  -0.009  0.012  -0.089  -0.043  0.008  0.025  -0.044  -0.075  -0.003  -0.39  48 Stratified random  12 36 48  Ellipsoidal  a  Two-stage Systematic with fractional interval selection  a  Two-stage Unequal Probability  -0.59  0.14  -0.59  -0.73  0.12  0.10  -0.05  -0.84  -0.03  -0.006  -0.007  0.019  -0.092  -0.034  0.007  0.033  -0.025  -0.074  0.007  -0.23  -1.11  0.23  -1.11  -0.57  0.10  0.29  -0.29  -0.83  0.07  0.006  0.002  -0.009  -0.006  0.018  -0.091  -0.031  0.007  0.034  -0.020  -0.074  0.005  -0.61  -0.19  -0.21  -0.93  0.21  -0.93  -0.52  0.10  0.30  -0.23  -0.83  0.05  0.017  0.002  0.020  -0.007  0.002  0.040  0.052  -0.007  0.011  -0.004  -0.002  0.101  0.37  0.05  0.41  -0.15  -0.02  -0.41  -0.88  -0.08  0.09  -0.05  -0.02  36  0.003  1.06  0.015  0.019  -0.005  0.003  0.045  0.061  -0.005  0.013  -0.003  -0.001  0.113  0.33  0.07  0.40  -0.11  -0.03  -0.46  -1.03  -0.07  0.11  -0.03  -0.01  0.86  0.015  0.003  0.019  -0.004  0.005  0.046  0.064  -0.004  0.016  -0.001  -0.003  0.115  0.33  0.07  0.40  -0.09  -0.06  0.47  -1.08  -0.06  0.14  -0.01  -0.03  0.03  I  0.116  0.117  0.321  -0.318  0.304  -0.687  -0.860  0.576  -0.563  -0.196  0.110  -0.168  2.54  2.83  7.04  -6.77  -6.94  -14.43  8.55  0.367  0.071  -5.01  -2.07  1.22  -1.63  II  0.450  -0.434  -0.895  -0.355  -1.988  -0.249  -0.445  -0.280  -0.029  -0.189  8.05  1.71  9.87  -9.23  -10.67  -3.59  -33.36  -3.85  -3.96  -2.96  -0.32  0.110  0.258  0.375  -0.249  -1.83  III  0.404  -0.739  -0.905  0.670  -0.623  0.124  0.028  -0.060  2.41  6.23  8.22  -5.30  4.82  -7.46  -15.18  9.94  1.31  0.31  0.355  0.232  0.536  -0.58  IV  -0.366  -0.760  -0.469  -1.951  -0.280  -0.507  -0.076  0.031  0.152  7.79  5.60  11.75  -7.79  -9.06  -4.74  -32.73  -4.15  -4.15  -0.08  0.34  1.47  I  -0.018  -0.159  -0.179  -0.056  0.176  -0.054  0.029  -0.046  -0.195  0.023  -0.170  -0.39  -3.99  -3.98  -1.20  1.99  -0.56  2.34  0.40  -0.46  -2.27  0.23  -1.88  II  -0.018  -0.087  0.029  -0.002  -0.093  -0.019  0.019  0.040  -0.046  -0.246  0.002  -0.162  -0.30  -0.69  0.61  0.04  -1.09  -0.20  0.32  0.56  -0.42  -2.89  0.02  -1.79  III  -0.037  -0.028  -0.118  0.216  0.170  -0.014  0.111  0.046  -0.048  0.004  0.027  -0.102  -0.82  -0.68  4.00  4.39  1.93  -0.14  1.83  0.64  -0.45  0.05  0.20  -0.017  -1.13  IV  0.048  0.087  0.211  -0.056  -0.084  0.069  0.019  -0.046  -0.068  -0.003  0.008  -0.37  1.16  1.79  4.29  0.65  -0.86  1.14  0.27  -0.43  -0.78  -0.78  0.09  -0.014  -0.018  -0.013  0.078  -0.108  -0.037  -0.066  0.004  0.004  0.052  0.110  3.62  0.143  -5.54  12  0.023 0.50  -0.34  -0.38  -0.27  0.89  -1.10  -0.62  -0.99  0.04  0.04  0.57  1.06  36  0.013  -0.004  -0.014  -0.009  -0.038  -0.094  -0.019  -0.033  0.025  0.003  0.035  0.090  0.28  -0.10  -0.29  -0.19  -0.44  -0.94  -0.32  -0.49  0.22  0.03  0.39  0.93  0.004  0.004  -0.015  -0.007  -0.042  0.013  -0.004  -0.025  0.024  0.003  0.032  0.120  0.00  0.10  -0.32  -0.15  -0.49  0.13  -0.07  -0.37  0.21  0.03  0.35  1.15  48  a  -0.14  0.002 -0.13  12  48 Two-stage Systematic with a linear selection  -0.11  0.007 -0.48  I, II, III, and IV refer to sampling intensity of 10% and 20%, 10% and 40%, 20% and 20%, 20% and 40% of first-order branches and twigs, respectively.  Chapter 4. Results and Discussion  113  There were no noticeable differences among the average tree leaf area (TLA) estimates obtained by the different sampling intensities and sampling designs. Percent bias ranged from: -1.52 to 0.55% for simple random sampling, -1.11 to 0.3% for stratified random sampling, -1.08 to 1.06% for ellipsoidal sampling, -33.36 to 11.75% for two-stage systematic sampling with a linear selection procedure, -3.99 to 4.39 % for two-stage systematic sampling with a fractional interval selection procedure, and -1.10 to 1.15 % for two-stage unequal probability sampling. The biases of the two systematic selection procedures (i.e., linear systematic sampling and fractional interval systematic sampling) are presented for comparison purposes. However, only the results of the fractional interval systematic sampling procedure are discussed, as the estimates from the linear systematic sampling procedure were biased.  4.4.2  Root Mean Square Error and Sampling Variance  Since all biases were close to zero, the mean square error values and the sampling variance values were nearly the same. Thus, only the R M S E values are given and discussed in this section (Table 4.19). The R M S E values decreased substantially with increasing sample sizes for all sampling designs. When the sample sizes were increased from 12 to 36 twigs and from 36 to 48 twigs, the R M S E values were reduced by more than 1.73 and 1.14 fold, respectively (Table 4.19). Since sampling intensities (%, instead of fixed sample sizes) were used for two-stage systematic sampling, the number of sampled twigs for each tree was variable. Linear extrapolation between comparable sample sizes was used to calculate approximate R M S E values for a sample size of 36 twigs. This was used to compare the efficiency of two-stage systematic sampling with other sampling designs. Stratified random sampling resulted in the lowest R M S E values for all three sample sizes, followed by ellipsoidal, two-stage systematic, simple random, and two-stage unequal  Chapter 4.  114  Results and Discussion  Table 4.19: Root mean square error (m ) by sample size, sampling design, and sample tree. Tree leaf areas (m ) of test populations are given in brackets. The average numbers of sampled first-order branches and twigs for two-stage systematic sampling, respectively, are given in bold. 2  2  Sampling design  Sample size  Tree  Number  1 (4.56)  2 (4.14)  3 (4.56)  4 (4.70)  5 (8.39)  6 (9.90)  7 (5.96)  8 (6.74)  9 (11.24)  10 (9.45)  11 (9.03)  12 (10.31)  SRS  12 36 48  0.405 0.210 0.197  0.442 0.226 0.195  0.636 0.351 0.297  0.646 0.365 0.311  2.219 1.267 1.091  2.760 1.585 1.365  1.824 1.036 0.889  1.213 0.688 0.593  1.474 0.848 0.733  1.241 0.777 0.672  1.415 0.789 0.668  2.690 1.543 1.332  Stratified random  12 36 48  0.351 0.197 0.167  0.298 0.190 0.164  0.265 0.145 0.122  0.625 0.342 0.298  1.505 0.797 0.678  2.394 1.408 1.249  1.715 1.034 0.865  0.891 0.518 0.437  1.391 0.743 0.603  1.018 0.640 0.592  1.362 0.742 0.616  2.420 1.367 1.123  Ellipsoidal  12 36 48  0.345 0.197 0.164  0.330 0.187 0.158  0.567 0.313 0.266  0.618 0.348 0.298  2.118 1.209 1.045  2.638 1.503 1.293  1.742 1.021 0.890  1.172 0.667 0.574  1.314 0.813 0.705  1.202 0.689 0.595  1.389 0.788 0.689  2.681 1.538 1.327  Two-stage Unequal Probability  12 36 48  1.934 0.972 0.830  1.067 0.655 0.886  1.003 0.574 0.503  0.819 0.486 0.457  5.951 3.797 3.240  3.412 2.027 1.652  2.360 1.616 1.441  1.074 0.699 0.645  4.481 3.024 2.593  1.049 0.942 0.809  3.003 1.784 1.630  7.288 4.451 3.909  Two-stage  I  0.473  0.569  0.847  0.844  1.401  2.293  1.702  1.128  0.492  0.674  0.610  1.373  6/10  5/10  5/6  6/11  6/25  6/19  6/14  7/18  8/67  0.266  0.349  0.561  0.473  0.986  1.212  0.991  0.874  0.361  a  Systematic with fractional  II  6/22 III  0.297 12/19  interval  IV  0.164 12/40  TSYS  36  6  0.176 10/47  0.221  5/14  6/27  6/55  6/45  6/36  0.538  0.562  0.812  1.384  1.045  10/14  12/19  12/50  12/39  12/29  0.367 10/29  0.332  0.325  0.477  12/45  12/111  0.349  1.187  0.657 12/92  1.454  0.687 12/72  0.990  7/23  0.722 14/24  8/153  0.402 13/90  7/50  7/40  0.415  0.964  7/114  0.281  0.399  0.320  16/121  26/73  14/100  7/96  0.857 14/81  0.601  0.186  0.271  0.248  0.621  14/34  16/275  26/162  14/228  14/192  0.787  0.714  0.767  1.449  0.583  I, II, III, and IV refer to sampling intensity of 10% and 20%, 10% and 40%, 20% and 20%, 20% and 40% of first-order branches and twigs, respectively. Approximated RMSE values for two-stage systematic sampling for n=36 twigs.  a  b  0.184  5/23  0.304 10/22  13/40  Chapter  4.  Results  and  115  Discussion  probability sampling (Table 4.19). The poor showing of two-stage unequal probability sampling was due to the lack of a strong linear relationship between (1) the lengths of first-order branches and twig leaf area (average Pearson correlation coefficient of 0.10) and (2) twig lengths and twig leaf area (average Pearson correlation coefficient of 0.30). These relationships also varied by vertical and horizontal positions in tree crowns as evidenced by wide variation of A P C M in Section 4.2.1. On average, stratified random sampling was 1.28 times more efficient than SRS (Table 4.3). Thus, SRS will require 1.64 times (1.28 ) more samples than stratified random 2  sampling to achieve the same level of precision. The high relative efficiency of stratified random sampling over SRS is a reflection of the reduction in twig leaf area variability by using the three strata (Table 4.21). The highest variability of twig leaf area was observed in the third stratum (at the tip of the tree crown) for most sample trees. Trees between 50 to 60 years of age showed the highest twig leaf area variability, followed by trees older than 120 years, and then by younger trees (less than 16 years). Ellipsoidal sampling consistently had the second lowest R M S E value for the three sample sizes and for most sample trees (Table 4.19). There were no remarkable differences in R M S E values between stratified random and ellipsoidal sampling designs on average. Ellipsoidal sampling was 1.11 times more efficient than SRS (Table 4.20). This is due to the wide variation of twig leaf area among the three ellipses (Table 4.21). The highest variability was observed in ellipse I (at the outer ellipse of the tree crown) for most sample trees. Trees between 50 to 60 years of age showed the highest twig leaf area variability, followed by trees older than 120 years, and then by trees younger than 16 years. Since the R M S E values of stratified sampling (i.e., stratified random and ellipsoidal) were smaller than the R M S E values of SRS, stratifying the tree crowns did result in reduced variance of the estimated total. The stratified designs were 1.17 times more efficient than SRS on average.  The stratified sampling designs had noticeably higher  116  Chapter 4. Results and Discussion  Table 4.20: Efficiencies' of stratified random, ellipsoidal, two-stage systematic, and two-stage unequal probability sampling design relative to SRS. A sample size of 36 twigs was used for comparison. 1  a  Tree No.  Stratified random  Ellipsoidal  Two-stage unequal probability  Two-stage systematic  1 2 3 4 5 6 7 8 9 10 11 12  1.06 1.19 2.42 1.07 1.59 1.13 1.00 1.33 1.14 1.21 1.06 1.13  1.06 1.21 1.12 1.05 1.05 1.05 1.01 1.03 1.04 1.13 1.00 1.00  0.22 0.34 0.61 0.75 0.33 0.78 0.64 0.98 0.28 0.82 0.44 0.35  1.14 1.02 1.06 1.05 1.07 1.09 1.05 1.18 1.08 1.09 1.03 1.06  Average  1.28  1.06  0.55  1.08  Efficiencies were calculated as the ratio of the R M S E of SRS to the R M S E of the sampling design under consideration. Higher number denotes higher efficiency.  efficiency than the non-stratified sampling designs (i.e., SRS, two-stage systematic, and two-stage unequal probability) (Table 4.20). The first and second sampling intensity options of the two-stage systematic sampling (i.e., 10% first-order branches) resulted in less precise estimates than SRS for trees younger than 15 years of age. This could be ascribed to 1) large variations of twig leaf area among the selected first-order branches and 2) small sample sizes of twigs. However, the fourth sampling intensity option (20% first-order branches and 40% twigs) had lower R M S E values for most sample trees than SRS owing to its large sample sizes (Table 4.19).  Chapter 4. Results and Discussion  117  Table 4.21: Standard deviations of twig leaf area (cm ) by stratum. 2  Stratum  Ellipse  Tree No.  I Base  II Middle  III Tip  I Inner  II Middle  III Outer  All combined  1 2 3 4  31.11 20.29 22.93 97.33  26.85 23.96 38.14 47.25  33.12 23.12 43.74 29.56  44.26 25.31 70.26 41.32  22.26 17.17 64.19 50.58  19.74 25.87 63.37 26.12  32.01 28.63 72.99 42.13  5 6 7 8  72.74 59.26 82.16 33.68  66.45 89.41 119.41 45.32  69.44 118.21 31.82 49.94  42.25 128.14 91.71 32.35  69.96 94.84 86.85 52.46  72.17 65.01 85.35 53.76  69.11 105.36 87.67 49.25  9 10 11 12  14.11 25.99 14.23 23.18  15.23 9.31 23.11 39.21  23.41 11.04 31.46 73.90  14.86 22.48 22.43 51.98  18.51 27.71 20.28 49.54  21.54 24.30 20.61 40.85  18.00 25.41 21.19 48.03  Chapter 4. Results and Discussion  118  Also, when the sampling intensity of first-order branches was increased from 10% to 20%, the R M S E values decreased by more than 40% for most sample trees. The R M S E values decreased by 35% on average when the sampling intensity for the second-stage (twigs) increased from 20 to 40% (Table 4.19). This indicates that the variation in secondary sampling units is less than that of the primary sampling units and the variance of tree leaf area estimate was dominated by the variance of the first-stage sampling units (i.e., first-order branches).  This can be ascribed to the wider variability of twig leaf area  among the three strata (by height) than among the three ellipses (by relative lengths) for most sample trees (Table 4.20). The R M S E values of two-stage unequal probability sampling were consistently higher than the R M S E values of SRS for the three sample sizes and for all sample trees. This can be ascribed to weak linear relationships between twig leaf area and first-order branch lengths, and between twig leaf area and twig lengths, as the selection probabilities were based on the lengths of first-order branches and twigs. This is reflected by (1) lack of noticeable linear trends in A P C M along the horizontal lengths, and (2) the finding of the highest A P C M values between the relative heights of 0.30 to 0.50 for most sample trees. The efficiency of two-stage unequal probability sampling could be improved by increasing the sample size and by selecting an auxiliary variable that has a strong positive linear relationship with twig leaf area. The R M S E values of two-stage systematic sampling were consistently lower than the R M S E values for two-stage unequal probability sampling for most sample trees (Figure 4.10 and Table 4.19). In two-stage unequal probability sampling, the probability selection favored longer first-order branches and twigs which are usually located at lower heights on the crown, whereas in two-stage systematic sampling every first-order branch and twig had an equal chance of being included in a sample. This ensured complete coverage of the crown.  Chapter  4.  Results  and  119  Discussion  Figure 4.10: Root mean square error ( m ) and tree number for simple random sampling (SRS), stratified random (STRS), ellipsoidal (ELLIP), two-stage systematic (TSYS), and two-stage unequal probability sampling (TSUEP) designs. A sample size of 36 twigs was used for comparison. 2  Chapter  4.  Results  and  120  Discussion  Table 4.22: Summary of average R M S E (m ) values by age classes and sampling designs. A sample size of 36 twigs was used for comparison. 2  4.4.3  Age class (years)  Simple random  Stratified random  Ellipsoidal  Two-stage unequal probability  Two-stage systematic  < 20 50 to 60 > 120  0.29 1.14 0.99  0.22 0.94 0.87  0.26 1.10 0.96  0.67 2.04 2.55  0.27 1.05 0.93  ALL  0.81  0.68  0.77  1.75  0.75  C h a n g e in Variance w i t h Tree Characteristics  The patterns between the R M S E values and tree size attributes (e.g., D B H , age, and tree height) for the five sampling designs tested were not linear. Trees between 50 to 60 years of age showed the highest R M S E values followed by trees older than 120 years, and then by younger trees (less than 15 years) for all sampling designs except for two-stage unequal probability sampling (Figure 4.10 and Table 4.22). This indicates that fewer samples are required to achieve a given level of precision on younger trees than on middle aged and older trees. Stratified random sampling resulted in the lowest R M S E values for all sample trees, followed by ellipsoidal, two-stage systematic, simple random, and two-stage unequal probability sampling. The R M S E differences among these sampling designs were more pronounced on trees older than 50 years of age. In two-stage unequal probability sampling, the R M S E values ranged from 0.46 to 7.29 m (Figure 4.10 and Table 4.19). 2  Chapter  4.4.4  4.  Results  and  Discussion  121  D i s t r i b u t i o n of the Estimates  The range of the estimated totals for the 1000 samples decreased as the sampling intensity increased for all sampling designs, as expected. The narrowest and widest ranges were obtained by the stratified random and by the two-stage unequal probability sampling design, respectively (Table 4.23). The skewness coefficient of the estimates ranged from 0.439 to 0.954 for SRS, -0.677 to 1.029 for stratified random, -0.237 to 0.661 for ellipsoidal, -1.098 to 1.085 for two-stage systematic, and -0.255 to 13.025 for two-stage unequal probability sampling. The kurtosis coefficients ranged from -0.790 to 0.975 for SRS, 0.907 to 0.738 for stratified random, -0.899 to 1.282 for ellipsoidal, -1.257 to 6.926 for two-stage systematic, and -0.558 to 205.925 for two-stage unequal probability sampling (Table 4.24). Two-stage unequal probability sampling had the widest range and the highest skewness and kurtosis coefficients (Table 4.24). This can be ascribed to the wide range of the Monte Carlo estimates obtained using this design (Table 4.23). There were no definite trends between the skewness coefficients and sample sizes or tree size variables. The average skewness and kurtosis coefficients did not show any trend with increasing or decreasing age class. The distribution of the estimates from SRS, stratified random, ellipsoidal, and twostage unequal probability sampling designs were reasonably approximated by the normal distribution. However, the estimates of most of the simulations showed departure from normality when the Shapiro-Wilk (1965) and Lilliefors (1973) tests were used (a = 0.05) (Table 4.25). This is in agreement with Conover's (1980 p. 367) assertion that almost all goodness-of-fit tests would result in rejection of normality when the number of observations is large since real data distributions are rarely the same as theoretical distributions. Neter et al. (1990) stated that the lack of normality is not crucial when the departure  Chapter  4.  Results  and  122  Discussion  Table 4.23: Minimum/maximum for the 1000 estimates of total leaf area using simple random, stratified random, ellipsoidal, two-stage unequal probability and two-stage systematic sampling. Tree leaf areas (m ) (TLA) of test populations are given in brackets. 2  Tree number (TLA)  Sample size  SRS  Stratified random  Ellipsoidal  Two-stage unequal probability  1 (4.56)  12 36 48  3.129/5.631 3.968/5.093 4.161/4.979  3.340/5.663 3.968/5.050 4.126/4.981  3.519/5.475 4.102/4.993 4.138/4.904  2.812/36.498 3.148/15.212 3.256/12.248  2 (4.14)  12 36 48  2.726/5.381 3.543/4.703 3.691/4.559  3.181/5.129 3.631/4.659 3.718/4.528  3.356/4.922 3.779/4.493 3.823/4.439  3 (4.56)  12 36 48  3.228/5.739 4.091/4.954 4.152/4.945  3.717/5.170 3.970/4.913 4.101/4.816  4 (4.70)  12 36 48  3.037/6.376 4.003/5.867 4.108/5.552  5 (8.39)  12 36 48  6 (9.90)  Intensity  Average Sample size'*  Two-stage systematic sampling  I II III IV  6/10 6/22 12/19 12/40  2.864/5.438 3.796/5.035 3.394/5.309 4.081/4.988  2.036/13.161 2.764/8.608 3.028/7.562  I II III IV  5/10 5/23 10/22 10/47  2.336/4.966 2.461/4.804 3.536/4.552 3.800/4.572  3.464/5.674 4.268/5.316 4.460/5.163  2.424/8.427 3.357/7.195 3.393/7.068  I II III IV  5/6 5/14 10/14 10/29  2.358/6.311 3.257/6.325 2.781/5.478 3.632/5.771  3.435/7.765 3.972/5.808 4.033/5.481  3.748/7.249 4.011/5.902 4.060/5.528  2.417/7.266 3.558/5.960 3.741/5.950  I II III IV  6/11 6/27 12/19 12/45  3.777/7.050 4.056/5.506 3.855/9.537 4.099/6.682  4.264/13.299 6.406/10.492 6.669/10.137  3.796/13.762 5.948/10.455 6.417/10.026  4.453/15.025 6.414/11.241 6.478/10.860  0.642/30.413 1.358/21.366 1.717/20.012  I II III IV  6/25 6/55 12/50 12/111  3.632/14.628 3.921/14.176 5.739/12.393 7.549/12.809  12 36 48  4.380/18.450 6.193/13.747 7.016/12.686  4.370/19.881 6.256/15.224 7.001/14.517  5.091/22.577 6.529/14.872 6.903/13.859  4.731/23.672 6.177/17.637 6.220/16.172  I II III IV  6/19 6/45 12/39 12/92  4.779/14.566 6.603/13.325 6.775/13.058 7.549/12.809  7 (5.96)  12 36 48  2.895/13.088 4.255/8.438 4.629/8.158  2.707/10.838 3.629/8.935 3.906/8.262  2.561/11.966 3.434/10.386 4.017/9.050  1.752/14.403 2.532/10.834 2.955/12.856  I II III IV  6/14 6/36 12/29 12/72  2.789/10.479 3.344/10.218 3.049/10.982 3.551/9.493  8 (6.74)  12 36 48  4.200/9.969 5.346/8.680 5.595/8.250  4.286/10.024 5.239/8.137 5.423/7.992  4.450/9.766 5.307/8.549 5.638/7.957  3.329/14.583 4.866/14.558 4.952/12.464  I II III IV  7/18 7/23 14/24 14/34  3.636/13.675 4.466/12.627 3.998/12.418 3.880/10.135  9 (11.24)  12 36 48  5.990/14.926 9.429/12.921 9.336/12.806  7.241/15.777 8.676/13.695 9.302/12.979  7.532/16.133 8.613/13.530 9.270/13.090  3.467/43.737 4.939/27.705 6.049/24.433  I II III IV  8/67 8/153 16/121 16/275  9.992/11.667 10.298/11.571 10.012/11.499 10.299/11.071  10 (9.45)  12 36 48  6.269/11.798 7.647/10.081 7.934/9.741  5.745/11.381 6.961/10.393 7.352/10.287  6.219/11.037 7.333/9.949 7.398/9.953  4.211/21.412 5.620/19.644 6.142/18.531  I II III IV  13/40 13/90 26/73 26/162  6.351/11.964 6.123/10.543 7.954/10.470 8.142/9.419  11 (9.03)  12 36 48  5.944/13.049 7.323/11.288 7.719/11.033  5.868/14.211 7.054/11.738 7.319/11.051  5.687/13.738 6.866/11.348 7.074/11.076  2.743/23.916 5.154/15.360 5.518/14.542  I II III IV  7/50 7/114 14/100 14/228  6.077/14.028 5.347/12.794 7.945/12.321 7.847/11.653  12 (10.31)  12 36 48  4.049/18.538 6.644/14.203 7.410/13.326  4.857/18.295 6.573/14.835 7.396/14.241  4.548/19.665 6.384/14.605 6.526/13.461  0.894/43.810 2.875/28.901 2.891/25.716  I II III IV  7/40 7/96 14/81 14/192  7.585/10.859 7.408/10.842 8.074/10.151 7.714/10.267  a  I, II, III and IV refer to sampling intensity of 10% and 20%, 10% and 40%, 20% and 20%, 20% and 40% of first- and second-order branches, respectively. Average number of first-order branches/twigs for two-stage systematic sampling. a  Chapter  4.  Results  and  123  Discussion  Table 4.24: Average skewness/kurtosis values for the 1000 estimates of total leaf area using simple random, stratified random, ellipsoidal, two-stage unequal probability, and two-stage systematic sampling. Tree leaf areas (m ) (TLA) of test populations are given in brackets. 2  Tree Number (TLA)  Sample size  SRS  Stratified random  Ellipsoidal  Two-stage unequal probability  1 (4.56)  12 36 48  -0.439/0.210 0.015/-0.544 0.109/-0.510  -0.032/0.182 -0.169/-0.354 -0.063/-0.673  -0.036/-0.285 0.063/-0.899 -0.063/-0.687  13.025/205.925 7.234/70.533 6.166/51.577  2 (4.14)  12 36 48  -0.155/-0.054 -0.158/-0.163 0.092/-0.352  -0.073/0.686 -0.231/-0.313 -0.275/-0.555  0.225/-0.163 -0.047/-0.424 0.001/-0.426  3 (4.56)  12 36 48  -0.120/0.041 -0.194/-0.387 -0.224/-0.038  -0.331/-0.175 -0.577/0.738 -0.677/0.482  4 (4.70)  12 36 48  0.096/-0.049 0.501/0.232 0.422/-0.210  5 (8.39)  12 36 48  6 (9.90)  Intensity  Average Sample size  Two-stage systematic sampling  II III IV  6/10 6/22 12/19 12/40  -10.924/1.225 -0.477/0.017 -0.921/0.633 -0.163/-0.646  2.448/16.505 1.635/7.702 1.420/4.727  I II III IV  5/10 5/23 10/82 10/47  -0.791/-0.123 -1.033/0.015 -0.286/-0.576 0.010/-0.490  -0.050/0.397 0.366/-0.525 0.349/-0.489  0.295/0.160 0.186/-0.056 0.113/0.301  I II III IV  5/6 5/14 10/14 10/29  -0.164/-0.484 -0.194/-1.135 -0.727/0.204 -0.680/-0.508  1.029/0.725 0.532/-0.229 0.311/-0.610  1.085/1.282 0.655/0.351 0.407/-0.438  -0.255/-0.033 0.416/-0.463 0.483/-0.558  I II III IV  6/11 6/27 12/19 12/45  1.328/1.137 0.079/0.956 2.467/6.926 1.657/2.348  -0.029/-0.221 0.138/-0.457 0.033/-0.496  0.037/-0.155 -0.377/-0.365 -0.447/-0.291  0.254/0.078 0.198/-0.068 0.106/0.028  0.763/0.132 0.502/0.002 0.373/-0.085  I III III IV  6/25 6/55 12/50 12/111  0.177/-0.638 0.211/-0.682 0.726/-0.371 0.951/0.211  12 36 48  0.455/-0.245 -0.176/-0.323 -0.121/-0.414  0.642/0.580 0.648/0.541 0.633/0.268  0.661/0.910 0.258/0.029 0.168/-0.167  1.234/1.338 0.738/0.362 0.577/0.147  I II III IV  6/19 6/45 12/39 12/92  0.268/0.426 0.315/-0.631 0.303/-0.261 0.560/-0.30  7 (5.96)  12 36 48  0.848/0.975 0.871/0.699 0.954/0.843  0.391/-0.693 0.232/-0.401 0.322/-0.437  0.692/0.176 0.573/0.490 0.635/0.149  0.850/0.223 0.618/-0.303 0.694/0.570  I II III IV  6/14 6/36 12/29 12/72  0.339/-0.920 0.608/-0.509 0.710/-0.490 0.394/-1.01  8 (6.74)  12 36 48  0.349/0.201 0.438/-0.266 0.361/-0.362  0.126/0.003 -0.213/-0.296 -0.196/-0.248  0.391/0.227 0.321/0.101 0.304/-0.273  0.601/2.583 2.643/16.838 2.586/20.774  I II III IV  7/18 7/23 14/24 14/34  1.147/0.975 1.039/0.275 0.691/0.560 0.452/-0.278  9 (11.24)  12 36 48  0.084/-0.113 0.108/-0.790 0.103/-0.697  0.055/-0.137 -0.054/0.104 -0.004/-0.009  0.272/0.357 -0.144/0.214 -0.228/-0.088  2.379/11.003 1.695/5.108 1.500/3.655  I II III IV  8/67 8/153 16/121 16/275  0.249/-0.076 0.563/1.190 0.109/-0.335 -0.564/-0.284  10 (9.45)  12 36 48  -0.021/0.035 -0.047/-0.281 -0.099/-0.534  -0.259/-0.386 -0.327/-0.771 -0.287/-0.907  -0.085/-0.214 -0.237/-0.177 -0.228/0.036  1.771/6.820 1.234/5.241 0.876/4.632  I II III IV  13/40 13/90 26/73 26/162  0.314/-0.965 0.123/-1.257 0.946/0.545 0.317/-0.601  11 (9.03)  12 36 48  0.404/-0.182 0.306/-0.350 0.478/-0.236  0.597/0.189 0.471/0.313 0.422/0.312  0.404/0.051 -0.079/-0.348 -0.039/-0.088  0.897/1.286 0.270/-0.374 0.270/-0.247  I II III IV  7/50 7/114 14/100 14/228  -0.322/-0.658 -0.731/0.389 -0.238/-0.656 -1.098/1.099  12 (10.31)  12 36 48  0.206/-0.350 0.084/-0.644 0.150/-0.616  0.364/-0.150 0.058/-0.283 0.164/0.138  0.689/0.366 0.130/-0.160 -0.154/-0.368  1.455/2.113 0.947/0.950 0.641/0.082  I II III IV  7/40 7/96 14/81 14/192  0.423/-0.099 -0.002/-1.122 0.031/-0.835 0.106/-0.818  a  °I, II, III and I V refer to sampling intensity of 10% and 20%, 10% and 40%, 20% and 20%, 20% and 40% of first- and second-order branches, respectively. Average number of first-order branches/twigs for two-stage systematic sampling.  Chapter 4. Results and Discussion  124  from normality is not extreme. This is the case for the estimates from SRS, stratified random, ellipsoidal, and two-stage unequal probability sampling designs (Figures 4.11 to 4.14). The estimates approach a normal distribution with an increase in sample sizes for the four random sampling designs (i.e., SRS, stratified random, ellipsoidal, and two-stage unequal probability sampling) (Figures 4.11. to 4.14). This is consistent with the central limit theorem that states that the distribution of an estimator approaches to the normal distribution with an increase of sample size. The convergence to normality would require a large number of sample twigs due to the high variation of twig leaf area values and the non-normal distribution of twig leaf area values in tree crowns. The distribution of the estimates of two-stage systematic sampling were non-normal (Figure 4.14). This can be ascribed to the lack of complete randomness of samples (i.e., lack of independence of sample observations), and the low sampling intensities taken from highly variable twig populations. This finding agrees with Wall (1986) who stated that the central limit theorem does not hold when the variables of interest are not independently distributed and vary widely.  Chapter 4. Results and Discussion  125  Table 4.25: Summary of the Shapiro-Wilk (1965) W test statistics of normality and the corresponding probability levels.  a  Tree no.  Sample size  Simple random sampling  Stratified random sampling  Ellipsoidal sampling  Two-stage systematic sampling  Sample size  Two-stage unequal probability sampling  1  12 36 48  0.983/0.0360 0.976/0.0001 0.978/0.0001  0.987/0.628 0.978/0.0001 0.972/0.0001  0.981/0.004 0.957/0.0001 0.971/0.0001  0.344/0.0001 0.501/0.0001 0.540/0.0001  6/10 6/22 12/19 12/40  0.716/0.0001 0.796/0.0001 0.829/0.0001 0.840/0.0001  2  12 36 48  0.970/0.0001 0.973/0.0001 0.971/0.0001  0.981/0.0030 0.975/0.0001 0.966/0.0001  0.978/0.0001 0.979/0.005 0.982/0.027  0.868/0.0001 0.909/0.0001 0.917/0.0001  5/10 5/23 10/22 10/47  0.867/0.0001 0.874/0.0001 0.879/0.0001 0.881/0.0001  3  12 36 48  0.940/0.0001 0.930/0.0001 0.927/0001  0.974/0.0001 0.969/0.0001 0.958/0.0001  0.977/0.0001 0.961/0.0001 0.965/0.0001  0.979/0.0001 0.983/0.02 0.986/0.50  5/6 5/14 10/14 10/29  0.885/0.0001 0.904/0.0001 0.904/0.0001 0.907/0.0001  4  12 36 48  0.979/0.0001 0.965/0.0001 0.965/0.0001  0.907/0.0001 0.955/0.0001 0.963/0.0001  0.911/0.0001 0.952/0.0001 0.958/0.0001  0.971/0.0001 0.957/0.0001 0.947/0.0001  6/11 6/27 12/19 12/45  0.909/0.0001 0.914/0.0001 0.914/0.0001 0.915/0.0001  5  12 36 48  0.975/0.0001 0.978/0.0001 0.974/0.0001  0.987/0.585 0.969/0.0001 0.964/0.0001  0.984/0.137 0.982/0.022 0.987/0.482  0.920/0.0001 0.964/0.0001 0.974/0.0001  6/25 6/55 12/50 12/111  0.918/0.0001 0.920/0.0001 0.921/0.0001 0.924/0.0001  6  12 36 48  0.983/0.036 0.982/0.018 0.972/0.0001  0.963/0.0001 0.951/0.0001 0.922/0.0001  0.966/0.0001 0.978/0.0001 0.982/0.018  0.877/0.0001 0.948/0.0001 0.964/0.0001  6/19 6/45 12/39 12/92  0.928/0.0001 0.931/0.0001 0.932/0.0001 0.936/0.0001  7  12 36 48  0.986/0.341 0.978/0.0001 0.984/0.120  0.952/0.0001 0.973/0.0001 0.969/0.0001  0.949/0.0001 0.966/0.0001 0.954/0.0001  0.926/0.0001 0.941/0.0001 0.957/0.0001  6/14 6/36 12/29 12/72  0.946/0.0001 0.947/0.0001 0.947/0.0001 0.949/0.0001  8  12 36 48  0.984/0.124 0.961/0.0001 0.964/0.0001  0.985/0.288 0.979/0.0001 0.981/0.003  0.975/0.0001 0.980/0.001 0.973/0.0001  0.986/0.360 0.892/0.0001 0.867/0.0001  7/18 7/23 14/24 14/34  0.950/0.0001 0.950/0.0001 0.953/0.0001 0.953/0.0001  9  12 36 48  0.984/0.143 0.964/0.0001 0.971/0.0001  0.984/0.147 0.964/0.0001 0.971/0.0001  0.983/0.039 0.982/0.025 0.977/0.0001  0.840/0.0001 0.883/0.0001 0.895/0.0001  8/67 8/153 16/121 16/275  0.954/0.0001 0.954/0.0001 0.955/0.0001 0.955/0.0001  10  12 36 48  0.986/0.5135 0.983/0.049 0.976/0.0001  0.987/0.5153 0.983/0.049 0.976/0.0001  0.982/0.011 0.978/0.0001 0.983/0.053  0.544/0.0001 0.742/0.0001 0.804/0.0001  13/40 13/90 26/73 26/162  0.957/0.0001 0.958/0.0001 0.959/0.0001 0.960/0.0001  11  12 36 48  0.967/0.0001 0.971/0.0001 0.961/0.0001  0.967/0.0001 0.971/0.0001 0.961/0.0001  0.975/0.0001 0.978/0.0001 0.979/0.0001  0.950/0.0001 0.974/0.0001 0.974/0.0001  7/50 7/114 14/100 14/228  0.960/0.0001 0.960/0.0001 0.962/0.0001 0.962/0.0001  12  12 36 48  0.975/0.0001 0.971/0.0001 0.971/0.0001  0.975/0.0001 0.971/0.0001 0.971/0.0001  0.954/0.0001 0.979/0.0001 0.979/0.0001  0.854/0.0001 0.929/0.0001 0.955/0.0001  7/40 7/96 14/81 14/192  0.965/0.0001 0.969/0.0001 0.974/0.0001 0.975/0.0001  a  Average number of first-order branches/twigs for two-stage systematic sampling.  Chapter  4.  Results  and  200,  126  Discussion  n=12  0.2  1501  o o  TJ  o  § 1001 cu  0.1 § TJ g> oo  50  -o= 10 Tree leaf area (square metres)  10.0 15 -,0.12 -0.10 -0.08  -0.04  TJ  jrtion  -0.06  TI o  TJ (0  —i  G>  -0.02 CD -1 0.0  Tree leaf area (square metres)  11  0.10 | Q03  | 3  7 8 9 10 Tree leaf area (square metres)  Figure 4.11: Distribution of the 1000 estimates of tree leaf area using simple random sampling for Tree 5 for sample sizes of 12, 36, and 48.  Chapter  4.  Results  and  127  Discussion  4.0 4.5 5.0 Tree leaf area (square metres)  4.0 4.5 Tree leaf area (square metres)  -i  1  1  ,0.2  r  n=48 o  TD O  0.1  a. °  T (03 GO  -r-r-nr  - f~H  4.0 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4. Tree leaf area (square metres)  LO  Figure 4.12: Distribution of the 1000 estimates of tree leaf area using stratified random sampling for Tree 5 for sample sizes of 12, 36, and 48.  Chapter  4.  Results  and  128  Discussion  5 10 15 Tree leaf area (square metres)  20  7 8 9 10 11 Tree leaf area (square rretres)  12  Tree leaf area (square rretres)  Figure 4.13: Distribution of the 1000 estimates of tree leaf area using ellipsoidal sampling for Tree 5 for sample sizes of 12, 36, and 48.  Chapter 4.  129  Results and Discussion  0  10 20 30 Tree leaf area (square rretres)  5  10  15  40  20  Tree leaf area (square metres)  0.14  n=48  0.12 0.10 0.08  )rtion  0.03  TJ (D  0.04  id  Ho-.  5 10 15 20 Tree leaf area (square metres)  T3 O TJ  -T  CD 01  -1002 0  Figure 4.14: Distribution of the 1000 estimates of tree leaf area using two-stage unequal probability sampling for Tree 5 for sample sizes of 12, 36, and 48.  Chapter  4.  Results  and  130  Discussion  *°  5 10 Tree leaf area (square metres)  -,0.16 -0.14 -0.12 TJ •§  -0.10 o  -0.06 o' -Q06  T>  CD  -ao4 i - 0.02 8 9 10 11 Tree leaf area (square metres)  12  8 9 10 11 Tree leaf area (square metres)  7  Jao 13  12 ° Q  Figure 4.15: Distribution of the 1000 estimates of tree leaf area using two-stage systematic sampling for Tree 5 for sampling intensities of I, II, III and IV.  131  Chapter 4. Results and Discussion  4.4.5  Cost Efficiency and Relative Ease  A two-person team averaged 4.17 minutes to select and cut a sample twig and 8.4 minutes to measure the leaf area (Table 4.26).  As expected, the large number of high-order  branches resulted in slower identification and labelling for older trees. The total costs for sampling and measurement varied by tree size and sampling design. Two-stage systematic sampling showed the lowest total costs, followed by stratified random, ellipsoidal, SRS and two-stage unequal-probability sampling (Table 4.27). The cost of sampling would vary also by sample size, site attributes, and the experience of the sampler. The sampling costs for SRS and two-stage unequal probability sampling design were noticeably higher than the sampling costs for stratified random and two-stage systematic sampling owing to slower identification of sample twigs, and measurements of first-order branch and twig lengths, respectively. Although statistical efficiency is important, the ease of employing a sampling design in the field must also be considered. Two-stage systematic sampling was considered the easiest for estimating T L A , followed by stratified random, ellipsoidal, SRS, and two-stage unequal probability sampling design (Table 4.28). This is due to the fact that two-stage systematic sampling does not require a detailed sampling frame and identification of the spatial location of strata or ellipses in the crown.  4.4.6  Overall Ranking of Designs and Summary  The overall ranking from best to worst of the sampling designs based on R M S E , cost, and relative ease was stratified random, two-stage systematic, ellipsoidal, simple random, and two-stage unequal probability sampling designs (Table 4.28). The estimates from two-stage systematic sampling have non-normal distributions (Figure 4.15). Hence, the standard error of estimates for T L A cannot be obtained by classical methods. Resampling  Chapter  4.  Results  and  132  Discussion  Table 4.26: Summary of sampling and leaf area measurement time (minutes) by sample tree.  Tree No.  DBH (cm)  Sampling time (minutes)  No. of twigs sampled in this time  Sampling time (minutes/twig)  Leaf area measurement time (minutes)  1 2 3 4  5.9 8.0 7.3 11.0  90 120 125 151  30 31 32 38  3.00 3.87 3.91 3.97  5 6 7 8  16.7 16.5 38.5 20.1  197 145 158 168  42 47 50 58  9 10 11 12  44.0 42.2 41.0 40.0  354 378 225 402 209.4  Average  No. of twigs measured in this time  Measurement time (minutes/twig)  70 110 90 120  9 11 10 13  7.78 10.0 9.00 9.23  4.69 3.09 3.16 2.90  145 140 132 171  17 17 15 19  8.53 8.24 8.80 9.00  60 51 57 96  5.90 7.42 3.95 4.19  192 170 245 230  22 20 38 36  8.73 8.50 6.45 6.39  49.3  4.17  151.25  18.92  8.39  Chapter  4.  Results  and  133  Discussion  Table 4.27: Summary of total costs (dollars) for sampling and leaf area measurement by sample tree and sampling design. A sample size of 36 twigs was used for comparison.  Tree No.  DBH (cm)  SRS  Stratified random  Ellipsoidal sampling  Two-stage systematic  Two-stage unequal probability  1 2 3 4  5.90 8.00 7.30 11.00  389.34 501.16 473.37 483.38  344.52 443.34 414.96 424.07  366.93 472.25 444.17 453.73  322.11 414.44 385.75 394.42  411.75 530.07 502.58 513.04  5 6 7 8  16.70 16.50 38.50 20.10  500.12 407.79 428.18 420.56  430.05 361.62 380.97 377.24  465.08 384.71 404.58 398.90  395.01 338.54 357.36 355.57  535.15 430.87 451.79 442.22  9 10 11 12  44.00 42.20 41.00 40.00  569.36 641.97 399.27 410.03  481.22 531.12 340.26 347.43  525.29 586.54 369.77 378.73  437.14 475.69 310.75 316.13  613.44 697.40 428.78 441.33  Average  24.27  468.71  406.40  437.56  375.24  499.87  134  Chapter 4. Results and Discussion  Table 4.28: Ranks of five sampling designs evaluated for estimating total tree leaf area.  Criteria  Ease of: a) sample unit: selection identification b) operations c) sample size control 1) overall ease (based on a and b) 2) sampling variance 3) cost efficiency 4) distribution of the estimates Overall Rank (based on 1, 2, 3, and 4)  Simple random  Two-stage systematic  Stratified random  Ellipsoidal  Two-stage unequal probability  3 5 4 1  1 1 1 4  2 2 2 2  4 3 3 3  5 4 5 5  4  1  2  3  5  4 4  3 1  1 2  2 3  5 5  1  5  2  3  4  4  2  1  3  5  .  "1" is most preferable and "5" is least preferable.  Chapter 4. Results and Discussion  135  techniques such as Monte Carlo sampling, the jacknife (Gregoire 1984), and the bootstrap method (Cook 1990) can be used to calculate S E E and the confidence interval of the estimates. The order of the sampling designs is based on the following assertions. 1. Identification of the spatial locations of height or strata (prior to sampling) is easier than identifying the spatial locations of ellipses on a standing tree. The presence of well-defined whorls in a hybrid spruce crown also justify the use of stratified random sampling by vertical position in estimating T L A . 2. Selection and identification of sample twigs from primary sampling units (PSU) for two-stage systematic sampling is easier than selection and identification of sample twigs in SRS, stratified, and ellipsoidal sampling designs. This is due to the relative ease of P S U selection prior to selecting sample twigs. 3. The two-stage unequal probability sampling was the most expensive design and was also ranked as the most difficult sampling design. This is due to the need to measure the lengths of all first-order branches in the tree crown and all twigs on each selected first-order branch prior to selecting sample twigs. Overall, stratified random sampling based on relative crown height was cost-effective (i.e., faster to locate and select sample twigs than the other sampling designs, except the two-stage systematic sampling) and relatively easy to implement. Also, stratified random sampling does not require measurement of additional variables and the distribution of its estimates were also nearly normal. Of the five designs tested, stratified random sampling is recommended as a practical and efficient sampling design to estimate T L A . Based on the results of the five designs, combinations of the five sampling designs may provide better estimates of T L A . Stratified two-stage systematic sampling, by first stratifying tree crowns by relative heights, and then employing a two-stage systematic  Chapter  4.  Results  and  Discussion  136  sampling within each stratum may provide more precise T L A estimates than stratified random sampling. This is due to reduced sampling costs by stratified two-stage systematic sampling, which in turn allows for measurement of more sample twigs for the same cost. Another alternative is to use two-way stratified random sampling. Two-way stratified random sampling would involve stratifying by relative heights and by relative horizontal lengths. Sukhatme and Sukhatme (1970) called this type of sampling design deep stratification. The wide variation of foliage distribution at various heights and horizontal lengths of tree crowns found in this and other studies (e.g., Maguire and Bennett 1996, Whitehead et al. 1990, Woodman 1971) can be captured more by two-way stratified random sampling than by one-way stratified sampling. However, the number of strata should be limited (i.e., four to nine), as misidentification of sample twigs to appropriate strata (misclassification error) increases with more strata. It should be noted that some strata may have no twigs, particularly those close to the bole of the tree. The three dimensional graphs indicate an increase in A P C M towards the tip of the tree and the tip of the first-order branches. This implies an inverse relationship between (1) the lengths of first-order branches and twig leaf area, and (2) between twig lengths and twig leaf area. Thus, selection of sample twigs by the probability proportional to the reciprocal of lengths of first-order branches and twigs may provide strong positive correlation between the selection probabilities and twig leaf area values. This, in turn, will improve the precision of two-stage unequal probability sampling. This approach will favor the selection of shorter first-order branches and twigs. A variant of this approach is selection of sample first-order branches using probability proportional to their lengths in the first-stage and selection of sample twigs by probability inversely proportional to their reciprocal lengths in the second-stage.  Chapter 5  Conclusions and Recommendations  Tree leaf area (photosynthetic surface area) can be used as a response variable in analyzing the impacts of silvicultural treatments such as fertilization and thinning. Single-tree leaf area estimates are more useful than stand level estimates when a study involves a small number of trees, the individual tree response (e.g., respiration or productivity per tree) is of interest, or the response of different crown classes within a stand is required (Neumann 1990). Knowledge of specific leaf area (leaf area/leaf weight) within tree crowns and their positional variation will contribute to the understanding of light dynamics (i.e., light extinction coefficients, radiation transfer models, photosynthesis, and respiration by foliage) in tree crowns and the complex nature of tree crowns. This study has provided useful information on variation of leaf area in tree crowns, leaf area distribution models, the relationship of leaf area distribution model parameters to tree attributes, and a cost-variance comparison of selected sampling designs using hybrid spruce trees. Three-dimensional analyses and displays of leaf area illustrated that leaf area per centimetre ( A P C M ) increases towards the tip of the tree and the tip of first-order branches. Model estimates of branch length and A P C M were more precise when branch position from the tip of the tree was expressed relative to tree height (zii) rather than relative to crown length (zii).  Thus, the reference of a branch using zii  instead of zii is suggested for single tree growth and yield modelling. The bivariate Weibull and beta distribution functions are suggested for A P C M prediction as these models resulted in more precise representations of A P C M within tree  137  Chapter 5. Conclusions and Recommendations  138  crowns than the univariate Weibull and beta distribution functions. The bivariate distribution functions, unlike the univariate distribution functions, provide insights into positional (vertical and horizontal) variability of leaf area distribution in tree crowns. It is also suggested that the seemingly unrelated regression (SUR) fitting method be used for parameter prediction of A P C M within tree crowns, as this fitting method resulted in the most precise A P C M estimates. Technical difficulties were encountered in reconstructing each tree crown to create test populations. These difficulties can be ascribed to: (1) limited references for creating and positioning higher-order branches hierarchically; (2) finding starting values that would converge and also result in global minimum parameters of non-linear models; and (3) maintaining a biological interpretation of prediction equations in light of (1) and (2). The evaluation of sampling designs on test populations that were comprised of measured and estimated data showed consistency as few biologically reasonable data were collected with accuracy and high precision. No significant differences were found between the sample and model total leaf area (TLA) estimates. Overall, it can be concluded the created test populations were a reasonable abstraction of the actual tree crowns. The ranking of the five sampling designs based on cost-variance considerations will help in reducing the costs of sampling given a desired level of precision for T L A . Stratified random sampling proved to be an efficient and practical sampling design and it is recommended for T L A estimation for hybrid spruce crowns. These findings also may guide researchers in identifying other approaches to improve the estimation of T L A . Cost factors (i.e., precision per dollar) and the availability of suitable variance estimators are two fundamental concerns in designing methods for estimating T L A . Based on the Monte Carlo simulations using the test populations, stratified random sampling was the most precise design, followed by ellipsoidal, two-stage systematic, SRS, and then the two-stage unequal probability sampling design. However, two-stage systematic sampling  Chapter  5.  Conclusions  and  139  Recommendations  was considered best using an overall rank of cost-variances. The stratified random and two-stage systematic sampling designs have good potential for operational use for other tree species and should be further tested. The ellipsoidal sampling design has two possible problems that could reduce its efficiency, namely 1) identification of ellipses, and 2) operational difficulty. However, this sampling design is effective for describing foliage distribution within tree crowns as it considers both the vertical and horizontal variability of foliage. Based on review of the literature, the experience acquired during data collection and analysis, and the results of this study, the following recommendations are made for future research. 1. Different silvicultural practices, species composition, and biogeoclimatic zones are known to impact the branching density, crown length, and foliage distributions within tree crowns. Thus, further studies on leaf area distribution of hybrid spruce trees grown under different conditions are recommended. 2. Large sample sizes of trees and random selection of trees within a stand provide a better picture of foliage distribution within tree crowns and also increase the depth and breadth of inferences.  Since sample trees and the stands were subjectively  selected, the results of this study mainly apply to the stands under consideration. Due to lack of test data, the various non-linear equations and the bivariate Weibull distribution functions were not validated.  Hence, further work on this subject  should involve validation of the estimated model parameters using a sufficiently large sample of trees. Also, applicability tests should be carried out to determine the accuracy of the models, the distribution functions, and the efficiency of the sampling designs for hybrid spruce in other biogeoclimatic zones or for other tree species in the same biogeoclimatic zones.  Chapter  5.  Conclusions  and  Recommendations  140  3. The amount of light received by tree crowns has a noticeable effect on the rate of tree growth. Silva et al. (1989) found that dominant trees that received fulloverhead light grew three times faster than overtopped (completely shaded) trees. Hence, studies on leaf area distribution and efficiency of sampling designs tested in this and other studies should be extended to various crown classes. 4. Orientation of leaves arising from first- or higher-order branches is an important factor in determining the amount of sunlight received by tree crowns (Oker-Blom 1986) and in describing leaf area distribution in tree crowns. However, this attribute was not considered in this study. Studies on the orientation of leaves and higher-order branches would contribute to modelling light transmission and light interception in tree crowns. 5. Fractal dimension demonstrates self-similarity on different scales and is reported to bring new logic to quantify irregular structure and growth of complex biological forms such as tree crowns (West and Goldberg 1987, Ziede and Pfeifer 1991). The box-counting method of fractal geometry uses samples of natural units such as twigs, first- or higher-order branches.  Regression equations that contain fractal  dimension as their slope could provide a measure of foliage density. This, in turn, could be used to describe foliage distribution and its attributes such as leaf area and leaf weight. Thus, studies on the potential use of fractal geometry for quantifying foliage distributions within tree crowns and for estimating T L A deserve attention in tree crown research. 6. Leaf area sampling and measurement are onerous, destructive and time consuming (costly). Improved field-portable leaf area meters or digital cameras that can digitize and measure individual twig leaf area in tree crowns would make these  Chapter  5.  Conclusions  and  Recommendations  141  operations easier and cheaper. Such cameras could also be attached to a computer for direct downloading of measurement values. The alternative approaches outlined in Section 4.4.6 and the above recommendations for future research should improve estimation of T L A . Accurate estimates of T L A are needed to assess responses to silvicultural treatments, to refine and/or extend existing tree growth models, and as input to light extinction models. This study lays the foundation for future joint research between the disciplines of plant physiology, morphology, and forest mensuration in an attempt to quantify morphological attributes and physiological processes of tree crowns. 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Xylem Structure and the Ascent of Sap. Berlin. 143 pp.  Springer Verlag.  Appendix A  Leaf area per centimetre at different relative horizontal lengths  158  Appendix  A.  Leaf  area  per  centimetre  at different  relative  horizontal  lengths  159  Direction  0.2 0.4 0.6 0.8 Relative horizontal length  1.0  A West + South x North o East  10  TO cr  OJ  E  8 6  o  E  _o  4  Direction  75  oo tu  n°  o  &  2 +  t  ox  +  0.0  0.2 0.4 0.6 0.8 Relative horizontal length  1.0  A West + South x North o East  Direction  0.2 0.4 0.6 0.8 Relative horizontal length  1.0  A West + South X North o East  Figure A. 16: Leaf area per centimetre (cm /cm) at different relative horizontal lengths by the four cardinal directions for Tree 1, 2, and 4. A relative horizontal length of one indicates values at the tip of the first-order branch. 2  Appendix  A.  Leaf  area  per  centimetre  at different  relative  horizontal  lengths  160  Direction  0.2 0.4 0.6 0.8 Relative horizontal length  1.0  A West + South x North o East  Direction  0.2 0.4 0.6 0.8 Relative horizontal length  1.0  A West + South x North o East  Direction A West + South x North o East 0.0  0.2 0.4 0.6 0.8 Relative horizontal length  Figure A. 17: Leaf area per centimetre (cm /cm) at different relative horizontal lengths by the four cardinal directions for Tree 5, 6, and 8. A relative horizontal length of one indicates values at the tip of the first-order branch. 2  Appendix  A.  Leaf  area  per  centimetre  at different  relative  horizontal  lengths  161  10 E  o  a 5 i _  ro z: cr E  8  6  o  'cm  —  4  O  *t fio  CO (D  iC  Direction  4*?"  + +x  o£  ^ f it" x  A + x o  ro 2 ro OJ  _i  0 0.0  0.2 0.4 0.6 0.8 Relative horizontal length  West South North East  Direction  0.0  0.2 0.4 0.6 0.8 Relative horizontal length  1.0  A West + South x North o East  Direction  0.2 0.4 0.6 0.8 Relative horizontal length  1.0  A + x o  West South North East  Figure A . 18: Leaf area per centimetre (cm /cm) at different relative horizontal lengths by the four cardinal directions for Tree 9, 11, and 12. A relative horizontal length of one indicates values at the tip of the first-order branch. 2  Appendix B  L e a f a r e a p e r c e n t i m e t r e at different r e l a t i v e  162  heights  Appendix  B.  Leaf  area  per  centimetre  at, different,  relative  heights  163  Direction  0.4 0.6 Relative height  1.0  ^ + x o  West South North East  Direction  0.4 0.6 Relative height  0.8  1.0  ^ + x o  West South North East  Direction  0.4 0,6 Relative height  0.8  1.0  a + x o  West South North East  Figure B.19: Leaf area per centimetre (cm /cm) at different relative heights by the four cardinal directions for Tree 1, 2, and 4. A relative height of one indicates values at the tip of the first-order branch. 2  Appendix  B.  Leaf area per centimetre at different relative heights  164  Direction A West + South x North o East 1.0  0.4 0.6 Relative height  0.2  0.0  Direction A West + South x . North o East 0.0  0.2  0.4 0.6 Relative height  0.8  1.0  10 8 6 4  •**Y  2  A  lull! - K  03  0.0  0.2  A  J  *  ft 1  Direction A West + South x North o East  + £Q  * %  0.4 0.6 Relative height  0.8  1.0  Figure B.20: Leaf area per centimetre (cm /cm) at different relative heights by the four cardinal directions for Tree 5, 7, and 8. A relative height of one indicates values at the tip of the first-order branch. 2  Appendix B. Leaf area per centimetre at different relative heights  165  Direction  1.0  0.4 0.6 Relative height  A + x o  West South North East  Direction  0.2  A + x o  0.4 0.6 Relative height  West South North East  Direction  0.2  0.4 0.6 Relative height  0.8  1.0  A + x o  West South North East  Figure B.21: Leaf area per centimetre (cm /cm) at different relative heights by the four cardinal directions for Tree 10, 11, and 12. A relative height of one indicates values at the tip of the first-order branch. 2  Appendix C  L e a f a r e a p e r c e n t i m e t r e at v a r i o u s h e i g h t s a n d h o r i z o n t a l l e n g t h s  166  Appendix  C.  Figure C.22:  Leaf  area  per  centimetre  at various  heights  and  horizontal  lengths  167  Leaf area per centimetre ( c m / c m ) at different relative heights and relative 2  horizontal lengths for Tree 2 (young tree). A relative height of zero indicates values at the tip of the tree. A relative horizontal length of one indicates values at the tip of the first-order branch.  Appendix  C.  Leaf  area  per  centimetre  at various  heights  and  horizontal  lengths  168  Figure C.23: Leaf area per centimetre (cm /cm) at different relative heights and relative horizontal lengths for Tree 3 (young tree). A relative height of zero indicates values at the tip of the tree. A relative horizontal length of one indicates values at the tip of the first-order branch. 2  Appendix C. Leaf area per centimetre at various heights and horizontal lengths  169  Figure C.24: Leaf area per centimetre (cm /cm) at different relative heights and relative horizontal lengths for Tree 4 (young tree). A relative height of zero indicates values at the tip of the tree. A relative horizontal length of one indicates values at the tip of the first-order branch. 2  Appendix C. Leaf area per centimetre at various heights and horizontal lengths  170  Figure C.25: Leaf area per centimetre (cm /cm) at different relative heights and relative horizontal lengths for Tree 6 (middle age tree). A relative height of zero indicates values at the tip of the tree. A relative horizontal length of one indicates values at the tip of the first-order branch. 2  Appendix  C.  Leaf  area  per  centimetre  at, various  heights  and  horizontal  lengths  171  Figure C.26: Leaf area per centimetre (cm /cm) at different relative heights and relative horizontal lengths for Tree 7 (middle age tree). A relative height of zero indicates values at the tip of the tree. A relative horizontal length of one indicates values at the tip of the first-order branch. 2  Appendix  C.  Leaf  area  per  centimetre  at various  heights  and  horizontal  lengths  172  Figure C.27: Leaf area per centimetre (cm /cm) at different relative heights and relative horizontal lengths for Tree 8 (middle age tree). A relative height of zero indicates values at the tip of the tree. A relative horizontal length of one indicates values at the tip of the first-order branch. 2  Appendix  C.  Leaf  area  per  centimetre  at various  heights  and  horizontal  lengths  173  Figure C.28: Leaf area per centimetre (cm /cm) at different relative heights and relative horizontal lengths for Tree 9 (old tree). A relative height of zero indicates values at the tip of the tree. A relative horizontal length of one indicates values at the tip of the first-order branch. 2  Appendix  C.  Leaf  area  per  centimetre  at various  heights  and  horizontal  lengths  174  Figure C.29: Leaf area per centimetre (cm /cm) at different relative heights and relative horizontal lengths for Tree 11 (old tree). A relative height of zero indicates values at the tip of the tree. A relative horizontal length of one indicates values at the tip of the first-order branch. 2  Appendix  C.  Leaf  area  per  centimetre  at, various  heights  and  horizontal  lengths  175  Figure C.30: Leaf area per centimetre (cm /cm) at different relative heights and relative horizontal lengths for Tree 12 (old tree). A relative height of zero indicates values at the tip of the tree. A relative horizontal length of one indicates values at the tip of the first-order branch. 2  

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