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Examination and modelling of tree form and taper over time for interior lodgepole pine Muhairwe, Charles K. 1994

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EXAMINATION AND MODELLING OF TREE FORM AND TAPER OVERTIME FOR INTERIOR LODGEPOLE PINEByCharles K. MuhairweB. Sc. (For), Makerere UniversityM. Sc., University of EdinburghA THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIESFOREST RESOURCES MANAGEMENTWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAJune 1993© Charles K. Muhairwe, 1993In presenting this thesis in partial fulfilment of the requirements for an advanced degree atthe University of British Columbia, I agree that the Library shall make it freely availablefor reference and study. I further agree that permission for extensive copying of thisthesis for scholarly purposes may be granted by the head of my department or by hisor her representatives. It is understood that copying or publication of this thesis forfinancial gain shall not be allowed without my written permission.Forest Resources ManagementThe University of British Columbia2075 Weshrook PlaceVancouver, CanadaV6T 1Z1Date:(‘)/9AbstractForest management, concerned with maintaining or increasing the output of the differentforest resources, is increasingly becoming more intensive. Therefore, relevant, accurate,timely, and cost-effective forest resource information on the current inventory and futuregrowth potential is critical. This information must also be utilized over time in the mosteffective way for planning purposes. Such information can he provided via forest inventoriesand growth and yield studies. Understanding and modelling of taper changes over time willprovide some of the necessary information in an efficient manner.The two objectives of this study were: (1) to investigate changes in tree form and taperover time as affected by changes in tree, stand and site variables, and (2) to develop adynamic taper function for dominant and codominant trees of interior lodgepole pine (Finnscontorta Dougi.) based on the results of this investigation. To meet these objectives, twodifferent sets of data were used: permanent sample plot data from Alberta and detailed stemanalysis data from Interior British Columbia. The permanent sample plot data were usedto develop a model to predict stand density and to select models to predict total tree heightand diameter at breast height. The stem analysis data were used to examine tree form andtaper changes over time, and to select, fit, and test the dynamic taper model. Instead ofdeveloping an entirely new taper model, existing taper functions were investigated as possiblecandidates for both objectives. Two static taper models, the simple ta.per equation (Huschet al. 1982, p. 99) and the variable-exponent taper equation by Kozak (1988), were selected.The first objective of this study was achieved. Tree shape and taper were found to changealong the stem at one time and over time with changes in tree and stand variables, such asthe ratio of diameter at breast height to total tree height, crown length and crown ratio,and stand density. It was also found that trees have a simple parabolic shape at young age.11However, as trees age or increase in size, different portions of the stem take different shapesbecause of unequal diameter growth along the stem. Stand density and crown size appearto the determining factors in tree shape and taper changes and any changes in these twofactors will determine tree shape and taper.The second objective of this study was also achieved. By incorporating tree. stand, andsite factors into a simple static variable-exponent taper model by Kozak (1988), a dynamictaper function was developed. This dynamic taper function tracked the behaviour of verycomplex tree shape and taper changes over time with reasonable accuracy. The function wastested using a validation data set and it provided consistent estimates of diameter insidebark along the stem over time. The model was fitted using ordinary linear and nonlinearleast squares. Feasible generalized least squares was considered, but not used, because ofthe difficulties in obtaining a consistent estimate of the error covariance matrix.The dynamic taper modelling approach will be a useful tool in forest management becausethe taper models will enable forest managers to simulate stand development in order toachieve specific objectives. Dynamic taper modelling appears to be a feasible and practicalidea, and it is recommended that dynamic taper models for other species and crown classeshe developed to incorporate in individual tree growth models.111Table of ContentsAbstract iiList of Tables vList of Figures viAcknowledgement vii1 Introduction 155891315Variation 1818191920212222242 Background2.1 Tree Growth2.1.1 Height Increment2.1.2 Radial Growth in Forest Trees2.1.3 Tree Growth Simulation Models .2.2 Variation in the Form and Taper of Trees .2.2.1 Theoretical Bases for Tree Form and Taper2.2.1.1 Nutritional Theory2.2.1.2 Water Conduction Theory2.2.1.3 Mechanistic Theory2.2.1.4 Hormonal Theory2.2.1.5 Comparison of Theories2.2.2 Factors Affecting Form and Taper Variation2.2.2.1 Site Characteristics . .2.2.2.2 Tree and Stand Characteristicsiv2.3 Tree Form and Taper Estimation 292.3.1 Static Tree Taper Estimation 292.3.1.1 Simple Taper Functions 312.3.1.2 Segmented Taper Functions 362.3.1.3 Variable Form and Exponent Taper Functions 402.3.1.4 Taper Functions based on Other Methods . 432.3.2 Dynamic Tree Taper Estimation 442.4 Alternative Methods for Fitting Taper Functions 462.4.1 Ordinary Least Squares and Nonlinear Least Squares 492.4.2 Generalized Least Squares 512.4.3 Fitting Dynamic Taper Functions Using FGLS 533 Methods 583.1 Data Preparation 583.1.1 Permanent Sample Plot Data 593.1.2 Stem Analysis Data 613.2 Examination of Tree Form and Taper Variation 733.3 Dynamic Taper Model Development 763.3.1 Height Prediction Model 773.3.2 Prediction Model for Diameter at Breast Height 803.3.3 Relative Height Function 813.3.4 Variables Selected for Estimating the Form Exponent 833.3.5 Dynamic Taper Model Assembly and Fitting 843.3.5.1 Optimization of q 873.3.5.2 Fitting the Taper IViodels using OLS and NLS 873.3.5.3 Fitting the Taper Models using FGLS 883.3.6 Dynamic Taper Model Testing and Evaluation 94v4±1.4 Age at Breast Height4.1.1.5 Site Index4.1.1.6 Predicted Quadratic Mean Diameter at Age 504.1.1.7 Crown Length and Crown Ratio4.1.2 Variation in Form and Taper within Individual Trees over4.1.2.1 Form and Taper Variation along the Stem4.1.2.2 Form Variation with Tree Age and Height4.2 Dynamic Taper Model Building4.2.1 Height Prediction Model4.2.2 Prediction Model for Diameter at Breast Height4.2.3 Relative Height Prediction Model4.2.4 Dynamic Taper Function Selection4.2.5 Taper Models Fitted4.3 Taper Function Evaluation5 Discussion 1325.1 Variation in Tree Form and Taper5.2 The Dynamic Taper Functions0 Conclusions and Recommendations 1434 Results4.1 Variation in Tree Form and Taper4.1.1 Variation of Tree Form and Taper with TSS4±1.1 Total Tree Height4.1.1.2 Diameter at Breast Height4.1.1.3 Dbh to Height Ratio98989899999999104104105Time . . . 1051051071111111141151151161171171284.3.14.3.2Model Evaluation for PredictionsModel Evaluation for 1/ Prediction132138viLiterature Cited 148Appendices 159A Glossary of Variables Used 159B Quadratic Mean Diameter Curves 160C Form Variation with Tree Age 162D Form and Taper Variation along the Stem 166E Form Variation with Height and Age 172F Tree Volume Prediction 176viiList of Tables2.1 Some static taper models developed to date 323.2 Summary statistics for the permanent sample plot data 623.3 Fit and prediction statistics for QD models 713.4 Summary statistics for the stem analysis data 733.5 Number of trees by height and dbh classes 744.6 Fit and prediction statistics for height prediction models 1124.7 Fit and prediction statistics for dbh prediction models 1144.8 Fit and prediction statistics for relative height predictioi models 1164.9 Fit and prediction statistics for OLS dynamic taper models for 1174.10 Coefficient estimates for d1 for all taper models 1184.11 Fit and prediction statistics for all taper models for 1204.12 Mean biases for prediction for all taper models 1214.13 Mean biases for volume (rn3) prediction for all taper models 1294.14 Mean biases in volume prediction for different age classes 131viii2.1 The geometric division of a tree stem 333.2 Quadratic mean diameter for SI 20 m by breast height age3.3 Data distribution by age classes4.4 1004.5 1014.6 1024.7 1044.8 1064.9 1084.10 1094.11 1104.12 1234.13 1244.14 1264.15 127Quadratic Mean Diameter Curves for Site Index 15Quadratic Mean Diameter Curves for Site Index 10163163164164List of Figures7274Form exponent by tree, stand, and site variables at 0.1H .Form exponent by tree, stand, and site variables at 0.5H .Form exponent by tree, stand, and site variables at 0.8H .Form exponent by age for tree 13.2 at different relative heights .Form exponent by tree crown size at 0.1H, 0.5H, and 0.8HForm exponent by relative height for tree 13.2Form exponent by relative diameter for tree 13.2Form exponent by height and age for tree 13.2Mean biases for at different heights for all treesMean biases for at different heights for dbh class 1 . .Mean biases for at different heights for dbh class 2 . .Mean biases for at different heights for dbh class 3B.16B.17C.18C.19C.20C.21161161Form exponent by age at 0.1H, 0.5H, and 0.8H for tree 16.2Form exponent by age at 0.1H, 0.5H, and 0.8H for tree 25.2Form exponent by age at 0.1H, 0.5H, and 0.8H for tree 42.1Form exponent by age at 0.1H, 0.5H, and 0.8H for tree 43.2ixC.22 Form exponent by age at 0.1H, 0.5H, and 0.8H for tree 46.1 165D.23 Form exponent by relative height for tree 16.2 . .D.24 Relative diameter by relative height for tree 16.2 .D.25 Form exponent by relative height for tree 25.2 . .D.26 Relative diameter by relative height for tree 25.2 .D.27 Form exponent by relative height for tree 42.1 . .D.28 Relative diameter by relative height for tree 42.1 .D.29 Form exponent by relative height for tree 43.2 . .D.30 Relative diameter by relative height for tree 43.2 .D.31 Form exponent by relative height for tree 46.1 . .D.32 Relative diameter by relative height for tree 46.1 .E.33 Form exponent by height and age for tree 16.2 . .E.34 Form exponent by height and age for tree 25.2 . .E.35 Form exponent by height and age for tree 42.1 . .E.36 Form exponent by height and age for tree 43.2 . .E.37 Form exponent by height and age for tree 46.1 . .Predicted volume by age for tree 14.1Predicted volume by age for tree 17.1Predicted volume by age for tree 25.1Predicted volume by age for tree 41.2167167168168169169170170171171173173174174175177178179180F.38F.39F.40F.41xAcknowledgementOne of the great benefits of llndertaking a research such as this is an opportunity to work withsome of the best people in their respective fields. This is especially true of the members ofmy committee: Drs. Antal Kozak, Valerie LeMay, Peter Marshall, A.Y. Omule, and MichaelPitt. Their help and encouragement is greatly appreciated. I am specially grateful to Dr.Valerie LeMay, my major professor, for her guidance and encouragement. Thanks also go toDr. Peter Marshall for his thorough editorial work. I have also profited from working withthe other graduate students in the Biometrics Hut. The discussions we had were fruitful.My appreciation and thanks go the Canadian Commonwealth Fellowships Plan for providing me with funding, without which I would not have made up it to here. Dr. Karl Klinkaand the Alberta Forest Service provided the data used in this research and I appreciate theirhelp.Finally I would like to thank my mother and beloved sister, Jane, who has always beenthere for me when I needed her most. I love you both, God bless you. Also, I thank Winifredwho provided me with all the support and the environment at home I needed.Chapter 1IntroductionForest management is concerned with maintaining or increasing the output of different forestresources. Managing the forest requires the forest manager to manipulate stand variables,such as age, density, and site quality, to achieve the intended objectives. In order to do thiseffectively, the forest manager requires accurate and up-to-date information on the currentgrowing stock (inventory) and future growth potential.Numerous growth and yield methods have been developed as sources of information.Early estimates of forest growth and yield were based on yield tables. In order to providedata to construct these tables, sample plots were established over a range of age and siteclasses for single species, even-aged stands. The selected stands were also normally stocked(neither understocked nor overstocked). The plot data were tabulated and summarized todevelop a series of alignment charts that were subsequently used to provide yield estimatesby the conventional age/site index format (Spurr 1952). Unfortunately, these tables werenot representative of actual stand conditions, which led to the development of stand and,later, individual tree models. Moser (1980) gave detailed historical steps in the developmentof growth and yield modelling.With forest management becoming information intensive, more information is requiredthan can be provided by stand models. Individual tree models have been developed forthis purpose. All individual tree growth models incorporate equations for predicting futurevalues of diameter outside bark at 1.3 m above ground (dbh), total tree height (height) (seeAppendix A) and volume. In some models, the equations for predicting dbh and heightare used together with a taper function to obtain estimates of future volume (Arney 1985).1Chapter 1. Introduction 2These taper (stem profile) equations have been developed to describe how diameter or thestem cross-sectional area changes as a function of height along the stem using dbh, height,and height above ground as independent variables (e.g., Kozak et al. 1969; Ormerod 1973).When these equations are integrated, they provide estimates of stem volume for any portionof the tree. If a taper function is used as a component of a growth model to obtain the stemprofile and volume of a tree in the future, an implicit relationship is defined for diameterincrement along the stem. It is usually assumed that changes in dbh would adequatelyreflect the change in upper stem diameter. This assumption may not hold (e.g., Kramer andKozlowski 1979; Loetsch et al. 1973).A dynamic taper function can be defined as a taper function which gives the diameter(usually inside bark) at any point along the stem over time. Except for a few quantitativestudies examining diameter growth rates at various positions along the stem throughout thetree’s life and their relationship to tree taper (e.g., Arney 1974; Mitchell 1975; Clyde 1986),little has been done to model the tree form and taper change along the stem over time.Clyde (1986) developed a dynamic taper model. She tried to improve stern profile andvolume modelling by including diameter increment variation along the stem based on the firstderivative of the taper model with respect to time. She found that this approach did not workwell. The poor performance of Clyde’s taper prediction model could be due to the fact thather height and dbh models were functions of age alone. She never included important tree,stand, and site variables known to affect tree growth in her diameter and height predictionmodels. Also, the taper models she used assumed that tree shape is constant over time. Adynamic taper model based on precise dbh and height growth prediction functions, alongwith accounting for changes in form and taper over time, should provide better predictionsthan Clyde’s model.The two principal objectives of this research were: (1) to investigate changes in tree formand taper over time as affected by changes in stand, tree, and site characteristics, and (2) todevelop a dynamic taper function for dominant and codominant trees of interior lodgepoleChapter 1. Introduction 3pine (Pinus contorta var. latifolia Engeim.) based on the knowledge gained from completingthe first objective.In order to achieve these objectives, the following analyses were performed:1. Based on a simple taper function (Husch et al. 1982, p. 99), variation in stem formand taper along the stem at one particular time and as time changes with differenttree, stand, and site factors was investigated using detailed stem analysis data.2. Using permanent sample plot (PSP) data, models for prediction of quadratic meandiameter, dhh, and height as functions of plot site index, stand density, and age weredeveloped and tested.3. Using the measured site, stand, and tree variables associated with stem form and taper,.an equation to predict the form exponent of Kozak’s (1988) taper function was selected.4. Models for dbh, height, and the form exponent were used to refit Kozak’s (1988) taperfunction to account for dynamic changes in form (shape) along the stem (i.e., makethe static taper model dynamic). The fitting methods used included ordinary leastsquares and nonlinear least squares. Feasible generalized least squares was consideredand is discussed, but was not used.5. Finally, the dynamic taper model fitted using ordinary least squares and nonlinear leastsquares was tested by comparing it with Kozak’s (1988) original taper function, fittedas both as a dynamic and a static model for predictive abilities based on mean bias,standard error of estimate. root mean square biases, fit index squared, and absolutebias using a reserved detailed stem analysis data set.This research contributes to increasing the body of knowledge on growth modelling inthe following ways:1. A detailed study of how tree shape (form) changes over time was conducted. Thebenefit of this detailed study is that it has increased our understanding of this areaChapter 1. Introduction 4(basic change in tree form as the tree ages) and how various factors (site quality, standdensity, crown length, etc) affect tree form.2. Taper functions already exist and are in use for estimation of current volume. Compatibility between future and current inventory is expected if future volumes are predictedby a dynamic taper function.3. Since the taper functions were developed using unmapped tree data, the dynamictaper function could be incorporated into a distance-independent individual tree growthmodel, which is relatively simple to understand, easy to calibrate, and easy to use. Thisshould provide increased accuracy in predicting future volumes over the use of statictaper functions.Chapter Two contains a summary of the literature on growth of trees as related to treeform and taper; a review of theory about how tree form and taper change with changes instand, tree, and site factors over time, and a discussion of how different researchers have triedto incorporate such changes during tree taper modelling are included. A description of thedata used in building the taper model, the modelling process, and the methods used to compare the dynamic taper fitting techniques are provided in the third chapter. Chapters Fourand Five contain the presentation and discussion of the results, respectively. Conclusionsand recommendations are given in the final chapter.Chapter 2BackgroundChanges in tree form and taper are a result of changes in diameter over the stern and heightgrowth. Therefore, in order to model tree form and taper changes, an understanding ofhow trees grow in diameter and height is needed. A summary of the background knowledgeabout tree growth, which includes the processes behind the growth of trees, the efforts ofmodellers to describe tree and stand growth, and a brief outliie of growth models, is givenin Section 2.1. The theory behind the changes in form and taper of trees and the factorsthat affect such changes is outlined in Section 2.2. The different taper models that havebeen developed for volume estimation and growth modelling are then discussed in Section2.3. In the final section, possible alternatives to the ordinary least squares fitting methods,applicable to detailed stem analysis data used for modelling taper over time, are examined.2.1 Tree GrowthGrowth may be defined as an irreversible change in volume (or other attributes of interest),which may be accompanied by a change in form (Thomas et al, 1973). During growth, cellsmultiply, enlarge, and differentiate into growing parts. There is a complex interplay of manygenetically determined metabolic and biophysical processes. Husch (1963) defined growthas the gradual increase in the size of an organism, population or an object over a periodof time. This increase or increment consists of the difference in size between the beginningof the growth period and its termination. Biologically, growth means more than jllst anincrease in size. For example, when a dry board is placed in water, it will swell and increasein size. However, this would not be called growth in the biological sense, since biological5Chapter 2. Background 6growth involves both an increase in size and the formation and differentiation of new cells,tissues, and organs.Tree growth is the increase in the sizes of individual trees and stands. Growth takesplace simultaneously and sometimes independently in different parts of a tree and can bemeasured by many parameters (e.g., change in diameter, in height, in crown size, and in bolevolume).Tree growth is influenced by the genetic capabilities of the species concerned, interactingwith the environment in which it is growing. Environmental influence is manifested throughclimatic factors, such as air temperature, precipitation, wind, and radiation; soil factors,such as physical and chemical characteristics, soil moisture, and soil microorganisms; andtopographic characteristics such as slope, elevation, and aspect. It is the sum effect of theseenvironmental and site factors that, for the rest of this thesis, will be referred to as the sitequality. The site quality will be good if it provides favorable growth conditions and poor ifit is inhibitive to the inherent growth capacity of a given species (Spurr 1952).The total tree growth in wood and bark consists of longitudinal and radial growth ofthe stem, roots, and branches. Longitudinal growth results when the stem or roots arelengthened by forming new tissues at their tips. Radial growth is brought about by divisionsof the cambia producing new cells, which become the new wood and bark between the oldwood and old bark. Generally speaking, the pattern of growth for a tree during its entire lifefollows a sigmoid-shaped trend (Spurr 1952; Husch et al. 1982). That is, a tree grows slowlyin all dimensions in its early years, called the youth stage, rapidly for a period of time after itbecomes well established on site, called the maturity stage, with its rate of growth graduallyfalling off, the senescence stage, and finally becoming almost negligible at physical maturity.This growth trend is found in individual cells, tissues, organs and individuals for both plantsand animals. Such “S-shaped” or sigmoid shaped curves are usually called growth curves.The cumulative growth curves (yield) for individual trees have typical characteristicswhich hold for any of the dimensions of a tree. However, the exact form, shape, or positionChapter 2. Background 7in reference to the curve axes differs, depending on the growth variables. Plotting cumulativeheight, diameter, basal area, volume, or weight (mass) over age will give differing specificcurves, but all will have the general sigmoid shaped trend, showing the four typical stagesdescribed above (see Spurr 1952, p. 212).A tree is a complex system with relatively few types of structures: 1) leaves, which arethe productive machines of trees, take in carbon dioxide from the air, light energy from thesun, and water from the soil and combine them in a process of photosynthesis to producecarbohydrates; 2) the stem and branches, which serve the function of supporting the leaves,transporting water and minerals to the leaves and transporting the manufactured materials(photosynthates) from leaves to the rest of the tree where they can be used for maintenanceand growth; and 3) roots, which anchor the tree firmly in the ground and absorb the waterand minerals from the soil.Tree growth occurs in three dimensions: 1) the extension of each growing point formingthe shoots of the crown and the roots, called primary growth; 2) the expansion of stem androot diameters (secondary growth); and 3) a combination of these processes which gives eachspecies its characteristic aerial structure and form.In order to understand the growth of stands and trees, it is necessary to first analyzethe pattern of growth and the resulting shape of individual trees. Trees make their annualgrowth by extending their shoots (height increment) and by thickening of the stems androots (diameter increment). Change in diameter can be converted into area increment alongthe stem. According to Pressler’s growth law (Mitchell 1975), volume growth is determinedby bole area increment along the stem which is a function of, or is proportional to, foliagevolume. Thus, tree volume increment along the stem is the product of area and heightincrement.Chapter 2. Background 82.1.1 Height IncrementAnnual shoot growth differs with species, genotype, and climate. The magnitude of theannual height increment fluctuates, depending on the weather conditions, competition, andimpact of pests and diseases. In determinate species like spruce, the dllration of the annualheight growth depends on the weather conditions of the growth year, and particularly onweather conditions of the previous year, especially when buds are formed (July to September)(Assmann 1970). This is called the period of shoot elongation. Tree height growth, likegrowth of other living organisms, follows a regular pattern in conformity with the naturallaws of growth (Assmann 1970).Height increment is influenced by a number of factors, including species type and whetherthe species is light demanding or shade tolerailt. Light demanding species reach their maximum current annual increment earlier than shade tolerant species. Trees on good sites haverapid annual height growth until the age when culmination occurs. After this age, heightgrowth slows down during the maturation period until senescence when the growth rate issimilar for both good and poor sites. On the other hand, trees on poor sites do not showrapid early growth, but they maintain slow growth for longer periods. That is, the site treeson better sites culminate earlier than site trees on poor sites. The intrinsic constitution ofindividual trees and their changing positions in the social structure of the stand will cause alot of deviations in height growth. In some species such as lodgepole pine, stand density (theavailable growing space) can affect height increment. Competition during the early years ofgrowth suppresses height increment and delays the culmination age.In conifers, particularly in lodgepole pine, height growth is knowfl to increase with anincrease in site quality and decrease with increasing stand age and rarely with density abovesome level. The effects of stand age and density (often represented by crown competitionfactor) on lodgepole pine height growth were reflected in the density-correlated site indexcurves developed by Alexander et at. (1967). These show that stand density is as importantChapter 2. Background 9a factor as age and site for determining height growth in lodgepole pine.2.1.2 Radial Growth in Forest TreesIt is well recognized that radial growth in trees of a particular species is influenced bymany factors, including climatic fluctuations, site, various stand conditions (including standtreatments), and defoliation (Larson 1963; Mott et al. 1957). Many researchers have completed descriptive studies to examine radial and longitudinal variation in diameter growthin conifers. (See reviews by Larson 1963, Gray 1956 and Assmann 1970). Duff and Nolan(1953, 1957) studied the distribution of radial increment in red pine (Pinus resinosa Ait.).From trees 15 to 30 years of age, records were taken of internodal lengths and the width ofall annual rings at all nodes. When plotted, the assemblage of ring measurements from asingle tree revealed an orderly design of ring widths which was best described in terms ofthree sequences (see Duff and Nolan 1953, Figure 5). Mott et al. (1957) used different namesfor the same descriptions (in parentheses).Type I (Oblique) sequence. This is the longitudinal variation in annual diameter increment. Starting from the tip of the tree, diameter increment (growth) increases to amaximum near the crown base, that is, at the area of maximum branch development(Farrar 1961; Larson 1963). Below this point, diameter increment remains constant ordecreases downwards along the clear bole and somewhere near the base of the bole, itstarts to increase again giving a butt swell. This regular pattern, repeated by successive rings as they are laid down, is attributed by Duff and Nolan (1953) to nutritionalgradients in the axis, arising from the distribution of foliage and incidence of light.Some random extrinsic factors can cause fluctuations in Type I sequences, but in mostcases these are usually masked by the characteristic “pattern” due to intrinsic growthfactors.Chapter 2. Background 10Type II (Horizontal) sequence. This is the radial diameter increment at a given height.Starting from the centre of the tree, this sequence increases towards a maximum in thefirst few rings and gradually declines in successive rings toward the bark with increasingage. Since diameter increment depends on the location along the stem and age, a betterunderstailding of change in diameter increment can be obtained by considering spatialand temporal patterns of increment variation simultaneously.Type III (Vertical) sequence. This sequence consists of annual rings laid down by cambiurn of uniform age, along the stem. It is ullsystematic variation in the mean ringwidth (growth) of a constant number of rings from the pith towards the bark alongthe stem at each internode from the apex downward. This sequence is based on radialincrement of the year in which the leader of the tree was formed. Thus the progressionin time does not involve radial growth or cambial age, but does involve the formation ofsuccessive rings of the sequence in successive years which involves apical activity. Theunpatterned growth in this sequence is associated with configuration variation, factorssuch as site quality and stand density, and randomly distributed variations such aswind and weather type. Tepper et al. (1968) graphically showed that the variationin the Type III sequence clearly expresses the influence of environmental factors oncambial growth (see Figures 3, 6, and 7 of Tepper et aL). Duff and Nolan (1953)also recognized that site quality and stand density have a systematic effect on radialgrowth, producing what is called “configuration” in the Type III sequence, and thatrandom variations, such as weather and defoliation, produce irregular fluctuations inthe growth curve.Farrar (1961) studied the distribution of diameter growth along the stern for trees inseveral positions in a stand. He found that the thickness of the outermost annilal ring ina healthy dominant tree in a stand of polewood size (medium-aged) timber with a mediumdensity varied as follows:Chapter 2. Background 111. The ring was narrow at the tree top, but increased in width with the descent throughthe crown, to a maximum near the branches with the most foliage.2. The annual ring width decreased in the lower crown and for a major part of the bole,and widened again at the base of the tree.3. The annual ring width of a dominant tree followed the same pattern as that of anopen-grown tree, but the open-grown tree had wider rings (more growth) throughoutthe stem.4. A crowded tree was characterized by less cross-sectional growth than the dominanttree, and had a maximum ring width closer to the tip, which is probably due to asmaller canopy, less thickening at the base of the tree, and a less efficient lower part ofthe canopy.Fayle (1973) represented radial increment at variolls heights and ages graphically in atopographic map using contour lines to connect equal diameters increments. Fayle (1973),Thomson and Van Sickle (1980), and Julin (1984) constructed three-dimensional surfacesof diameter increment and area increment which enabled them to see when the conditionswere not favorable for diameter growth (i.e., “troughs” and “valleys” running parallel to theheight axis). Likewise, “ridges” formed when conditions were favorable. Valleys and ridgesare caused mainly by disturbances such as fires, defoliation, climatic variation, silviculturaltreatments and release from nearby trees (Thomson and Van Sickle 1980; Mott et al. 1957and Stark and Cook 1957). Julin (1984) claimed that area increment represents growthmore closely than diameter increment. However, area increment depends on the presentdiameter increment as well as the previous year’s diameter (Assmann 1970). Since presenttotal diameter is the sum of past diameter increments, area increment combines the effectsof present and past disturbances and reflects cumulative size over time. This results inautocorrelation of area increment in successive time periods.Chapter 2. Background 12The form of the Type II sequence at different positions along the stem changes in a fairlyregular manner with increasing height, with most of the growth occurring lower on the stem.In older trees, the Type II sequences near the top of the tree tend to become flat (Clyde1986). Another trend is that Type I sequences tend to become flatter as the tree ages,indicating that diameter increment becomes more evenly distributed along the stern. Atearly ages, most of the diameter increment from the longitudinal series is distributed alongthe upper part of the stern. In all trees examined by Clyde, diameter increment continuedto increase toward the tip of the tree, without decreasing again after reaching a maximum(Type I sequence). This was in disagreement with what was reported by Duff and Nolan(1953, 1957), Farrar (1961) and Fayle (1973). Clyde attributed this result to measurementinterval, length of the live crown, and site quality. She compared three species for diametergrowth and found that lodgepole pine reached the maximum diameter increment at a givenheight fairly rapidly and then quickly decreased (Type II sequence). With white spruce(Picea glauca [Moench] Voss.) and black spruce (Picea mariana [mill] B.S.P.). diameterincrement reached a maximum much later than in lodgepole pine and did not decline asrapidly after reaching the maximum. The difference was probably due to the difference inshade tolerance for the species, since lodgepole pine is shade intolerant while the spruces areshade tolerant. Also, spruce lives three to four times as long.Smith (1980) separated radial growth along the stem (Type I sequence) into early woodand late wood. Heger (1965) postulated that temperature was an important factor governingradial growth and that differences in growth along the stem may be due to air temperaturegradients, but provided no evidence. The different sizes of the early wood and latewoodlayers reflect the respective spring and summer environment energy gradients. Smith foundthat maximum growth of earlywood and latewood occurred near the base of the full-growncrown for Douglas-fir (Pseudotsuga menziesii [Mirb.] Franco.). Koch (1987) found that diarneter growth in lodgepole pine was negatively correlated with stand density, but positivelycorrelated with site quality. He noted that even on significantly better sites, high standChapter 2. Background 13densities will limit the average stand diameter to be achieved at a given age.Differences in diameter increment among trees were observed within species in differentcrown classes (Duff and Nolan 1953; Mott et al. 1957; Clyde 1986). Dominait trees of allspecies are able to maintain growth near the level of the maximum diameter increment ata given height for a longer period of time than codominants, intermediates and suppressedindividuals.2.1.3 Tree Growth Simulation ModelsGrowth modelling is basically concerned with how individual trees or groups of trees changeover time. Growth simulation models are mathematical relationships that quantify tree orforest development over some range of time, conditions, and treatments. The intent andobjectives of a model, availability of good data, philosophy of the modeller, and many otherpractical considerations cause the model structure to vary from one model to another. Nomodel can perfectly represent the real situation in nature being modelled. Therefore, nothingcan be gained from proving that a model is not an exact copy of the real system. However,the model should be able to help the modeller and the user gain a better understanding ofwhat might happen, and help the manager/user to make rational decisions.Some of the more important reasons listed by Goudie (1987) as to why forest growthshould be modelled include:1. to help the modeller and the user to better understand the dynamic processes of treeand stand growth;2. to provide short-term projections of inventory plots;3. to identify critical information needs, such as determining harvest levels/allowable cut;4. to provide means to bridge gaps in field information so that good prediction of futuregrowth and yield can he made; andChapter 2. Background 145. to provide means of investigating the effect of applying alternative silvicultural treatments to the stand.Growth models can be classified as stand- or individual tree-level models (Munro 1974,1984). Stand growth simulation models are usually easy to use aild provide estimates fornatural stands or for certain management regimes. Individual tree growth simulation modelsinvolve modelling each tree separately, by adding increment to simulated tree boles. Theincrements are accumulated over time and then summed to produce stand tables or summaries. These models can either be diameter-growth or height-growth driven. Individualtree simulation models are divided into distance-independent and distance-dependent modelsdepending on whether tree spatial data are used in modelling. This basic distinction betweenindividual tree models relates to how competition among trees is estimated. If the estimation is based on measured or mapped distances from each subject tree to all trees within itsspecified zone of competition, then the model is distance-dependent (see Table 5-1, p. 100of Davis and Johnson 1987). Common examples of these types models include: PTAEDA(Daniels and Burkhart (1975) summarized by Davis and Johnson 1987, p. 141-145), Treeand Stand Simulator (TASS) (Mitchell 1975), and FOREST (Ek and Monserud 1974). If thecrown competition index is based only on the subject tree’s characteristics and the aggregate stand and site characteristics, the model is distance-independent. Common examples ofthese models include: Prognosis (Stage 1973) updated by Wykoff (1984, 1986) and Wykoffet al. (1982), Stems and Tree Evaluation and Modelling System (STEMS) (Belcher et al.1982), and Stand Projection System (SPS) (Arney 1985).Most individual tree growth models handle tree growth modelling by either modellingdiameter growth first and then modelling height growth (diameter-driven models), or modelling height growth first and then modelling diameter growth (height-driven models). Theseapproaches have problems in that total and merchantable volumes have to be estimated separately using other functions, such as separate taper functions, and logical compatibility ofchapter 2. Background 15the height and diameter estimates is not assured. The approach used in TASS (Mitchell1975) of modelling tree crown development and then predicting the resultillg bole changeover time results in a complicated and large model that is difficult to incorporate into existinginventory systems.One possible alternative to modelling tree growth is to model taper over time (i.e., dynamic taper modelling) (Clyde 1986). The dynamic taper modelling of tree growth wouldresult in a growth model that is based on compatible functions for prediction of diameter,height, and volume (merchantable and total). Such a tree growth model could be incorporated into an individual tree growth model that would be relatively simple to understand,easy to calibrate, have a wide application, and be less cumbersome to handle. It would alsobe compatible with existing inventory information, as well as being portable (require lesscomputer power and storage).During tree growth, trees experience change in height and diameter. However, growth indiameter for a given tree is not proportional in all sections, consequently differences in taperand form result. Modelling tree growth using taper functions requires an understanding ofthe theory and manner in which tree form and taper changes along the tree at one time andover time.2.2 Variation in the Form and Taper of TreesStem form and taper have been studied for over a century now and still appear to be highpriority subjects in forest research. There are several possible reasons for this. First, no singletheory has been developed that adequately explains the variation in stem form, both withinaild among trees. Thus, it has not been possible to develop a satisfactory taper functionthat is consistently best for all estimated tree dimensions. all tree species, and uniformlyacceptable over a wide range of geographic conditions. Second, and more important froma practical point of view, a taper function that can accurately predict the diameter at anyChapter 2. Background 16given point on the stem from a few easily measured variables is essential for estimating thevolume of standing trees and constructing volume tables to different merchantable limits(Newnharn 1988). Most taper models have been based oniy on dbh, total height, and someheight above the ground; as a result, some noticeable bias still exists, particularly for buttand top sections (see Newnhani 1988, Figures 10 and 11; Perez et al. 1990, Figure 2). Thismeans that more information, either about the tree or about the stand and the site, shouldbe taken into account in taper estimation. That is, the one or two readily measured variablesare not the only essential variables for estimating the taper of standing trees.Taper and form have been often used interchangeably. However, based on Gray’s (1956)stem profile model, H — h = d2, (where H is the total tree height, h is any height aboveground along the main stem of a tree, d is the diameter inside bark at height h, and c isan unknown parameter for the maui stem of a tree), stem form and taper (profile) havespecific meanings. Form is the characteristic shape of the solid as determined by the powerindex of d in Gray’s stem profile model, while taper or stern profile is the rate of decrease(narrowing) in diameter (d) over a specified length or height. Therefore, tree taper is therate of decrease or narrowing in stem diameter with an increase in height up the stem of agiven tree form. Two sections of trees of the same form can have different taper and twosections with different forms can have same average taper (i.e., the same end cross sectionalareas and the same length).According to Gray’s (1956) model, form is expected to be constant, while taper variesby tree. However, both taper and form should vary by tree. That is, form is the geometricshape of the tree stem. The shape may he regular, as for a solid of revolution, or morecommonly irregular. The form may be measured by the form factor such as the ratio of thevolume of the tree to that of a cylinder of equal basal cross-sectional area and height. Thisratio depends on the bark thickness and the taper of the stem. The greater the taper, thesmaller the form factor. The degree of taper depends on diameter increment in differentparts of the stem. Even if a tree or a log has an estimated form factor of 0.5 (equal to thatChapter 2. Background 17of a paraboloid), concluding that the form is that of a paraboloid is unjustified, as the formis probably irregular and not constant over the entire stem.Trees have often been considered to be composed of three sections (Spurr 1952; Husch1963): a collical top section (which includes the crown), paraboloidal sections below the livecrown, and a neiloidal butt section. Osawa (1992) divided the tree into four sections fromthe apex: a cylinder at the top, then a cone, a paraboloid, and a swollen base. Within thesesections, various irregularities occur in tree form. These are caused mainly by: (1) an abruptchange of diameter at a node; (2) a deformity after injury to the cambium; (3) an abruptchange of diameter associated with heart rot; (4) a swelling from occlusion of branches; or(5) the influence of root swell, buttress, or stem flutes. These irregularities vary with standdensity, species, site, age, and other variables (Gray 1956; Larson 1963).Differences in tree taper can be summarized by the following statements.1. Above the region of butt swell, the greatest taper occurs in that portion of the stemwithin the live crown. Both ring width and rillg area increase with an increase indistance from the top of the stem, indicating that the stem is probably conical (oreven neiloid) in form in lower parts of the stern below the crown (Larson 1963, 1965).2. The maximum growth of the annual ring area occurs near the base of the crown, andthe minimum occurs at some point between the maximum butt swell and the base of thecrown. Both the minimum and maximum move upwards in dry years and downwardsin excessively wet years (Gray 1956; Larson 1963, 1965).3. Below the live crown, the rate of diameter growth is largely governed by the positionof the tree within the crown canopy. For free-growing trees, ring area may continue toincrease down the stem. For trees in the upper canopy, ring area may remain constantso that ring width will consequently decrease. Both ring width and area decline downthe stem for suppressed trees (Duff aid Nolan 1953 Larson 1963, 1965).Chapter 2. Background 18Four theories have been derived to explain the differences in form and taper of trees.These theories will be described iii detail in subsection 2.2.1. Factors affecting form andtaper changes will be outlined in subsection 2.2.2.2.2.1 Theoretical Bases for Tree Form and Taper VariationLooking at a tree as a system with only three components, leaves, shoot system (branchesand main stem), and root system, ca be helpful in understanding how a tree functions,but sheds little insight about tree form and taper. The form of trees results from theirdesign as the tallest and longest-lived plants. They have a large number of parts that arehighly organized (Wilson 1984). They have refined internal anatomy for physical strengthand to allow physiological processes to proceed efficiently. To try to explain the variabilityof tree shape (form), four theories have been put forward (Gray 1956; Larson 1963, 1965;and Assmann 1970): nutritional, water conduction, mechanistic, and hormonal.2.2.1.1 Nutritional TheoryThe nutritional theory was first proposed in 1883 by Hartig (referenced in Larson 1963). Heenvisioned stern growth in terms of an equilibrium between transpiration and assimilation.Transpiration was assumed to be the primary factor deterniining the amount of conductivetissue or early wood in sterns. Therefore, a large tree with high transpirational requirementswould produce a strongly tapered stem with a high proportion of early wood to satisfy theserequirements. As the crown decreases, transpirational demands also would decrease andboth the total amount and the proportion of early wood in the growth increment wouldbe reduced. Suppressed trees have extremely low trauspirational requirements, resultingin little or no early wood in the lower bole, and thus have less taper. Thinning increasescrown size which in turn increases transpirational requirements; therefore, thinning increasesearly wood and taper. Pruning decreases crown size which in turn decreases transpirationalrequirements and decreases early wood and taper.Chapter 2. Background 192.2.1.2 Water Conduction TheoryUnlike Hartig (1883, referenced in Larson 1963), Jaccard (1912, 1919 referenced in Larson1963) held a strong quantitative and mechanistic view about water conduction. He considered the development of the crown (i.e., the organs of transpiration and sunlight absorption),and roots (i.e., the organs of water absorption) to be related and proportional in their development (growth). Thus, the tree stem size was assumed to be determined by the requirementfor water conduction. This means that a cylindrical bole would be required for equilibriumwater transport between crown and roots. This theory rests on the assumption that cross-sectional area growth is uniform over the branch-free bole. Butt swell was explained bycomparison to a capillary system. This theory assumed that any expansion in crown sizeresulted in increasing the root system. These changes in crown size resulted in a change inthe stem form due to an adjustment in relative water conduction area for the new crownsize. This theory has a problem in that trees growing on poor sites have been found to havea very high proportion of biomass in the root system compared to trees growing on goodsites (Kramer and Kozlowski 1979), yet trees on good sites may have higher taper.2.2.1.3 Mechanistic TheoryStems carry equal resistance to bending stresses imposed by the wind. Metzger (1894), Gray(1956) and Wilson (1984)), recognized two mechanical forces that influence the erect stem:1) the vertical force consisting of the weight of the stern itself plus the additional weightof winter snow and ice, and 2) the horizontal force imposed upon the stem by the wind.Wind was accorded the greatest attention in developing the theory, since it was thought todetermine the form and quality of growth on the stem.The weight of the stem and subtending branches contribute to stem form (i.e., the effectof its own weight). Based on Metzger’s (1894) d3 rule, which says that the crown size isrelated to cubic diameter (d3) (where d is diameter at a distance h from the ground), treeChapter 2. Background 20form should follow static laws since the stern is a carrier of the crown and must also resistexternal forces applied to it. A difficulty for all stern form theories is the predictable variationin form due to wood constituents, tree age, nutrition, water supply and pressure relations ofthe cambium.Gray (1956) demonstrated that the dimensions of the main stem conform to a quadraticparaboloid, d2/h, where h is height along the stern above the ground, and d is diameter at h,rather than to Metzger’s (1894) cubic paraboloid. A stem of this shape would be consistentwith the mechanical requirements of a tree in regards to, not only horizontal wind-pressure onthe crown, but also to other forces acting on the stem. Anchoring of the stem was suggestedas the possible cause of butt swell. Wind was taken to be the prime factor determining theform and distribution of growth on the stem. According to the theory, the stem carries equalresistance to bending, and the force of the wind responsible for the bending action is greatestat the center of gravity or midpoint of the crown.Stem form or taper is dependent upon variations in total height, as well as stem diameter.Density changes within limits do not usually affect height growth, but they do affect diameter.Thus, density impacts on taper as the diameter/height ratio changes. Regardless of theintricacies and subtle effects of wind and light on the diameter/height ratio, it is evidentthat the final tree height resulting from variations in stand closure has a profound influenceon the taper of the individual tree stems.2.2.1.4 Hormonal TheoryAccording to Larson (1963), the nutritional theory accounts for much of the variation in stemform, as well as the distribution of earlywood and latewood in the growth rings. However,various studies have shown that plant growth hormones (auxins) produced in tree apicalareas, activate growth when transported to other parts of the tree. Also, growth is knownto fail when hormone-producing tree parts are removed (Wilson 1984; Larson 1963). Thehormonal theory bridges the gaps of the other theories. The water conduction theory holdsChapter 2. Background 21oniy for ideal stems; stem form is explained on a functional basis, not from the physiologicalaspect. The mechanistic theory is a functional concept also, and although this theory adequately interprets stern form within reasonable limits, it in no way provides a physiologicalexplanation for the observed facts. However, the hormonal theory on its own cannot explainall the tree form and taper changes because hormones just carry out regulatory functions.2.2.1.5 Comparison of TheoriesEach of the stern form theories appears to explain some aspects of stern form variability. Thismeans that they are either all applicable under certain conditions or, more likely, there areparts of these theories which hold some truth. All the theories seem to include the followingtwo points:1. Butt swell is very variable, but appears to have a support function.2. Crown size, particularly crown length, is the most important single factor which determines tree form or taper. It plays a decisive role in determining the stem form.Larson (1963) and Kramer and Kozlowski (1979) felt that the more recently developedhormonal theory does not supplant earlier developed stem form theories, but may he considered as an adjunct to them by providing the physiological basis. Kozlowski (1971) statedthat the formation of wood along the stem is governed more by the physiology of the treethan its strength requirements; the fact that the stem is also mechanically efficient may hefortuitous. However, under most conditions, tree stems are known to respond to stress with,for example, the formation of the compression wood (Larson 1963).Most of the theories that describe stern form are in qualitative terms. Only the mechanistic theory, largely developed by Metzger (1894; referenced by Larson 1963 and Assmann1970) and subsequently modified by Gray (1956), attempts to develop a functional relationship between stem diameter and height. Because Metzger postulated that wood formationChapter 2. Background 22in the stern was governed by its requirement for strength, he viewed tree form mechanicallyas a cantilever beam of uniform resistance to bending. That is, he described the stem as abeam of uniform resistance to bending (particularly to forces brought about by wind), withone end fastened in the soil. Such a beam would have the form of a cubic paraboloid. Hewas able to show that, below the centre of gravity of the crown, diameter along the treestem, cubed (d3), plotted against height (h) above ground along the stem was more or lessa straight line. Gray claimed that the cubic paraboloidal form represented an overexpenditure of material for the strength requirements of the stem since the stem was firmly heldat its base. A quadratic paraboloidal form, in which diameter squared is linearly correlatedwith height (Ii), would be more efficient. Newnham (1965) and Bllrkhart and Walton (1985)found that the d2 against h relationship held well for that portion of the stem between 15and 80 percent of the total height and used it to study the variation in taper with age andthinning regime in coniferous species. It should be noted that two of the most commonlyused formulae for calculating log volumes (Smalian’s and Huber’s equations (Husch et al.1982, p. 101)) assume that the stem has the form of a quadratic paraboloid.2.2.2 Factors Affecting Form and Taper VariationGenerally, variation in tree form and taper is caused by differences in (1) site characteristics(e.g., water, nutrient, weather, etc), and (2) tree characteristics (e.g., age, crown size, canopyposition, species) and stand characteristics (e.g., density, stand age, etc).2.2.2.1 Site CharacteristicsGrowth of tree crops and their productive performance over some unit of time depends on thesite capability. Site influences stern form and taper through its effects on crown development(Larson 1963, 1965). Trees on poor sites (i.e., trees growing on sites deprived of nutrients orlackiiig water) show the greatest taper and least desirable forms (Metzger 1895, see Larson1963, p. 9). Smith and Wilsie (1961) found that the annual increment along the stemChapter 2. Background 23increased downwards (increased taper) in wet periods and decreased downwards (decreasedtaper) in dry periods. These difference ill stern forms and taper can be traced to the well-known growth relationship with site quality. Height growth diminishes for trees of the samediameter as site quality decreases, thus increasing stern taper and changing stem form.According to Kunz (1953 referenced in Larson 1963, p.9), trees growing on poor sitesrepresent an exception to the general rule of taper changes. Although much of the variationin stem form can be assigned to differences in height growth, the relative distribution ofdiameter growth on the stem does vary widely with site quality. On good sites (i.e., sitesthat are sufficiellt in nutrients and available water) growth is concentrated in the uppercrown of the lower stem classes, whereas oi poor sites the growth tends to be more uniformlydistributed along the bole. Newberry and Burkhart (1986) found both taper and form todecrease as the combined crown ratio-site index term decreased. In some exposed areas, windhas the effect of reducing the increase in height per unit of volume as the tree gets older andincreasing the taper. Newnham (1965) used Gray’s (1956) taper function to show that siteindex and stand age have no significant effect on stem taper for Douglas-fir. However, thislack of effect has not been investigated thoroughly.According to Smith (1980), trees that grow rapidly have a greater degree of taper thantrees that grow slowly, due to differences in the distribution of the woody material throughoutthe growth ring. Young trees growing in the open on good sites will have a greater annualring growth rate (cambial division) from the tip through the crown, and thus will have agreater degree of taper which results in a lower form factor.Tree form is highly related to the environment (Larson 1963). Dry interior pines, compared to the wet coastal Sitka spruce (Picea sitchensis [Bong.] Carr.) and true firs (Abiesspp.), tend to have short. compact, rounded, and bushy crowns. This type of form is thoughtto be related to the high temperature and moisture stress of their environmeiit. Tall excurrent growth would severely expose the crown to strong, dry winds.Chapter 2. Background 242.2.2.2 Tree and Stand CharacteristicsIn young trees, the greater height growth and the steeper slope of the Type I sequence in theupper part of the stem result in a conic form (more taper). As height growth declines, a moreconstant diameter increment is added over most of the stem with increasing age (i.e., theType I sequence approaches an asymptote), except at the base and the tip. As the tree getsolder, the main bole does not change as mllch and it becomes more cylindrical. However, thebutt swell becomes more pronounced because there is still an increase in diameter incrementnear the base of the Type I sequence, even in older trees. In lodgepole pine, the Type Isequence becomes constant over most of the stem earlier than in spruce and other shadetolerant species, so that the main stem becomes less tapered (Larson 1963; Newnham 1965).In addition to size and distribution of live tree crowns, species type has much to dowith stem form. Some tree species have less taper and more cylindrical boles even whenthey are notably dominants and codominants. Such trees have high form factors (i.e, arecylindrical in shape). Gray (1956) explained the difference due to dominance by suggestingthat dominance is characterized by relatively greater diameter than height growth. Whentwo trees with the same diameter and height but different crown length are compared, theone with the longest crown will exhibit the greatest taper on the lower stem.Some insight into the development of stem taper in lodgepole pine trees can be gainedby examining the diameter increment trends using the Duff and Nolan (1953, 1957) conceptdeveloped for red pine. Koch (1987) noted that lodgepole pine is noted for its minimal taperamong the conifers. Butt swell in conifers occurs over a period of time when the diameterincrement at any point near the base of the tree is greater than diameter increment at anotherpoint above it. Lodgepole pine develops less butt swell and this butt swell takes longer todevelop than in the spruces (Clyde 1986). Thus, more shade tolerant, longer crowned species,such as the spruces, will have a more pronounced butt swell and taper (Larson 1963; Clyde1986) than lodgepole pine. This is mainly attributed to the influence of crown size.Chapter 2. Background 25There is a diversity of opinions as to the relative role of heredity in tree form (Larson1963). However, trees of the same species growing in the same environment may vary in formand taper if some become dominants and others are suppressed. Trees of the same speciesgrowing under identical conditions are believed to have the same stem form and taper atequal relative tree heights, but forms and taper will differ at different absolute tree heights.Trees of different species may have different forms and taper. Many authors (e.g., Metzger1896; Petterson 1927; Fischer 1954 as referenced in Larson 1963 and Gray 1956) observedthat trees of a single stand with the same species also tend to show differences in form, whichis mostly attributed to genetic difference. Dominant trees are taller and larger in diameterand take a more conical form than the intermediate and suppressed trees, which tend to hemore parabolic. This would seem to indicate that the simple shapes assumed by the volumeestimation systems (cone-paracone1-parabola) are approximations whose performance willvary within a species (Reed and Byrne 1985).The strong selection pressures of snow, ice, and wind force trees to adopt a conical form ofgrowth. This excurrent tree form is an expression of strong apical control (Assmann 1970).MacDonald and Forslund (1986) examined the geometric form of five species (dominantsand codominants only): balsam fir (Abies balsamea (L.) Mill.), black and white spruce,white birch (Betnia papyrifera Marsh.), and trembling aspen (Popnlns tremuloides Michx.).From their analysis, they concluded that balsam fir was close to a paracone in form, blackand white spruce and aspen more parabolic, and white birch more conical in form. However,from Figure 1 on page 312 of MacDonald and Forsiund (1986), it appears that the formsassigned to the species result in large biases at the base and toj) of the tree. This shows thatthe shapes change from the base to the tip of the tree.Valinger (1992) studied the effect of wind sway on stem form and crown developmentof Scots pine (Finns sylvestris L.). He found wind to be an important factor influencing‘A paracone is a term introduced by Forsiund (1982) to mean a tree form which is neither paraboloidnor a cone but somewhere in between.Chapter 2. Background 26the distribution of radial growth in scots pine. Increased bending stresses imposed by windpromoted unequal diameter growth for different parts of the stem, with more diameter growthin more stressed parts of the tree. This led to different stem forms for different stem parts,with free-swaying trees having more taper and the stayed trees haviig less taper.Tall open-grown trees with deep live crowns have conical shapes and high taper, whereastrees grown in dense stands (stand-grown trees) with shallow crowns and trees that have beenhighly pruned tend to have low taper. As the stand closes and natural competition sets in,the lower branches die and a progressively longer branch free bole is produced with a decreasein taper. According to Gray (1956), it is evident that trees become more cylindrical in formwith an increase in stand density and a decrease in crown length. This is in accordance withPressler’s growth laws (i.e., tree growth increases from the tree tip to an area of highestbranch length jllst near the base of the crown and then equal increment is distributed tothe rest of the stern) (Mitchell 1975). Suppressed trees have narrower growth rings withdiameter growth decreasing from upper to lower stem parts, sometimes with missing rings.Stem form is a composite reflection of both stand density and canopy position.Depending on the canopy position of the tree, the form and taper of the tree changeswith time (Gray 1956, p. 47-50). A dominant tree, if it remains dominant, will increase itstaper which likely results from change in form due to increased crown size. An intermediateor suppressed tree, if it remains in this canopy position, will have a relative decrease or nonoticeable change in taper over time. However, trees are expected to change in form andtaper with age, because, at a young age, the social structure of the stand is constant (i.e.,no dominance or suppression before intra-specific competition begins). If a suppressed treeincreases in height with time while its crown size decreases relative to height, the crown ratiodecreases, giving a tree more cylindrical in form with less taper.Newnham (1958), in his studies of form and taper of forest-grown and open-grownDouglas-fir, hemlock (Tsnga heterophylla [Rat.] Sarg.) and western red cedar (Thuga plicataDonn.), found that most open-grown trees are conical in shape (i.e, they have a conicalChapter 2. Background 27form) and most forest-grown trees are neiloidal in form from ground up to about 15 percentof the total height. From 16 to 80 percent of total height, forest-grown trees are quadraticparaboloidal in form, and conical in form for the last 20 percent of the total height from theground. Using a limited subsample of hemlock, Newnharn also found that taper increasesthroughout the lifespan of a tree as long as the tree maintains a dominanting canopy position.However, as soon as it changes canopy position (i.e., becomes dominated by the surroundingtrees), its taper will begin to decrease or remain constant with increasing age depending onthe new canopy position it acquires. Osawa (1992) found stand grown trees to have fourdistinct sections (cylinder, cone, parahola and swollen base). However, he found a variationfor open-grown and young trees which lacked the parabola and the swollen base sectionsrespectively. Smith (1965) indicated that open-grown trees have a cylindrical form factor ofabout 0.33 compared to 0.40 for forest-grown trees.Figure 4 of Baker (1950) shows the differences in depositing woody material in ponderosapine (Pin’us ponderosa Laws.), between an open-grown tree with a long live crown, and anintermediate forest-grown tree about twice the age of the first, having a live crown extendingonly about one-third the length of the stem. This figure demonstrates the striking differencein pattern of diameter increment along the stern between an open-grown and a forest-growntree. Since open-grown trees often retain their live branches close to the ground (i.e., theyoften have nearly 100 percent crown length), the degree of taper of an open-grown tree isalways greater than that of a forest-grown tree. Taper is affected by crown size which inturn depends upon whether the tree is open-grown or forest-grown.Smithers (1961) noted that boles of open-grown lodgepole pine trees taper noticeably inan almost conical form. In extremely dense stands, the stern is whiplike and hardly thickerat ground level than at the top. He further observed that in more mature stands, densityaffects appearance, so that in very dense stands (for example, 25,000 stems per ha at 90years), trees have very little taper and are rarely over 6 mel ers in height and 76 mm indbh. At medium densities (2,500 to 7,500 stems per ha), the form class, defined as the ratioChapter 2. Background 28between breast height diameter outside bark and the diameter inside hark at the top of thefirst 16-feet log (Koch 1987), is high, averaging 70 to 75 percent. In low density stands (250to 2,500 mature trees per ha), the bole has considerably more taper and a form class of65 to 75 percent. In even-aged forests, taper tends to decrease after canopy closure, beforeleveling off and, thereafter, altering very little. The change in taper at the time of crownclosure follows changes in the depth of the live crown (Newnharn 1965; Larson 1963).Larson (1963, 1965) pointed ollt that most variations in bole form are attributed tochanges in the size of the live crown, its distribution along the stem, and the length of thebranch-free hole. In studying the effects of branch length on diameter growth of loblolly pine(Pinus teada L.), Labyak and Schumacher (1954) found that the number of branchlets andthe location of a single branch on the bole determined its contribution to diameter growthof the main stem. Kozlowski (1971) cited a variety of studies which verify the concept thatcrown size affects radial growth below the crown, but has less effect on growth within thecrown. Burkhart and Walton (1985) used Gray’s (19.56) taper function to find the importanceof the crown in explaining tree taper. From their investigations, it can be concluded that thecrown size measure (crown ratio) of loblolly pine trees in unthinned plantations is related toform and taper parameters of the models of Kozak et al. (1969), Gray (1956), and Ormerod(1973).The major stand treatments which will alter both tree crown size and average standcanopy closure, as well as the stand density, include thinning, pruning and fertilization.Thinning reduces stand density and allows individual trees more space to expand theircrowns. For heavily thinned stands, trees grow like open-grown trees which means they havehigh taper and conical shapes. Pruning does the opposite (Larson 1963, p. 15—17, 1965).Pruning a tree reduces the crown size for a given height, which is similar to increasing thestand density. Thus, pruning decreases taper and makes trees look more parabolic in shape.Fertilizing a tree increases tree vigour; the tree will put on more branches, increase its crownsize, and will have a higher taper as a result. However, taper was found to be only slightlyChapter 2. Background 29affected by heavy fertilization by Thomson and Barclay (1984). Similar results were found byGordon and Graham (1986) working with radiata pine (Pinus radiata D. Don) and Tepperet at. (1968) working with red pine.2.3 Tree Form and Taper EstimationTaper estimation functions can be divided into two major divisions, static or dynamic. Astatic taper model is a model which predicts the diameter along the tree stem at a particulartime. Whereas, a dyllamic taper function is a model that predicts the changing diameteralong the tree stem over time.2.3.1 Static Tree Taper EstimationMost of the literature on taper modelling is characterized by attempts to model static taper(e.g., Behre 1923; Kozak et at. 1969; Matte 1949; Ormerod 1973; Amidon 1984; Walters andHann 1986). The taper of a tree can be characterized by measuring diameter at successivepoints along the stem. If sufficient measurements are taken, it is possible to develop averagetaper tables which show estimated diameter at chosen heights along the stem (Spurr 1952).The intent is to portray the actual form of the trees which can be used in the calculation ofvolume.An alternative to taper tables is taper equations. Taper equations or curves are functionsfor estimating stem diameter at a given height from known variables such as dbh, height,distance from the tip, and crown size. While tree physiologists have been trying to discover asatisfactory theory for the complex stem form, forest mensurationists have developed mathematical functions that describe the stem profile from the ground to the tip.Stem taper (profile) functions are used to provide (Kozak 1988):1. predictions of inside hark diameters at any point on the stem;2. estimates of total tree stem volume;Chapter 2. Background 303. estimates of merchantable volume and merchantable height to any top diameter andfrom any stump height; and4. estimates of individual log volumes.The desirable features of any taper function are that it should be possible to directlyestimate height for any stem diameter (useful for determining merchantable height to a givenupper diameter limit), and that the taper function should be capable of being integrated togive a compatible volume function. If neither of these conditions is fulfilled, time-consumingiterative procedures have to be used. Taper functions which do not integrate exactly suchas Kozak (1988) and Newnham (1992) may be used if they improve volume estimation.Munro and Demaerschalk (1974) discussed the advantages of compatible volume and taperfunctions. The usual approach is to develop the taper function first and then the volumefunction. However, some modellers (e.g., Demaerschalk 1973; Amateis and Burkhart 1988;and Alerndag 1988) have proceeded in the opposite direction by deriving taper functionsfrom existing volume functions.Different static taper functions have been developed. These can he divided into fourmajor groups, depending on the philosophy of the modellers, as follows:1. Simple single functions describing diameter change from ground to top (e.g.. Hojer 1903and Jonson 1911 referenced in Gray 1956 and Larson 1963; Behre 1923; Matte 1949;Kozak et al. 1969; Demaerschalk 1972 and 1973; Amidon 1984; Rustagi and Loveless1991). These are relatively easy to fit, and generally they do not require extensivecomputer capabilities when applied (Alberta Forest Service 1987).2. Different equations are used for various parts of the stem and joined in such a way thattheir first derivatives are equal at the points of intersections (e.g., Max and Burkhart1976; Demaerschalk and Kozak 1977; Bennett et al. 1978 Cao et al. 1980; Walters andHann 1986; Ormerod 1986; Flewelling and Raynes 1993: Flewelling 1993). These areChapter 2. Background 31sometimes called segmented taper equations. They require special fitting approachesand more computer power than single equations.3. Variable-form functions, which are one continuous function describing the shape of thebole, with changing exponents from ground to top to compensate for the changingshape from neiloid at the base of the tree, to paraboloid in the middle of the stem,and to a cone in the crown (e.g., Newberry and Burkhart 1986; Kozak 1988; Newnham1988, 1992; Perez et al. 1990). These functions can be considered a modification ofsimple taper equations.4. A more recent approach to taper function modelling is the use of simultaneous equations, mixed linear models in a polar co-ordinate system, and other methods (e.g.,Sloboda 1977; Kilkki et al. 1978; Kilkki and Varmola 1979, 1981; Lui 1980; Lappi1986; Ojansuu 1987; Swecla 1988).Table 2.1 is a summary of taper functions for each of these groups. Some of these functionswill be described in detail.2.3.1.1 Simple Taper FunctionsThe earliest efforts to express tree taper by mathematical functions began with relativelysimple formulas, similar to the taper equation given by Husch et al. (1982, p. 99):y=K./ (2.1)where y is the radius of the stem at distance x from the tip of the tree; K is a constant fora given form (i.e., K =.-fi where Rb is radius at the base of the tree and H is tree totalheight); and r is a form exponent which changes for different geometric solids. When r is1. a paraboloid is obtained by rotating the curve of this equation around the x axis, whenr is 2.. a cone is produced, when r is 3, a neiloid is produced, and when i’ is 0, a cylinder isChapter 2. Background 32Table 2.1: A summary of some static taper models developed to date.Simple Taper Segmented Taper Variable-Form or Other Taper ModelsEquations Equations -Exponent Taper (including MixedEquations Linear Models,____________________Splines, PCA)Hojer (1903) Heijbel (1928) Reed and Byrne Fries (1965)Jonson (1911) Ormerod (1973, (1985) Fries andBehre (1923) 1986) Newberry and Matern (1966)Matte (1949) Max and Burkhart (1986) Kozak andGray (1956) Burkhart (1976) Newnham (1988, Smith (1966)Bruce et at. (1968) Demaeschalk and 1992) Sloboda (1977)Kozak et at. (1969) Kozak (1977) Kozak (1988) Kilkki et at. (1978)Demaeschalk (1973) Bennett et at. (1978) Perez et at. (1990) Liu andBennett and Cao et at. (1980) Keister (1978)Swindel (1972) Byrne and Kilkki andMunro and Reed (1986) Varmola (1979, 1981)Demaeschalk (1974) Walters and Liu (1980)Goulding and Hann (1986) Nagashima et at.Murray (1976) Czaplewski (1989) (1980)Clutter (1980) Czaplewski et at. Biging (1984)Forslund (1982) (1989) Brink andGordon (1983) Flewelling (1993) voi Gadow (1986)Laasasenaho (1983) Flewelling and Lappi (1986)Amidon (1984) Raynes (1993) Kilkki andReal and Lappi (1987)Moore (1987) Nagashima andAlemdag (1988) Inada (1987)Allen (1991) Ojausuu (1987)Forsiund (1991) Nagashima (1988)Rustagi and Sweda (1988)Loveless (1991) Real et at. (1989)Allen et at. (1992)ap is principal component analysis.Chapter 2. Background 3:3PARABOLOIDr=1NEILODr=3Figure 2.1: The geometric division of a tree stem (Db (2Rb) is the diameter at the tree base,d is the diameter at height (h) above ground, and the other symbols as defined for Equation2.1).produced (Husch et al. 1982, Figure 8.1). However, trees are rarely cones, paraboloids, orneiloids, but generally they are a combination of all of them (see Figure 2.1).In 1903, Hojer (quoted by Behre 1923) developed the first recorded taper equation, toexpress the diameter of a tree at any point on the stem based on the measurements of Norwayspruce (Picea abies (L.) Karst). The function was of the form:d a+x—=Clog +e (2.2)D awhere d is the diameter inside bark at height h above ground or at distance x from the treetip; D is dbh C and a are constants; and e is the error term.In 1923, Behre used ponderosa pine data to determine whether Hojer’s (1903) equationcould be improved by introducting a new term to better fit the conditions, or if a different.equation could be found which would more nearly describe the average form of tree stems.As a. result, a new equation was developed, which more consistently described the shape.The equation was hyperbolic in nature as follows:CONEr=2Chapter 2. Background 34d a.+e (2.3)D a+bxwhere constants a and b vary with the form quotient., but in all cases a + b = 1. Conformityto the equation is indicated if a plot of (-) on .x produces a. straight line. The constantscan be found graphically or by least squares.Kozak et al. (1969) developed a taper function based on the assumption that the treebole is a quadratic paraboloid. The equation was a second order quadratic polynomial:(d)2= b0 + b1Z + b2Z + e (2.4)subject to the restriction b0 + b1 + b2 = 0, where Z = h/H; b0, b1 and b2 are constants; h isheight above ground to diameter, d; and H is total tree height. For some of the species onwhich the model was tested, coastal Sitka spruce and western red cedar in British Columbia(B.C.), negative estimates of upper stem diameters were obtained. For those species, thefollowing conditioned function was recommended:(d)2= b1(1 — 2Z + Z2) + e (2.5)This equation showed some systematic bias2 for both butt and tip sections of the tree fordiameter estimation (see Table 3 of Kozak et al. 1969).Amidon (1984) used bole diameter and height measurements to predict volume to avariable top diameter accurately and precisely for several conifer species. Since he did notentertain the idea of multiple bole sections. Amidon used a single equation to predict taperabove breast height and estimated the volume for the section below breast height usingSmalian’s formula. The taper function was:d = b1(D xH bh +b2(H — h2)(h — bh)/H2+ e (2.6)2For this thesis, bias is defined as the difference between the actual (or measured) observation and thepredicted value. If the bias is positive it means the variable (e.g., diameter) is underestimated and whennegative it means the variable is overestimated.Chapter 2. Background 35where d is the predicted diameter inside bark at height h; bh is breast height; and b1 andare regression coefficients to he estimated. Seven alternative models were selected forcomparing diameter predictions with the above model. The hyperbolic taper function byBehre (1923) and a segmented function by Ormerod (1973) were immediately discardedbecause both had standard errors more than twice that of Amidon’s model. The remainingfive models were by Bennett and Swindel (1972), Demaerschalk (1972), Kozak et al. (1969),Max and Burkhart (1976) (a segmented model), and Bruce et al. (1968). From Table 2 ofAmidon (1984), it can be seen that Amidon’s model was the best, followed by the modelsof Max and Burkhart, Bruce et al.. Demaerschalk, Bennett and Swindel, and Kozak andothers.Rustagi and Loveless (1991) developed simple, fully compatible, cubic volume and stemtaper equations based on individual trees. They developed equations to predict a cylindricalform factor based on the ratios between total height above breast height and total heightusing varying form values (1/2, 2/3, and 3/4), and showed that it is related to the geometrictheoretical sections of Figure 2.1 and therefore to stem taper. Based on Pressler’s (1864)theory (referenced in Loetsch et al. 1973), the tree stem was assumed to be divided intotwo parts, the portion above breast height (BH), and the portion below breast height (buttsection). Taper and volume equations were developed for the above breast height section andthe section below breast height was approximated as being a cylinder. Modifying Ormerod’s(1973) taper function, Rustagi and Loveless’ taper function became:H-ld1 = (co + ciD) H — 1.37 + e (2.7)where d1 is the diameter at height 1 above breast height; c0 +c1D is a function for predictingdiameter inside bark at breast height, where c0 and c1 are coefficients; and p was the formexponent computed from form factor function (p = b0 + b1R), where R represented theheight ratios of H — 1 to H — 1.37 at 1/2D, 2/3D and 3/4D. Based on validation datafor 20 Douglas fir trees, Rust agi and Loveless compared the performance of their taperChapter 2. Background 36model with that of Kozak’s (1988) variable exponent taper function and Walters and Hann’s(1986) segmented taper model for volume prediction. From Figures 3 and 4 of Rustagi andLoveless, this taper model appears to have out-performed both Kozak’s, and Walters andHann’s models. It was also noticed that Kozak’s taper function tracked the actual stemprofile more accurately than the other two for lower form factors (less than 0.4), but as theform factor increased Kozak’s taper model became the poorest. Since Rllstagi and Loveless’compatible stem profile has no inflection points, it can model change in geometric shapesalong the stem, but its form exponent is continuous, meaning that the stem form changesas height increases from the ground. This taper model has a weakness in that it assumesthat all trees are cylindrical below breast height. If butt swell extends above breast height,this model will overestimate diameters in lower part of the stem above breast height andunderestimate diameters for upper sections towards the tip of the stem. Another weaknessis that the R are needed and these have to be measured.The earliest and the majority of taper functions developed to date belong to the categoryof simple taper models (see Table 2.1). However, the unsatisfactory predictions by thesemodels have kept many taper modellers looking for better models to improve prediction.This has lead to development of more complicated models, such as the segmented and othertypes of taper functions.2.3.1.2 Segmented Taper FunctionsAs pointed out by Heijbel (1928), Grosenbaugh (1966), and other researchers, stem profilemay be best modelled in sections. Grosenbaugh observed that tree stems assume an infinitenumber of shapes and that it was difficult to develop a single, accurate equation to describethe taper of the stem. He said that each stem had a number of inflection points and thetraditional conoid, paraboloid, and neiloid are merely convenient instances in a continuumof short monotonic shapes, increasing from tip to stump, with a possibility of having manyinflection points along the stem. Several authors have recognized that other geometric formsChapter 2. Background 37are possible. The form of a tree stem does not change abruptly from one geometric form toanother; it is continuous, as has been recognized by taper modellers such as Kozak (1988)and Newnham (1988).The introduction of computers in forest research in the early 1960s and the increasedavailability of appropriate software, coupled with the failure of simple taper curves to tracethe multiple inflection points along the stem bole or failure to fit the swollen butt logs orboth, led to the development of complex taper functions. Segmented taper models describeeach of several sections of a tree bole with separate equations. The method commonly usedto describe these shapes is to fit each section with a simple or polynomial equation, usuallyquadratic, and then mathematically provide conditions for a continuous curve at the twojoin points of the segments (Max and Burkhart 1976; Demaerschalk and Kozak 1977: Cao etal. 1980; Ormerocl 1973, 1986). Studies by Cao et al. (1980), Martin (1981, 1984) Amidon(1984), and Walters and Hann (1986), and others, have shown that complex taper equations,such as segmented taper functions, provide better fits of the stem profile than simple tapermodels, especially in the high volume butt region.Since Heijbel (1928) (referenced by Larson 1963) proposed a taper model composed ofthree submodel curves, one for the base, one for the clear middle part, and one for thetop, several other models in this class have been developed (Ormerod 1973, 1986; Max andBurkhart 1976; Demaeschalk and Kozak 1977; Cao et al. 1980; Walters and Hann 1986; etc)(see Table 2.1). Ormerod (1973) was the first to develop a two-section taper function. Hestarted with a simple model of the type:H-li.d=dj[Hhr+e; p>Owhere d is the measured diameter at a fixed height li above ground, and p is the fixed exponent. This model could not provide an adequate description of the changing tree bole, dueto change in form along the length. Ormerod believed that a better description may resultfrom the use of a modified exponent of the above equation as a step function. Therefore,Chapter 2. Background 38separate exponents had to be fitted for each stern section as:H-id=(di_ci)[H] +C1e; pi>O (2.8)where H is the height at the intercept (section i); C is diameter at H; d is the sectionmeasured diameter at fixed height h above ground; and p is the fitted exponent on theclosed interval, [h = H_1,h = H] (see Figure 1 of Ormerod for a two-section case). Ormerodshowed that his taper function was more accurate than that of Kozak et al. (1969).Perhaps, the most commonly used model in this class was developed by Max and Burkhart(1976). Their segmented polynomial taper model, developed for loblolly pine natural standsand plantations, divides the stems into three sections. Separate conditioned polynomialequations were calculated for each section. The location of the joining points was selectedby the model to give the best fit to the stem profile. Max and Burkhart found that the mostsatisfactory equation system was a quadratic-quadratic-quadratic model of the form:= b1(Z— 1) +b2(Z — 1) + b3(a1 — Z)21 + b4(a9 — Z)219 + e (2.9)where a1 and a2 are the relative distances from the top of the tree of the upper and lowerjoining points respectively, that is11 =1, 0<Z<a1=0, a1<Z<112 =1, 0<Z<a20, a2<Z<1and b1, b2, b3, and b4 are regression coefficients. The Alberta Forest Service (1987) tested 15taper and volume functions, for accurate estimation of log and merchantable volume. TheMax and Burkhart taper function was foumici to he the best overall and was recommended forgeneral application in Alberta. Although this fumiction requires nonlinear fitting methods,most statistical software packages have nonlinea.r regression procedures. However, a considerable amount of computing time and money may be required, particularly if initial estimatesChapter 2. Background 39of the parameters are not close to the actual values (Neter et al. 1985). The advantages ofthis function are that it can be directly integrated to give both total and merchantable treevolumes as well as log volumes and it can be transposed to directly estimate merchantableheight.Based on Ormerod’s (1973) taper model, Byrne and Reed (1986) decided to define p,the form exponent, as a segmented equation similar to the segmented taper functions ofMax and Burkhart (1976). In order to make this form exponent correspond to the acceptedtheoretical tree sectional forms, a two-segment equation was used to define the coefficient p,depending on the relative height from the ground and the fitted coefficient, a, as follows:= 3/2 - (Z/ai) - [1- (Z/ai)]Ii + [] Ii (2.10)where j is the predicted p value; and I = 1 if Z a1, otherwise I = 0. This madep continuous along the stem bole. The p coefficient equation was constrained so that itwas equal to 3/2 at the tree base, then decreased to 1/2 in the middle at a point wherea1 = Z, and increased again to 1 at the top. As Byrne and Reed pointed out, a linearlysmooth transition from p = 3/2 to p = 1/2 without passing through p = 1 (for a cone) isimpossible. This taper function cannot be integrated to an exact form, therefore numericalintegration has to be used for volume estimation. The taper equations were calibrated usingred pine and loblolly pine data. From the comparison of results of this model with that ofCao et al. (1980) and Max and Burkhart (1976), Byrne and Reed concluded that this modelwas more accurate and precise for prediction of diameter at any height (h) above ground,and total and merchantable tree volumes.Flewelling and Raynes (1993) developed a three segment taper function for western hemlock. This function is very complex; it requires nonlinear fitting methods, and 26 parametershave to be estimated simultaneously. No substantial improvements were noted over the simplier model by Kozak (1988) (see Table 13 of Flewelling and Raynes). Flewelling (1993)Chapter 2. Background 40improved on the taper function by Flewelling and Raynes by including more variables (upper stem measurements) in addition to dbh and height. However, his function seems tooverestimate volume, particularly for the lower and upper segments compared to the function by Flewelling and Raynes.In general, segmented taper models are more complicated and more difficult to fit, particularly those involving nonlinear equations. Those requiring nonlinear fitting methods maytake many steps before converging or may not converge at all. Therefore, some easier-to-fitmodels were later developed, the variable form/exponent taper models.2.3.1.3 Variable Form and Exponent Taper FunctionsIn 1985. the idea of a single model to describe tree taper resurfaced. But this time, itwas known as “variable-form” taper function which described tree taper with a continuousfunction using a changing exponent to compensate for the changes in form of the differenttree sections. Based on Forslund’s (1982) idea of a paracone, Reed and Byrne (1985) usedOrmerod’s (1973) simple taper function and derived separate i values for each tree depending on the tree’s total height and breast height diameter. The resulting taper function wassimple, and variable, that is, it accounted for form change along the tree bole. Evaluation ofthis taper function showed that the taper curve tended to slightly underestimate diametersin the lower stem and overestimate diameters in the upper stern.Newnham (1988) assumed that instead of Ormerod’s (1973) taper model having fixedvalues of p for each section (see Figure 2.1 for the tree sections), the geometric form of thetree stem should vary continuously along its entire length. His model appears very simple,but the exponent k (p= ) is complex, as shown below:(H-h)211— (H — 1.30) . )where is inside bark diameter at breast height. In order to develop a model for k,Chapter 2. Background 41Newnham transformed Equation 2.11 logarithmically aiid calculated values for k. The valuesfor k varied between 0.5 and 4. In order to make k continuous along the stem, he fitted sixdifferent models for k based on transformations of relative height ((H — h)/(H — 1.30)),D/H, and 1/h. He substituted these function for k into Equation 3.11 and tested the tapermodels for their accuracy based on their prediction statistics. The best taper model selectedfor diameter estimation was:d = x (S)h/[b0+ /H)+b2Sx(D/H)+b3(u//i)’41 x e (2.12)where S = (H — h)/(H — 1.30); is inside bark diameter at breast height; b0, b, b2, b3and b4 are coefficients to be estimated; and the last part of the exponent (fraction 1/h) wasincluded in the model to account for butt swell. The weaknesses of this taper model include:1) bias a.t the butt and tree top (i.e., high positive values): 2) it cannot be easily used toestimate height (h) at a given diameter (d); 3) it is not possible to integrate directly fortotal and merchantable volume estimation; 4) the accuracy tests are based on predictionsfor d/D instead of d, therefore, the results might he misleadingly, and 5) must beknown, which is usually unavailable or difficult to measure on standing trees.Kozak (1988) used a similar approach to Newnham (1988) in that one continuous functionwas used to describe the shape of the bole, with a changing exponent from ground to treetop to compensate for the various geometric forms over the tree stem. Kozak defined hismodel as:d=DIXMC Xe (2.13)where M = , q is equal to (HI/H), HI is the height of the lower join point abovethe ground, and DI is the diameter inside bark at the join point. This function providedthe general shape needed to describe the change of diameter inside bark from ground to treetop. As with other variable exponent taper functions, this taper function can he viewed asa modification of the Husch et al. (1982) taper function (Equation 2.1 of this thesis). Kozakmodified X and K in equation 2.1 to account for the sharp change in diameters near theChapter 2. Background 42base and c represents r of Equation 2.L Based on Demaerschalk and Kozak (1977), the joinpoint ranges from 20 to 25 percent of the total height from the ground; the relative heightof this join point is fairly constant within a species regardless of tree size for most of thetree species in B.C. The exponent c should vary with Z and D/H. After trying differenttransformations of Z on the data, Kozak came up with the following function for estimatingC.c = b1Z2 +b2ln(Z + 0.001) +b3/+b4exp(Z) +b5(D/H) + e (2.14)where ln is the natural logarithm; exp is the exponential (i.e., 2.71828Z); and b1 to b5 arecoefficients. Substituting the changing exponent c into the original equation, the equationbecomes:d = DI x 21n(Z+O.OO1)+b3’/ b4exp(Z)+b5(D/H) e (2.15)Since the diameter at the join point (DI) cannot be measured readily, Kozak developed aregression relationship for it using dhh. The equation was: DI = aoDa1 a. By rearrangingthe function and carrying out logarithmic transformation, the function became:ln(d) = ln(ao) + a1 ln(D) + D ln(a2)+b1(Z2)ln(M) + b2 ln(M) ln(Z + 0.001)+b3/(ln(M)) + b4 ln(M)eZ + b5 ln(M)(D/H) + inc (2.16)This function was fit using ordinary least squares regression. It has the properties thatdiameter (d) equals 0 when h = H, and d equals estimated DI at h = HI.Kozak’s (1988) taper model has only three independent variables (k, H, and D) withseveral transformations, and as a result, the chances of multicollinearity are very high. Multicollinearity could result in the coefficient estimates from least squares being unstable andinconsistent in sign, with inflated variances. Such coefficients would be difficult to interpret.In order to reduce multicollinearity, Perez et al. (1990) tested Kozak’s model to find outif all the independent variables were needed using Oocarpa pine (Finns oocarpa Schiede)data from central Honduras and dropped the transformations that were not significant. TheChapter 2. Background 43selection of the reduced model was based on: 1) mean square error (MSE); 2) coefficientof determination (R2); and 3) prediction sum of squares (PRESS) for predicted ln(d). Thebest model was either the one with high R2 or the lowest value for one or more of the othercriteria. The selected reduced model was:ln(d) = ln(ao) +a1 ln(D) +b1 ln(M)Z2+b2 ln(M)(Z+O.OO1) +b5ln(M)(D/H) +lne (2.17)The equations by Kozak, Perez et al., and by Max and Burkhart (1976) were tested byPerez et al. on an independent data set. They found that Max and Burkhart’s model had arelatively large bias in the upper sections of the tree, while the models by Kozak and Perezet al. had a very small bias. Since Newnham (1988) compared his taper model with thatof Max and Burkhart and found his taper model only as accurate, it is probable that theKozak’s taper function is more accurate than the Newnharn’s taper model. It should benoted that none of these models were tested by Perez et al. for volume prediction abilities.In 1992, Newnham acquired data from the Alberta Forest Service and refitted his 1988taper model for jack pine (Pimus banksiana Lamb.), lodgepole pine, white spruce, and trembling aspen. The comparisons of the biases in estimating d/D for his models and Kozak’s(1988) taper equation seemed to show no real differences. However, Newnham assumed thatdiameter inside bark at breast height (D) is known, which is not always the case. Also, inhis comparisons, he used (d/D) which is not a practical variable to a forest manager whois going to use the model.2.3.1.4 Taper Functions based on Other MethodsOther approaches used for tree taper and form description include the use of multivariate statistical techniques, simultaneous equations, splines, mixed linear models, polar coordinates,and models developed from growth functions.Fries (1965), Fries and Matern (1966), Kozak and Smith (1966), and Real et al. (1989)used principal component analysis (PCA) to study the form and taper of different treeChapter 2. Background 44species. Kilkki et al. (1978) and Kilkki and Varmola (1979) used simultaneous equations fordetermining taper curves, while Liu (1980) introduced the idea of splines. In 1986, Lappideveloped a system of mixed linear models for analyzing and predicting variation in the stemform of Scots pine. This concept was later used by Ojansuu (1987) and Kilkki and Lappi(1987) to model the growth of tree size and changes in stem form. The stem was definedin the polar coordinate system, pioneered by Sloboda (1977), as the logarithmic lengths ofrays at different angles d(n), where n is the angle at which the diameter was measured, andtree size was defined as a weighted mean logarithmic dimension (mean diameter or height).Nagashima et al. (1980), Nagashima (1988), and Sweda (1988) employed the Mitscherlichgrowth process to produce stem taper functions. Also, Biging (1984) and Brink and vonGadow (1986) used growth and decay functions to model stem profiles.These types of taper functions have not yet been used in many countries. One possible reason why this type of taper modelling philosophy has seen limited application is thecomplicated fitting process.2.3.2 Dynamic Tree Taper EstimationOrmerod (1973) stated that his taper function would be better fitted using exponents asfunctions of important physical and biological parameters such as dbh, total height, density,age and site, however, he claimed there were no suitable data available at the time. In hisreview of the available literature on tree taper and form, Sterba (1980) pointed out that thetaper curve theory had reached a stage where further research would only be worthwhile ifthe influence of site and silvicultural treatments on relative stem taper curve were considered.Also, Kilkki and Varmola (1981) stated that the present static taper curve models lookedimprobable because the stem form was a result from the lifelong influence of the environmentand stand characteristics. Consequently, the influence of stand density, for example, cannoteasily be taken into account in static models.When a static taper model is used as a component of an individual tree growth model,Chapter 2. Background 45usually only dbh and total height are predicted (projected into the future) to obtain futurestem diameters and volume. However, the change in dbh over time may not iecessarily hea good reflection of the change in diameters at other positions along the stem (Kramer andKozlowski 1979). Clyde (1986) pointed out that when projecting growth, it would be ideal iftaper models predicted diameter increment along the bole consistent with observed patternsof diameter growth. Unfortunately, most taper equations are empirical models with littlebiological basis. These models are based on cumulative dbh and total tree height ratherthan the growth in diameter which represents the underlying growth processes. Therefore,it should be possible to improve the existing taper functions by looking at the variationin shape along the stem, projecting diameter change along the stem, and including someimportant site, stand and other tree characteristics in the prediction model (Sterba 1980).Newberry and Burkhart (1986) developed and evaluated procedures for incorporatingboth form and taper changes into stem profile models for lohiolly pine using a modified version of Ormerod’s (1973) taper function and Gray’s (1956) taper parameter. Their variable-form stem profile equation allowed the taper and form parameters to vary as functions ofdbh, crown ratio, age and site index. A two-stage fitting procedure was used to obtain astem profile that accounted for both form and taper changes along the stem. In the firststage, the parameters of a base model were estimated for each tree. The parameters werethen related to tree or stand characteristics in the second stage. From these models, theywere able to determine that taper decreases as dbh decreases and form decreases with anincrease in total tree height (see Newberry and Burkhart (1986) for their definition of taper and form). They concluded that variable-form taper models can be constructed if stemprofile models have parameters which can he related to tree form as defined by Gray (e.g.,Equation 2.1; Ormerod 1973). Their form parameter was related to several tree and standcharacteristics that have a biological meaning. The taper measure was fit as a function ofdbh and a product of site index and crown ratio, and the form measure was fit as a functionof height to crown base, height, tree age, aiid a product of site index and crown ratio. ThisChapter 2. Background 46variable-form taper model is considered to be dynamic because changes in diameter. height,crown length with time (age) will change the tree form and taper.Clyde (1986) used the Chapman-Richards function (Pienaar and Turnbull 1973) to modeldbh and height growth as functions of tree age. She substituted these models into the taperequations by Kozak et al. (1969), Max and Burkhart (1976), Bennett et al. (1978) andAmidon (1984) to make them dynamic. Clyde found that the taper models by Bennett et al.and Max and Burkhart predicted the pattern of diameter increment for the Type I sequencesmore reasonably than the simple models by Kozak et al. and Amidon. However, the predicteddiameters for all functions were poor for ages greater than 100 years, and there were someinconsistencies between the observed and predicted Type II sequences for all of the tapermodels examined. She suggested that in addition to dbh and height predictions, some othervariables or more complex models should he considered to obtain predictions of diameterchange over time along the stem that were consistent with the growth and development of atree.Although the idea of dynamic taper modelling existed as early as 1973 (e.g., Ormerod1973), it was not until 1986 that a dynamic taper model was produced. However, it shouldbe noted that apart from the works of Clyde (1986) and Newberry and Burkhart (1986), notmuch has been accomplished in the area of dynamic taper modelling.2.4 Alternative Methods for Fitting Taper FunctionsMost tree taper modellers have used ordinary least squares (OLS), or nonlinear least squares(NLS) fitting procedures to obtain parameter estimates for taper functions. For example,Clyde (1986) and Kozak (1988) used OLS, Newnham (1988) used both OLS and NLS methods, to fit the taper functions. However, proper fitting of these taper models could requirespecial techniques. Because the data used to fit these models are not identically distributedand not independent (i.e., non-iid), ordinary least squares is not efficient. If compatibleChapter 2. Background 47volume and taper predictions are desired, then volume and taper equations used should befitted as a system of equations (Burkhart 1986; Reed 1987). Simultaneous fitting proceduresshould be used for fitting the system because they ensure numeric consistency among thecomponent equations and more accurate and precise volume and taper estimations (Reedand Green 1984; Byrne and Reed 1986; Judge et al. 19985, p. 591-630).Two principal sources of data for dynamic taper modelling are permanent sample plots(PSPs) and detailed stem analysis data. PSP data are characterized by repeated measurements on the same individual (a tree or plot) over time (time-series or serial measurements).The resulting data are serially correlated or have autocorrelated error terms (i.e., are notindependent of each other). To use PSP data for dynamic taper modelling, measurement ofdiameter inside bark (di) along the stem over time would also be needed. Since these di’sare taken on a single tree, they will be correlated. This type of correlation, called cross-sectional or contemporaneous correlation, is the correlation between different error terms forobservations taken from the same individual at the same time. If di’s were available on PSPdata, PSP data would be the most appropriate data for dynamic taper modelling becauseit would combine tree and stand information measured over time. However, such data arerarely available for modelling dynamic taper.Detailed stem analysis data are obtained by taking diameters at several positions alongthe same tree at one time (called cross-sectional data). These data would have contemporaneously correlated error terms. If the cross-sectional data are also measured over time (i.e.,diameters at given positions along the tree stem are taken over time), panel or longitudinaldata result which generally are both contemporaneously (related over the stem) and seriallycorrelated (related over time) (Dielman 1983, 1989). These data lack tree and stand measurernents for previous time periods before felling (i.e., basal area, number of stems, crownsize), which are usually available for PSP (lata. However, stem analysis data are the mostoften available for use in taper modelling.Both PSP and stem analysis data have another characteristic. It was shown earlier thatChapter 2. Background 48diameter growth varies for different parts of the tree. As a result, trees will have varyingcumulative diameters for the different sections along the tree stem. Therefore, error variancesalong the stem will likely be different (heterogeneous variances), due to the variation indiameter increments. Thus, stem analysis or PSP data used in dynamic taper model fittingcan be characterized by having heterogeneous, contemporaneollsly correlated, and seriallycorrelated error terms.One model that could be used for fitting linear dynamic taper functions can be generallywritten as (Judge et al. 1985, p. 514):K= /3ijtO + /3tkXtk + (2.18)k=1where i corresponds to tree number; j corresponds to cross-sectional measurements which arecontemporaneously correlated; and t corresponds to time measurements which are seriallycorrelated. is the value of the dependent variable for tree i, section j and time period t;Xkt is the kthl independent variable for tree i, sectioi j, and time period t; 3jj is the kthcoefficient, associated with the kth independent variable, to be estimated which would varyover tree i, section j, and time period t; and the error terms (et) are assilmed to have zeroexpectation, to have heterogeneous variance o, and to be contemporaneous and seriallycorrelated.The model for fitting nonlinear functions would he:Y=f(X;e)+e (2.19)where Y is a column matrix of dimension Tj x 1: X is an Z x Kmatrix for the independent variables; e is an Z x 1 vector for error terms; eis an K x Tjj matrix of coefficients associated with the independent variables; Nrepresents the sample trees; M equals the number of sections for a given tree i; is thenumber of serial observations for section j and tree i; and the other symbols are defined forEquation 2.18.Chapter 2. Background 49These models assume that the time periods are equal and that the model coefficients arechanging with tree, section, and time. However, for stern analysis data, models with changingcoefficients for different sections and trees is not practical for taper modelling because it isimportant that a model with one set of coefficients be developed for a population (e.g., agroup of species in a region) (Kozak 1988). The models most appropriate for dynamic taperfunctions (linear and nonlinear) are:K= !3o + /3kXiitk + (2.20)kr1andY = f(X; 6) + e (2.21)where i3 to and 6 (a K x 1 matrix) are coefficients to be estimated for all trees, sections,and time periods of all population; and the rest of the symbols are as defined for Equations2.18 and 2.19. These models assume that the coefficients are constant for all trees, sections,and time periods and that any variations will be captured by the error terms.2.4.1 Ordinary Least Squares and Nonlinear Least SquaresIf OLS and/or NLS procedures are used with stem analysis data to estimate the coefficients( and 0) for the dynamic taper models, the coefficients are estimated as follows:b = (X’X)1X Y (2.22)= (F.’F.)’F.’Y (2.23)for OLS and NLS, respectively; where Y is the column vector of the dependent variable; Xis a matrix of independent variables; and F. represents a matrix of the first derivatives ofindependent variables with respect to the parameters (see Judge et al. 1985, p. 196—201).Equations 2.20 and 2.21 have error terms which are non-iid. If these equations are fittedusing OLS and/or NLS, assuming iid error terms, the parameter estimates will be unbiasedChapter 2. Background 50and consistent (Greene 1990; Kmenta 1971). However, the usual estimates of the varianceswill be biased as shown below for OLS.V(b— ,3) = E[(b 3)(—= E[(X’X)’X’ee’X(X’X)1]=u2(X’X)X’X(X’X ’where e is the matrix of error terms; ‘I’ is a positive definite matrix whose diagonal elementsare not equal to one and the off-diagonal elements are not zero (see Greene 1990, P. 383 for adefinition); and 2 is the common variaiice for the error terms. Since this new variance (17(b))is different from the usual OLS variaice estimate (â2X’X)’). the usual OLS estimate isbiased. Therefore, hypothesis tests and confidence iiterva1 construction cannot be properlyconducted because the usual t and F distributions based on OLS variance estimator will hebiased and misleading (kmeita 1971, p. 247—304; Gregoire 1987). Also, the OLS varianceswill no longer be the lowest in the class of linear, unbiased estimators. The coeffficientestimator b will no longer be the Best Linear Unbiased Estimator (BLUE). In the case of 0for nonlinear models, when the error terms are non-iid, the usual estimator for the variancefor will also be biased and 0 will be inefficient. If statistical inferences (constructingconfidence intervals and hypothesis tests) about the coefficients are important, alternativefitting methods to OLS and NLS must be selected.‘Vith error terms (et) that are assumed to be contemporaneously correlated amongsections for a given tree and time, serially correlated over time for a given tree and section,and with heterogeneous variances among sections of a given tree and time, generalized linearor nonlinear least squares or maximum likelihood estimates are more appropriate fittingtechniques.Chapter 2. Background 512.4.2 Generalized Least SquaresA Generalized Linear Least Squares (GLS) fit allows for efficient estimation of the parametersof the following model (Kmenta 1971, p. 499—508; Judge et al. 1985):I3G [X’’X]’X’Y (2.24)where Z is the variance-covariance matrix for the error terms e. The GLS estimator IG isunbiased, consistent and most efficient (Kmenta 1971). The associated generalized nonlinearleast squares (GNLS) fit is consistent, and asymptotically unbiased and efficient. The GLStechnique is the general approach for obtaining the BLUE of the parameters of any linearmodel. It is called genera1ized’ because it includes other models as special cases. Forexample, OLS is a special case of GLS in which the variance-covariance matrix has diagonalelements with the same value and the off-diagonal elements equal to zero.If the residuals are multivariate normally distributed, the log-likelihood for the maximumlikelihood (IVIL) function for the sample islnL = —.1n(2ir)— lnjZI —where n is the sample size; and e is an Y — X matrix. If e is replaced by Y — X,3, ln Lis differentiated with respect to , and the resulting equations are equated to zero (Kmenta1971, p. 504), solving for yields3ML = (X’rX)’(X’W1Y).This implies that the GLS estimator is also the ML estimator when the residuals are normallydistributed. Therefore, the GLS estimator has the same characteristics as the ML estimator.This also follows for GNLS when the error terms are normality distributed.The GLS parameter estimator (Equation 2.24) assumes that the variance-covariance matrix () of errors is known. However, in most cases (including this thesis), 1 is unknown,so the GLS estimator is no longer a feasible choice. Instead a consistent estimator of 1, (O)Chapter 2. Background 52could be found and substituted into Equation 2.24. This leads to a two-step Aitken (1935)estimator or Estimated/Feasible Generalized Least Squares (EGLS or FGLS). Generally,contains n(n + 1)/2 unknowns, if there are no prior restrictions on any of its elements, wheren is the total number of observations. When the number of unknowns is larger than thenumber of observations, assumptions about the structure of 1 must be made to restrict thenumber of unknown elements so that a consistent estimate can be found.For the two-step Aitken (1935) estimator, first the ordinary least squares estimator band its corresponding residuals are found. These residuals are used to obtain a consistentestimate of . In the second stage, the estimator is substituted into Equation 2.24 toobtain the FGLS estimator, 13F as follows:= [X’’X]’X’r1Y (2.25)where F is a column vector (K x 1); K is the number of parameters to be estimated; Yis a column vector (n x 1) for the dependent variable; n is the total number of observationsin the sample; and X is a matrix of n x K for the independent variables. Because FGLS isbased on , a consistent estimator of, /3F is also consistent and is asymptotically efficient.FGLS is more efficient than OLS when 1 is not a diagonal matrix with all diagonal elementsequal. If the 1 matrix is close to the requirement for OLS, OLS may be more efficient thanFGLS in small samples. If the error terms are iid, the FGLS aiid OLS estimate both will beefficient.For a nonlinear model, the appropriate estimator is the generalized nonlinear least squares(GNLS) (Gallant 1987, p. 127-139) estimator using the error covariance matrix, F. WhenI’ is not known, then an estimator I’, with feasible nonlinear least squares (FGNLS) is used.To find the FGNLS estimator, the two step procedure used for FGLS is followed by firstfitting NLS to estimate the residuals, which are then used for estimating F.Since dynamic taper modelling involves the use of stem analysis data, the error termsare non-iid. The error covariance matrix (i or F) will be characterized by heterogenousChapter 2. Background 53variances among sections for a given tree and time, contemporaneous correlation amongsections for a given tree and time, and serial correlation among time periods for a given treeand section. The methods of OLS and NLS for estimating parameters are inappropriate ifinferences are needed. The alternative choices available are:1. Use a combination of transformations (three transformations) to remove heteroskedaticity, and serial and contemporaneous correlations and then fit the equation using thetransformed data by OLS or NLS, or2. Use an FGLS or FGNLS estimator.Transformations to remove heteroskedasticity and autocorrelation are available (see Judgeet al. 1985), but a transformation method to remove contemporaneous correlation was notfound in the literature. Instead, the use of an FGLS estimator was considered for thisresearch.2.4.3 Fitting Dynamic Taper Functions Using FGLSTo fit dynamic taper functions of the form in Equations 2.20 and 2.21 with the FGLStechnique, some assumptions about the error terms (residuals) have to be made.First, it would be assumed that the variances associated with the different trees vary, buttrees are independent. For a given tree i, the following assumptions would then apply:1. The E[e] = 0.2. The E[eite3J 0, for t s. It is assumed that for a given tree and section, the firstperiodic measurements will be more correlated with observations taken five years laterthan those taken 10 years later, etc. Thus. the autocorrelation is considered to be offirst order. This gives:(a) = pijeij.t + uj, where Pu is the slope of the line.Chapter 2. Background 54(b) ztt N(O, o-), for each tree i and section j.(c) = 0, for all t s.3. The ==(heteroscedasticity), where is the variance for a givensection j on tree i.4. The E[eteit] 0. Instead it is equal to oi for j 1 for tree i and time t, sincesections j and 1 are contemporaneously correlated.5. The first value of has the following property: e1 N(0,6. In addition to these properties of the for the stem analysis data, the number ofserial measurements differ among the sections in a given tree. This results in somefitting problems.To obtain consistent estimates of 1 for the stern analysis data, when fitting dynamictaper models using FGLS for linear models, the following steps would be required:1. OLS would be applied to pooled data for all trees, and the corresponding residuals(e) for all trees by section and time period would be obtained. Since these residualswould be based on unbiased coefficient estimates, they would be consistent estimatesof the actual error terms. They could be used to obtain for a given tree and sectionas follows:= Lt=2 (2.26)2L.d=22. The jjs wollld then be used to transform the stem analysis data as follows:(a) The dependent variable would be transformed as:= —(2.27)Chapter 2. Background 55(b) The independent variables would be transformed as:= Xtk—(2.28)where k is the kill independent variable.(c) The first observation would also be transformed, in order to improve on optimalefficiency, as suggested by Dielman (1989, p. 15) as:= (1- 5)h (2.29)and predictors asX = (1—(2.30)(d) The residuals would be transformed as:= — (2.31)3. The resulting transformed values for the predictors and the response variable would heanalyzed by OLS as given by the equation below:= /3X +... +/3kXtk + ijt (2.32)where the new estimated residuals (jt) are serially uncorrelated. These residualswould then be used to estimate the variances and covariances as follows:(a) Sectional variances (e.g., sectioll j) for ujj’s= T K 1=2(2.33)where K is the lumber of parameters to be estimated.(b) Covariance between sections j and 1 for the ztj1’sT0=— K(2.34)This model assumes Tjj is equal to T1.Chapter 2. Background 56(c) Sectional variances (e.g., section j) for the e’s=(2.35)where & is the variance of àjj; and &jj is the variance of(d) Covariances between sections j and 1 for the e’s= jUl (2.36)1— PijPilAfter estimating Pu, then a consistent estimator of (see Kmenta 1971, p. 511—514for definition of P) would he estimated as:1 Ii1 Ii(2.37)‘T—1 T—2 T-3 1Pu Pu PuThese Pj are substituted into,a consistent estimator of T1. would be estimated asgiven below:ii’ ill il2 i12 0i1M1 i1MñJ21.r21 0i21-1i2M(2.38)-‘pO•ijl ij1 0ij iiiJuMlPuiIj &uJf2PjM2 OM.lJVfvwhere is the estimated variance for section j on tree 1; â1 is the estimated covariancebetween sections j and 1 where j 1.After obtaining the j’s and the calculated variances and covariances (using Equations2.35 and 2.36), would be obtained by substitution as follows:Chapter 2. Background 57L o •.. oo 2 0 (2.39)owhere 0 is a matrix of zeros. For a given section j on a particular tree i, the error terms arecorrelated over time.After is estimated, the FGLS estimator would be obtained as follows:= (2.40)The resulting coefficient estimate (/F) will be consistent and asymptotically efficient ifthe assumptions made about 2 are correct, but might not be efficient for small samplesizes (Greene 1990). Since taper models involve the use of large quantities of data, theasymptotic properties of FGLS (Kmenta 1971) would apply. This means that the coefficientestimates for a dynamic taper model would be asymptotically efficient when fitted usingFGLS compared to the inefficient coefficient estimates resulting from the fit by OLS. Also,the resulting standard errors for the coefficient estimates will be consistent estimates. Aswell, the coefficients will be asymptotically normally distributed. For the nonlinear models,the resulting FGNLS coefficient estimates would be consistent and asymptotically normal.Chapter 3MethodsMost existing taper functions are static, which means they are oniy able to predict treeshape at a particular time. Common to all these static taper models is that they tend toslightly underestimate diameters in some parts of the stem and overestimate diameters inother parts. This indicates that: (1) either the individual trees are not the exact shapesassumed or that these shapes are not constant over time, and (2) the models used are eitherrnisspecified or some important variables are not included in the models.The central objectives of this research were to examine changes in tree shape along thestern at one time and over time using a static taper function, aiid then to develop a dynamictaper function based on the relationships found between the form exponent of Kozak’s (1988)variable-exponent taper equation and the site, tree, and stand variables.The data used in the study will be described in detail in Section 3.1. The methods usedto determine the factors that influence tree shape are then described in section 3.2. Finally,the methods used to develop and test the dynamic taper model are discussed in Section 3.3.3.1 Data PreparationIn order to meet the objectives of the study, permanent sample plot (PSP) and detailed sternanalysis data for interior lodgepole pine were obtained. The PSP data were used to developmodels to predict stand density, tree dhh, and height. PSP data were used to develop heightand dbh prediction models. because the stein analysis data lacked a stand density measure atprevious time periods before felling and also, the stem analysis data set was small, only 135trees in 50 plots. The stem analysis data were used to examine the changing form exponent58Chapter 3. Methods 59over time, and to develop, fit, and test the dynamic taper model. The stem analysis datawere used instead of PSP data for these purposes because they provided diameter insidebark measurements along the stem over time, which were absent from the PSP data.3.1.1 Permanent Sample Plot DataThe PSP data were provided by the Alberta Forest Service, Timber Management Branch.The “Permanent Sample Plot Field Procedures Manual” (Alberta Forest Service 1990) givesa good description of the data collection procedures used by the Alberta Forest Service fortheir PSP measurements. Only plots with more than 80 percent pine by basal area wereused for this study. All plots which had noticeable incidences of natural (e.g., windthrow,disease) or man-made (e.g., cutting) interference during the remeasurement periods weredeleted. The data were from Western Alberta, which is dominanted by lodgepole pine; jackpine, a closely related species, occurs in Eastern Alberta.Remeasurements in each plot were planned for every five years for conifers less than 80years old or greater than 130 years old, and at every 10 years for stands between 80 and 130years old. However, the actual remeasurement schedule varied from three to seven years.The plots were established between 1960 and 1965. The number of remeasurements variedfrom one to four.Plot sizes varied from 0.1 to 0.34 ha depending on density. In each plot, all standing trees(dead and live) 9.1 cm dbh were tagged. Dbh was measured to the nearest 0.1 cm using ametal diameter tape. A minimum of 30 heights per species was measured using a clinometerand a 30 or 50 m measuring tape. Crown length was measured on three trees per plot forrecent remeasurements (starting 1983) only. To measure plot age, three trees adjacent tothe plot were felled, and cuts were made at stump and breast height. Both stump (0.3 mabove ground) and breast height ages were measured by counting the number of rings onboth sides of the cut disks and averaging them.For each plot, the following variables were calculated:Chapter 3. Methods 601. number of stems per hectare (SFH);2. average plot age at breast height, calculated by averaging the ages for three trees fellednear the plot;3. average height, calculated by averaging all dominant and codominant tree heights inthe plot;4. average dbh, calculated in a similar manner to average height;5. site index (SI), the average height of all dominant and codominant trees in the plotat a reference age, was calculated using average plot age and height. The followingfunction provided by Alberta Forest Service (1985, p. 3—5) for calculating SI at 50years was used:SI = 1.3 + 10.9408 + 1.6753(H — 1.3) 0.3638(ln(AGE))2+ 0.0054(AGE)(ln(AGE)) +8.82281() — 0.2569(H — l.3)(ln(H— 1.3)) (3.41)where SI is the estimated site index (m) at 50 years breast height age. H is the meanheight per plot for the dominant and the codominant trees, AGE is the mean breastheight age per plot, and ln is the natural logarithm;6. basal area per hectare (BA), the cumulative cross-sectional area for all measured treesat 1.3 m above ground;7. quadratic mean diameter (QD), the dbh for the tree of average basal area, calculatedas described in Davis and Johnson (1987, p.80—81);8. relative density (RD), calculated as the ratio of basal area per hectare to the squareroot of the quadratic mean diameter (Davis and Johnson 1987, p. 81); and9. stand density index (SDI), the number of trees per unit area (ha or acre) that a standwould support at a standard average dhh (Husch et al. 1982).Chapter 3. Methods 61For calculating average dhh, BA, SPH, QD, RD, and SDI, all trees in the plot which haddbh’s > 9.1 cm were used. After calculating the above stand statistics, oniy dominant andcodominant trees and the associated stand level statistics were selected for use in this thesis.This was done because the stem analysis data to be used in this research were composed ofonly the dominant and codominant trees.The PSP tree data were checked for outliers1 by plotting pairs of variables. No obviousoutliers were found. The PSP data consisted of 1908 trees on 613 plots. These 1908 trees werestratified into five-centimeter dbh and five-meter height classes and within each class, thirtypercent of the data was randomly selected for validation purposes (574 trees on 185 plots)(Table 3.2). Seventy percent of the data (1334 trees on 428 plots) was used for developingmodels to predict QD at 50 years, dbh, and height. The rationale for data splitting was tovalidate the fitted models, because the usual t and F tests would not be applicable whenusing the dependent PSP data.3.1.2 Stem Analysis DataThe stem analysis data were collected by Dr. Q. Wang, who was a graduate student atthe time in the Department of Forest Sciences, under tile supervision of Dr. Karl Klinka.The data were obtained from North of Burns Lake, South of Anaheim Lake, and Eastand Southeast of Prince George in the Interior of B.C. Physiographically, these areas occurwithin the Sllb-Boreal Spruce and Sub-Boreal Pine-Spruce Biogeoclimatic Zones of B.C.(Meidinger and Pojar 1991). Stands selected for sampling were dominated by lodgepole pine(greater than 80 percent of the crown cover), even-aged, fully stocked, and relatively free ofdisturbance after establishment.Fifty 0.04 ha fixed area plots were selected. For each plot, the elevation, slope, aspect, soilparent materials, and other features were recorded. The dhh of each tree greater than 1.0cm‘An outlier can he loosely defined as an observation which in some sense is inconsistent with the rest ofobservations or which disproportionately influences the conclusions drawn from the data set.Chapter 3. Methods 62Table 3.2: Summary statistics for the permanent sample plot data.aThe dbh and height (H) statistics are based on individual tree information, whereas all other statisticswere based on plot information.bThe statistics in brackets are for the validation data set (574 trees or 185 plots) and the unbracketedstatistics are for the model development data set (1334 trees or 428 plots).was measured. Then, for each plot, two or three dominant and codorninant trees (totalling147 trees) were destructively sampled. After felling, height to live crown and total tree heightwere measured on these trees. Trees were sectioned at 0.3 m, 0.6 m, and 1.3 m above ground,and at subsequent one metre intervals after felling. For small trees (trees with dbh 8.0cm), subsequent sections after 1.3 m above ground were cut at 0.5 m intervals. The numberof sections per tree varied from nine to 23. Disks were cut off the top of each section. Thesedisks were returned to the laboratory and measured for annual radial increment width usinga light microscope with a horizontal bar upon which the disk was placed. This was connectedto a microcomputer with a program written in FORTRAN for recording annual incrementmeasurements for the average radius on each disk. The procedure for measurement was:1. the widest diameter (Di) and narrowest diameter (D9) inside bark were measured;2. the geometric mean diameter (Da) was calculated as Da = D1 x D9, and convertedto radius as R =Variable Mean Standard Minimum Maximumdeviationclbh (cm)a 23.32(23.38)b 6.2 (6.4) 8.4 (8.6) 47.0 (48.8)H (m) 19.31 (19.24) 3.97 (4.13) 7.2 (8.2) 30.8 (29.3)AGE (years) 76.7 (76.8) 25.2 (25.7) 16 (17) 162 (160)SI (m) 14.6 (14.4) 2.7 (2.7) 6 (6) 23 (23)BA (in2/ha) 37.96 (37.10) 9.35 (10.2) 9.1 (9.09) 62.7 (62.0)SPH 2696 (2647) 2184 (2089) 256 (256) 18228 (17199)QD (cm) 11.6 (11.4) 3.1 (3.1) 5 (5) 21(22)Chapter 3. Methods 633. a riler was rotated around the cut disk with the zero at the pith, until a radius equalto the average was located. This was marked on each disk and the disk was placedon the horizontal bar in such a way that it was directly below the microscope duringmeasurement; and4. annual radial increments were measured along the transit from the pith outwards.For each plot, the following variables were calculated using the same methods as for thePSP data: SPH, average breast height age (AGE), SI, average height, average dbh, QD, BA,RD, and SDI. Site index was calculated by first averaging height and breast height age for thedominant and codominant pine trees in each plot. It should be noted that these plots wereassumed to have come from even-aged stands. For plots with breast height ages between48 and 52 years, SI was approxima.ted as the average height of dominant and codominanttrees per plot, while for other ages, Equation 3.41 was used to predict plot site index at areference a:ge of 50 years at breast height. For each tree that was felled, crown length (CL)was calculated as the difference between total tree height and height to live crown, and crownratio (CR) was calculated as the ratio of crown length to total tree height.To prepare the pine data for analysis, diameter inside bark for each section was calculatedby summing up all the annual increment widths for the section. The age at the time of fellingof each section was the ring count from the pith to the bark. Diameter inside bark was alsocalculated over each 5 year period, starting from the bark. Tree height was calculated foreach 5 year age using a method developed by Carmean (Carmean 1972; Dyer and Bailey1987; Newberry 1991). Dbh (D) for each 5 year period was calculated using a bark factor(k) of 0.9 (i.e., D d/0.9) (Husch et al. 1982, p. 104—108). Husch et al. (p. 105) statedthat this lower-stern bark factor (k) ranged from 0.87 to 0.93 varying with species, age, andsite. However, the majority of the variation is accounted for by species. Therefore, it wasreasonable and convenient to assume a constant hark factor for lodgepole pine and a valueof 0.9 was used because the trees used were still young. Therefore, they were assumed toChapter 3. Methods 64have a lower bark thickness than mature trees.After reconstructing trees at five-year periods, a plot of diameter inside bark (d) againstheight above ground (h) was produced for each tree by time period. All together, over 1000graphs were produced. From these graphs, 11 trees were eliminated, because of suspectedmeasurement error. These trees had more than one sectional diameter well out of linewith the rest (i.e., outliers). Also, one tree without stump height (0.3 m above ground)diameter measurements was removed, since this is an important measurement for taperfunction development. Therefore, only 135 trees remained, representing 887 5-year periodicmeasurements (tree-measures) with 8584 sectional measures.In order to examine the variation in form and taper along the stem at a particular timeand as time changes, a representative measure had to be used. The form exponent for thesimple taper equation (Equation 2.1) was selected. This form exponent (r) was selectedbecause it can be associated directly with the known shapes of the stems (Figure 2.1). Theform exponents were calculated by rearranging Equation 2.1 as follows:Yijt ==(3.42)where Yijt is the radius at distance from the tree tip for tree i, section j and for 5 yearperiod t (hereafter called time t); x, is the distance from the tip for a given section on anysample tree (i.e., xj = — where is total height for tree i and for time t andis the height of measured radius Yijt from the point of germination); the measure oftaper, is eqilal to pa—; is the form exponent for tree i, section j, and time t; and Rb/is the radius at the base for tree i and time t.Rearranging eqilation 3.42 yields:Yijt— f Xij — Xjt 2By multiplying the nilmerator and denominator of the equation by two, 2Yijt becomes diameter (clt) at a given height and 2Rb becomes diameter (Db) at the base (i.e., at stumpChapter 3. Methods 65height). The shape of the stem from the point of germination to the stump height wasassumed to be a cylinder, and to carry negligible volume. This assumption simplifies volumecalculation, but might introduce some bias in the calculated volumes. The above equationalso can be written as:xij1Db — ——Solving for the equation becomes:2 x in []=(3.43)ln Ltiwhere in is the natural logarithm.In order to calculate periodic (five-years) form exponents for every tree (i) and section (j),the following periodic measurements were needed: total tree height (H), diameter insidebark (d) to a given height distance from tree top (x) (xjj = — and thediameter inside bark at the tree base (Db). At the base of the tree (in this case stumpheight), was undefined; therefore, it was not included in the sample for r calculation.The values for ranged from 0.7 to 5.1. The calculated values were plotted againstfor all sample trees to check for obvious outliers.The dynamic taper model developed as part of this research was based on Kozak’s (1988)taper function (see Equation 2.15). Therefore, in addition to calculating the exponent (r) ofEquation 2.1, the exponent (c) of Kozak’s equation was calculated for every tree by sectionand time. Kozak’s taper function is similar to Equation 2.1, except that Kozak used adifferent base diameter (diameter at the join point) rather than Db, and modified the basefor the form exponent. If a subscript for the tree, section, and time are included, Kozak’staper function becomes:= DI x (3.44)where DI is the diameter at the lower join point of tree i a.t time period t and =Chapter 3. Methods 66( 1Y ). This model is nonlinear, but can be intrinsically linear depending on the assumptions about the associated error terms. In order to come up with an appropriate joinpoint, was plotted against - for the last time period (time of felling) for a subset of 20randomly selected trees. From the plots, the visual join points (q) ranged between 22 to27 percent of total tree height. Therefore, an approximate value of 25 percent was used.This agrees with the results by Demearschalk and Kozak (1977), Kozak (1988), and Perezet al. (1990), who found that the value ranged from 15 to 30 percent of total height for mostspecies and that any value used within this range would not affect the results greatly.Based on q = 0.25, the form exponent for Kozak’s (1988) model was calculated for eachtree by section and time period as:ln— lnMSince DI was not measured during data collection, interpolation was used to calculate DIat HI = 0.25H Although a tree is not a straight line, linear interpolation was selectedbecause the length of the sections were very short. The equation used for interpolation was:— d1 — — h1d2 — DI— —which was rearranged to give:— (d2 —d1)(h2— HI2)‘‘i2t — ‘i1twhere d1 is the diameter below DIt; d2 is the diameter above DI; h1 and h2 are heightabove ground corresponding to diameters d1 and d2 respectively; and HI is the height ofDI above ground.The stand measures and the crown size measure included with the stem analysis datacould only be collected at the time of felling. As a result, stand density and crown sizewere not available for previous time periods. To overcome these problems, two options wereChapter 3. Methods 67available. The first option was to develop a taper model without stand density and crownsize measures. However, it can be seen from Section 2.2.2 that stand density and crown sizeare very important factors which are expected to influence tree form changes. The secondoption was to develop prediction models for stand density and crown size measures usingPSP data and then use these models to predict density and crown size for each plot of stemanalysis data over time. Unfortunately, the PSP data did not have enough measured crownlengths for dominant and codorninant trees. As a result, no measure of crown size was used.However, crown size measures such as crown length are highly correlated with stand density.Therefore, using predicted stand density would somewhat account for crown size changes.The model for predicting stand density for each plot of the stem analysis data was basedon the PSP information which included site index, average plot age, and average plot heightfor each PSP. The data consisted of 613 plots which were divided into two sets: the modeldevelopmeit data set (70 %) and the validation data set (30 %). In order to model standdensity, an appropriate measure of average stand density had to be selected. The standdensity measures considered included: SPH, BA, QD, RD, and SDI. (See Davis and Johnson1987, p. 79—84 for a discussion of the different stand density measures). Crown competitionfactor (see Davis and Johnson 1987, p. 86 for definitioll) was not considered because boththe PSP and stem analysis data lacked crown width.For a given stand age and site quality, SPH is a good measure of stand density. BAis more commonly used than SPH because it relates directly to stand volume. However, ifSPH and BA are combined, they give a better measure of stand density by indicating thesize and number of the trees together. QD, RD, and SDI combine both SPH and BA. Eachof these stand density measures was graphed against plot age and site index using the PSPdata. These graphs were used to ideiitify which stand density measure was most highlycorrelated with these variables. In addition, a graph of the form exponent r at the time offelling against each of the five measures of stand density was obtained. QD was found to bemore highly correlated with plot breast height age. site index, and the form exponent thanChapter 3. Methods 68RD and SDI.From the graphs of QD against SI and AGE, some transformations (AGE2,,SI2,in AGE, in SI) and interactions of these variables (SI x AGE, SI2 x AGE) were selected.As a result, a total of nine variables were used to find the best model to predict QD. TheRSQUARE procedure in SAS (SAS Institute, Inc. 1985) was used to select the best possiblemodels. Four models were selected based on mean squared error (MSE), multiple coefficientof determination (R2) and adjusted R2 (R), and Mallow’s statistic (C,,) (see Neter et al.1985, p. 423-427 or Judge et al. 1985, p. 863—6 for definitions of R and C,,). Two of thefour models were linear and the other two were intrinsically linear and were transformed tolinear models using a natural logarithmic transformation. The four models selected were:1. QD = b0 + b,(SI x AGE) +b251 +b3-2. QD = b0 + b,(SI x AGE) +b2SI + b3E3. QD = 51b1 x AGEb2 x exp(b3 + b451 +b5SI2)This model was transformed into:ln(QD) = b1 ln(SI) + b2 ln(AGE) + b3 + b451 +b5124. QD = b0 x x AGEb2This model was transformed into:ln(QD) = ln b0 + b1 ln(SI) + b2 ln(AGE)where b,, are the coefficients to be estimated.All the selected linear models were fit using the REG procedure in SAS (SAS Institute,Inc. 1985) (OLS). In addition, models 3 and 4 were fit using the NUN (nonlinear) procedurein SAS. The models were evaluated for their predictive abilities based on the validation dataChapter 3. Methods 69set (185 plots). Model selection was based on the fit and prediction statistics. The fit statistics included R2, the standard error of estimate (SEE), and the PRESS statistic (see Draperand Smith 1981, p.325—32’7). The prediction statistics used for model evaluation includedthe fit index (Fl), estimated SEE (SEE), root mean square error (RMSE), and mean andabsolute biases. These prediction statistics were based on the differences (residuals) betweenthe observed and the predicted values, that is:where ê is the residual or difference between the measured and the predicted value for agiven observation i, % is the observed value, and Y is the predicted value. The five predictionstatistics used in model evaluation were calculated as follows:1. Fit Index (PT) or Estimated Coefficient of Determination:fl2Fl 1— fl(y-where Y is the mean value of the dependent variable; and n is the number of observations in the validation data set.The value of estimated R2 (PT), like that of R2, indicates the degree of goodness-of-fitof an equation. A higher value for a model indicates a better fit.2. Estimated standard error of estimate (SEE) (Spurr 1952):-‘n 2 0.5SEE = Ldjj en—Kwhere K is the number of coefficients to be estimated.A model with high SEE would be a poorer model than a model with a small SEE.The SEE is a good indicator of the spread of the actual observations (Y) around thepredicted values (Y) (Spurr 1952). However, with a large number of observations, FTand SEE will rank models identically.Chapter 3. Methods 703. Mean Bias (Bias) (sometimes called Mean Differences):Zn.Bias =flBias is a good measure of the accuracy of the model’s prediction abilities since it revealshow well or poorly the model represents the actual observations on average. If Bias issmall, this indicates that the model predicts well for an independent data set. However,large negative and large positive biases could cancel each other out and resulting in asmall value for Bias.4. Mean square bias or error (MSB) and the root mean square bias or error (RMSE)10.5MSB =—> and RMSE = /]VISB=e]Neter et al. (1985) stated that if the MSB is fairly close to the MSE based on theregression fit to the model-building data set, then the error mean square MSE for theselected regression model is not gravely biased and gives an appropriate indication ofthe predictive abilities of the models. They stated that if MSB is much larger thanMSE, then MSB should be relied upon to indicate the predictive ability of the selectedmodel.5. Mean Absolute Deviations (Residuals) (MAD):MAD=where is the symbol for absolute value.MAD overcomes the problem of large positive and large negative biases cancelling out,like MSB, but without squaring the values. Small values of MAD would indicate goodpredictive ability of the selected model for the validation data set.Chapter 3. Methods 71Table 3.3: The fit and prediction statistics for the quadratic mean diameter (cm) predictionmodels.Model Fita (n=428 plots) Prediction (n=185 plots)R2b SEE PRESS FT SEE RMSE Bias MAD1 .7869 2.06 1832.671 .7337 2.11 2.08 -0.10 1.612 .7867 2.06 1835.162 .7330 2.11 2.09 -0.10 1.613 .7902(.8127) 2.05(0.35) —(7.916) .7431 2.07 2.05 -0.08 1.614 .7808(.8079) 2.09(0.14) —(8.060) .7245 2.15 2.13 0.02 1.64aThe fit statistics in brackets were calculated based on the natural logarithmic transformed models (Models 3 and 4) fitted using OLS.bR2 is the coefficient of multiple determination; SEE is the standard error of estimate in cm; PRESS isthe predicted sum of squares residuals (Draper and Smith 1981, p. 325—327); Fl is the fit index; SEE is theestimated standard error of estimate in cm; RMSE is the root mean square error in cm; Bias is the meanbias in cm; and MAD is the mean absolute bias in cm.From the fit and prediction statistics (Table 3.3), it can be seen that Equation 1 wasslightly better than Equation 2 (linear models) in terms of fit and prediction statistics.Overall, Equation 3 (a nonlinear model) had the best fit aild prediction statistics. Based onthese results, Equation 3 was selected for predicting plot quadratic diameter. The selectedmodel was:QD = S12826423 x AGE°690036 x exp[—4.290671— (0.360959S1) + (0.007929S12)] (345)This model was then conditioned for predicting quadratic mean at a given (reference)age, similar to the procedure used to develop anamorphic site index curves. Without priorinformation about stand densities for the plots, it was necessary to bring all the plots toa common age (to standardize density for the various ages of plots). Fifty years at breastheight was selected as the reference age. The predicted quadratic mean diameter at 50 yearsbreast height age (QD50) can be defined as the expected quadratic mean diameter for a givenstand growing on a given site at that age. As an example, Figure 3.2 shows QD.50 estimatesC.)z00•for a site index of 20 m (see Appendix B, Figures B.16 and B.17 for site indices of 10 and15 rn).Predicted quadratic mean diameter at breast height age 50 years (QD50) was calculatedfor each stem analysis plot using the following model:QD50 = QD — [(812826423 x exp(X1)) x (AGE°’690036 — 500.690036)1 (:3.46)where X1 = —4.290671— 0.36095981 + 0.007929812.The stem analysis data with all the associated measured and calculated variables (Table 3.4) were stratified into five-centimeter clbh and five-meter height classes and tbei. foreach class, the observations were randomly divided into model development and validationdata subsets. Seventy percent of the data (612 tree-measures, representing 5916 sectionmeasurements over time) were used for model development; model assessments were carriedout using the remaining data as a validation data set (275 tree-measures, representing 2668sectional measures over time). These data were considered to he representative of the whole72Chapter 3. Methods5030201000 50 100 150 200BREAST HEIGHT AGE (years)Figure 3.2: Quadratic mean diameter (cm) for site index 20 m by breast height age.Chapter 3. Methods 73Table 3.4: Summary statistics for the stem analysis data.Variable Mean Standard Minimum Maximumdeviationdbh (cm) 8.84 (8.82)a 4.76 (4.60) 1.0 (1.2) 23.5 (21.9)H (m) 8.48 (8.06) 4.44 (4.14) 1.6 (2.0) 22.8 (20.9)CL (m)b 6.99 (7.20) 1.91 (1.67) 2.8 (4.3) 12.7 (12.2)CR .582 (.624) .1320 (.1494) .342 (.378) .943 (.947)AGEC (years) 21.6 (22.7) 13.8 (13.6) 1 (2) 61(57)SId (m) 17.1 (16.1) 3.9 (4.1) 9 (9) 24 (24)BA (m2/ha) 21.55 (22.84) 12.12 (12.55) 2.66 (4.09) 53.24 (53.24)SPH 3,702 (3,851) 2,814 (3,032) 350 (350) 10,925 (10,925)QD50 (cm) 14.3 (14.0) 4.7 (5.0) 6 (6) 25 (25)aThe stem analysis data were divided into a model-development data set (612 tree-measures, the valueswithout brackets) and a model-validation data set (275 tree-measures, the values in brackets).bCL is crown length and CR is crown ratio (the ratio of crown length to height). These are tree variablesbased on the final felling data. The number in brackets are for the validation data set of 41 trees and thenumbers outside the brackets are for the development data set of 94 trees.CAGE is the tree breast height age (years).dSI (site index), BA (basal area per ha), SPH (number of stem per ha), and QD50 (quadratic meandiameter at age 50 years) are plot statistics based on .30 plots while the rest are individual tree statistics.population. Figure 3.3 and Table 3.5 show the distribution of the fit and validation data setsby age class and by height and dbh. The rationale for data splitting is discussed in section3.3.6.3.2 Examination of Tree Form and Taper VariationTree form and taper change along the stern continuously (Kozak 1988; Newnham 1988) atone time and over time (Larson 1963; Clyde 1986). These changes in form and taper areprocesses which depend on changes in other factors such as site, stand, and tree variables(Section 2.2.2). Instead of developing an entirely new taper model, existing static tapermodels (see Chapter 2) were investigated as possible candidates. Two static taper functionsChapter 3. Methods 741 ZPC11.0z::::ZEZEZZZz11 2345678910111213Age assesFigure 3.3: Fit and prediction data distribution by breast height age classes (years). Theage classes represent age class ranges: age class 1 represent 1-5 years, age class 2 represent6-10 years, age class 3 represent 11-15 years, ..., age class 12 represent 55-60 years, and ageclass 13 represent 61 years and over.Table 3.5: Number of tree-measures by height (m) and dbh (cm) classes.HEIGhT 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18,0 20.0 22.0+ TOTALDBH1 22(7) 1(1) 0 0 0 0 0 0 0 0 0 23(8)2 27(10) 18(5) 0 0 0 0 0 0 0 0 0 45(15)3 6(3) 34(18) 3(2) 0 0 0 0 0 0 0 0 43(23)4 0(1) 25(11) 12(5) 1 0 0 0 0 0 0 0 38(17)5 0(1) 13(9) 28(14) 5(2) 0 0 0 0 0 0 0 46(26)6 0 3(2) 33(11) 9(5) 2 0 0 0 0 0 0 47(18)7 0 1(1) 22(7) 23(7) 4(6) 0 0 0 0 0 0 50(21)8 0 0 6(5) 24(11) 14(6) 3(1) 0 0 0 0 0 47(23)9 0 0(1) 1(7) 13(9) 17(7) 10(4) 0(1) 0 0 0 0 41(29)10 0 0 0 9(1) 18(7) 12(7) 2 0(1) 0 0 0 41(16)11 0 0 1 5(2) 15(5) 13(4) 2(2) 1 0 0 0 37(13)12 0 0 1 3 6(7) 12(2) 8 2(1) 0 0 0 32(10)13 0 0 0(1) 1(1) 7(4) 6(2) 6(4) 5(1) 1 0 0 26(13)14 0 0 0 1(2) 1(1) 10(5) 4(3) 5(1) 2(1) 0 0 23(13)15 0 0 0 0(1) 0 5 7(5) 5(3) 1 2 0 20(9)16 0 0 0 0 2(1) 2 4(1) 5(1) 3(2) 0 0 16(5)17 0 0 0 0 1(1) 1(1) 0(1) 5(3) 4 2 0 13(6)18 0 0 0 0 0(1) 1 0 2(1) 2(1) 2 0 7(3)19 0 0 0 0 0 2 1 0 2(3)2 0 7(3)20 0 0 0 0 0(1) 0 0 1 1 2(1)1 5(2)21 0 0 0 0 0 0 0 1 0 1(2)0 2(2)22 0 0 0 0 0 0 0 0 0 1 1 223 0 0 0 0 0 0 0 0 0 0 1 124+ fl 0 0 (1 11 fl fl 0 0TOTAL 55(22) 95(48) 107(52) 94(41) 87(47) 77(26) 34(17) 32(12) 16(7) 12(3) 3 612(275)Note: The numbers in brackets are the values for the validation data set.Chapter 3. Methods 75were selected, the simple taper equation (Equation 2.1) and the variable-exponent staticequation by IKozak (1988) (Equation 2.15). Equation 2.1 was used because the values of rcan be associated with known stem shapes (Figure 2.1). Kozaks (1988) static taper functionwas selected for use in this research because it was shown to be consistent and precise inpredicting d along the stem and total tree volilme (Newnham 1992; Perez et al. 1990) and itrelates well to Equation 2.1. However, this taper function has a form exponent that cannotbe directly related to the theoretical tree shapes shown in Figure 2.1. Therefore, the simplestatic equation was selected to examine the relationship between the calculated values forthe form exponent and the actual tree geometric shapes.To meet the first objective of this research, changes in tree form and taper over time wereexamined in relation to changes in tree, stand, and site variables using all the stem analysisdata (887 tree-measures). The analysis involved the following steps:1. The form exponent (r) was plotted against measured and calculated stand, tree, andsite variables at specific relative heights (10 percent height, 50 percent height, and 80percent height) to identify important variables that appeared to be related to form andtaper changes. The variables examined included QD50, AGE, CL and CR based on thefinal felling data, height, dbh, the dbh over height ratio hereafter referred toas the D/H ratio), and SI.2. Six trees with a range of ages and sizes were selected arbitrarily to demonstrate thevariation in tree form and taper. The selected trees were categorized as large andold (tree number two in plot 16 (16.2)), large and young (tree number one in plot 46(46.1)), small and old (tree number two in plot 25 (25.2)), and small and young (treetwo in plot 43 (43.2)). Two additional trees were included, which were assumed to heof middle age and size (tree number two in plot 13 (13.2) and tree number one in plot42 (42.1)).3. For each of these six trees, the variation in form (rt) and taper were examined asChapter 3. Methods 76follows:(a) Plots of against breast height age at the three relative heights on the stem(O.1H, and O.8H) were drawn. These plots were intended to show howtree form varied with time at different positions on the stern.(b) Plots of against relative height were obtained. These plots were intended toshow how tree form changed along the tree for a given age (measurement period).(c) Relative diameter (the ratio between diameter inside bark at a given height (d)and dbh (D), i.e., hereafter referred as d/D) was plotted against relativeheight (height at a given diameter (h) to height, i.e., hereafter referredas h/H). This was intended to show tree taper variation as a function of relativetree height.(d) Finally there was a need to see how tree shape changed along the stem duringgrowth. Therefore, three-dimensional plots of against and AGE wereprepared for the same six trees.3.3 Dynamic Taper Model DevelopmentModel development was carried out in three steps: (1) finding the factors that affect the formexponent of taper function, (2) finding models for predicting total tree height, breast heightdiameter, and the selected variables correlated to the form exponent, and (3) substitutingthese models and the variables into Kozak’s (1988) static taper model to make it dynamicand refitting the model. The fitted dynamic taper model was then assessed for its predictiveabilities for diameter inside bark along the stem and total tree volume.Turning a static taper equation into a dynamic taper function involves specifying a function capable of predicting over time at given heights (h) above the ground. However,in order to predict dynamically, variables used to predict d which change with time suchChapter 3. Methods 77as the form exponent of Kozak’s (1988) function (c), height, and dbh would have either tobe measured, which is most times not possible, or predicted over time.As part of the dynamic taper model development and testing process (objective two ofthis research), prediction models for height, dbh, and the variables highly correlated withc were determined. The models built had to be biologically meaningful and to give goodpredictions. The biological criteria included making sure that variables which are knownto be biologically correlated with the variable being modelled were included in the models.Such variables included SI and age for the height model, age and density for the dbh model,and D/H ratio, age, and density for the form exponent. The statistical criteria used to assessthe models were PT, SEE, RMSE, mean and mean absolute biases (Bias and MAD) (seeSection 3.1.2 for definitions), and the characteristics of the residuals (i.e., plotting residualsagainst predicted values and against the independent variables used).Since height, dbh, and the form exponent could not be measured over time, they hadto be predicted. Therefore, models had to be developed based on either the stem analysisor PSP data, for predicting height and dbh. The stem analysis data set would have beenbetter, but it lacked stand density measured over time and was small. For that reason, thePSP data were used to select the best models for predicting height and dbh, and then theselected models were fitted using only the measurements of the stem analysis taken at timeof felling (final felling measurements).3.3.1 Height Prediction ModelHeight growth in lodgepole pine depends on the site quality, tree age, and stand densitywhen trees are still young and often at extreme densities (Section 2.1.1). A literature searchwas carried out for height prediction models based on these variables. Many models werereadily available which predict height as a function of site index and age, commonly calledsite index curves. Other models, often incorporated in growth models, predicted height as afunction of dhh. Two models were selected. one linear and the other nonlinear.Chapter 3. Methods 78The selected linear model was proposed by Alexander et ci. (1967). This model includedbreast height age, site index, and stand density as independent variables. Crown competitionfactor (CCF) was used as the measure of stand density. However, for this research QD50was used instead of CCF (see Section 3.1.2). The height prediction model was therefore:Hkrno = bo+blAGEkrn+b9AGEmHb3(.I xQD5Ok)+b4(AGEkm X SIk)+b5(AGErn x SIk)(3.47)where Hkmo is the predicted total tree height; k is the PSP number; in is the measurementperiod; o is the tree number in plot k; AGEkm is the average plot age at period in; SIk isthe site index for plot k; and the b0 to b5 are the coefficients to be estimated.In addition to using t.he above model with QD50. a reduced model (without a measureof stand density) was also tried. The reduced model was:Hkmo = b0 + blAGEkm +b9AGE + b3(SIk X AGEk) +b4(AGEL X SIk) (3.48)where b0 to b4 are parameters to be estimated and the other symbols are as defined above.The selected nonlinear model was developed by Goudie (1984; as referenced in Goudie etal. 1990). This top height prediction model is based only on site index and breast height agewithout a measure of stand density. In other words, Goudie’s model assumes that densitydoes not affect height growth of dominant and codominant trees. The model is a conditionedlogistic function as shown below:H —13 (SI 3 1.0+exp(co — (ciln(50)) — (c2ln(SIk —1.3)) 349kmo + k j X 1.0 + exp(co — (ci ln(AGEk)) — (c2 ln(SIk — 1.3))where c0 to c2 are the parameters to be estimated using NLS and the rest of the symbols areas defined above.The linear prediction models were fit by OLS, using the REG procedure in SAS (SASInstitute. Inc. 1985) and the nonlinear model was fit by NLS, using the NUN procedure inChapter 3. Methods 79SAS. For all fits, the PSP model development data composed of 1334 trees was used. Theparameter estimates given by Goudie (1984; as referenced in Goudie et al. 1990) were used asthe starting values for the nonlinear model. In order to ensure that the parameter estimatesresulting from the NLS fit were optimal (the global minimum residual sum of squares), thethree methods available in SAS NUN procedure were used with multiple starting values.These were the Newton-Gauss method which uses Taylor series, the Marquardt methodwhich uses the updating formula method, and the Secant or “Doesn’t Use Derivative” (DUD)iterative method (see SAS Institute, Inc. 1985, p. 585—6).The models could not he compared based on the t and F tests because they were ofdifferent types. Also, the error terms would not be iid because the observations are expectedto be serially correlated. However, the prediction abilities of the models would not be affectedbecause the coefficients estimated using OLS are unbiased and consistent (Kmenta 1971).Model selection was based on comparison of the fit statistics and the prediction statisticsbased on the 574 trees in the validation data (185 plots). The prediction statistics usedfor model evaluation included the R2, SEE, Fl, SEE, RMSE, Bias and MAD (see Section3.1.2, p. 69—70 for definitions).One problem with the height prediction models was that they were based on plot ageinstead of tree age. This meant that the selected model would predict average plot heightinstead of individual tree height. If average plot measures from the PSP data were used todevelop the height prediction model, the resulting taper model would be predicting averageplot tree form and taper. As a result, individual tree characteristics would he lost and thedynamic taper model would give biased predictions for individual tree diameters along thestem. To alleviate this problem, the PSP data were used for model selection and the finalfelling stem analysis data were used for model calibration. The resulting model was:H — 13 1 3 1.0+exp(co — (c1ln(50)) — (c2ln(SIk —1.3)) (350)+ k X 1.0 + exp(co — (c1 ln(Age)) — (c2 ln(SIk — 1.3))where is the predicted tree height: i is the tree number; t is the five-year measurementchapter 3. Methods 80period for the stem analysis data; 51k is plot site index at 50 years breast age; Aget is treebreast height age; c0 to c2 are the parameters to be estimated using NLS; and the rest of thesymbols are as defined previously. After calibration, the height prediction model was usedto predict heights for each tree over time.3.3.2 Prediction Model for Diameter at Breast HeightDiameter growth is highly correlated with stand density, AGE, SI, and a crown size measuresuch as CL (Section 2.1.2). However, CL or crown diameter, found by Sprinz and Burkhart(1987) to be an important variable in dbh prediction, was riot used in the dbh predictionmodel because CL was only measured at the time of felling for the stem analysis data andcrown diameter was not measured at all.The growth and yield literature was searched for an existing dhh prediction model. However, most models predict dbh increment. Alternatively, the relationships between the dependent variable dbh, with possible predictor variables were examined by plotting dbh againsteach predictor variable (H, AGE, SI, QD50), and transformations and interactions of thesevariables. These plots enabled assessing relationships. if any, that existed between dhh andthe variables. The RSQUARE procedure in SAS (SAS Institute, Inc. 1985) was used toselect a few candidate models based on minimum MSE, highest R2 and R, and lowest C.Three linear models were selected for further testing:Dkmo = b0 + blHkmo+b2QD5Ok+b3QDok+b4(QD5okX AGEkm) +b.(QD5okX AGEm) (3.51)1-krno = b0 + biHmo + b2SIk +b3(SIk x Q135ok) +b4QD5ok + b5(QDOk)2+b6(QD5ok X AGEkm) +b7(QD5ok X AGEm) (3.52)Dkmo b0 + bimo +b2Hmo +b3SIk +b4SI + b5QDok + b6QDok +b((QD5Ok x AGEk) + bsAGEk, (3.53)Chapter 3. Methods 81where Dkrno is the predicted dhh; and b0 to b8 are the coefficients to he estimated. Alltrees in the same PSP have the same SI, and QD50. AGEkm represented the plot age atmeasurement period m.These models were fit using OLS by the REG procedure in SAS using the PSP fit data.The models were then tested on the validation data set for their predictive abilities. Thestatistics used for model evaluation included R2, SEE, PRESS statistic for the fit data, andFl, SEE, RMSE, Bias and MAD for the validation data (see Section 3.1.2, p. 69—70 fordefinition of these statistics).The dbh prediction models had the same problem as the height prediction models in thatthey were based on plot predicted height and plot age instead of individual tree height andage. This meant that any model selected would predict average plot dbh, not individual treedhh. To alleviate this problem, the PSP data were used for model selection and the finalfelling stem analysis data were used for model calibration. The model that was calibratedwas:= b0 + + b2H + b3SIk + b4SI + b5QDok + b6QDok +br(QD5Ok x Age) +b8Age (3.54)where b7 is the predicted dbh for tree i and measurement period t for the stem analysisdata; b0 to b8 are the coefficients to be estimated; and the rest of symbols are as alreadydefined.3.3.3 Relative Height FunctionIn a number of top sectional measurements, the valile for 1ijt was higher than the predictedheight (i.e., at the tip of tree is greater that As a result:1. both 1 — and were no longer bounded by zero and one.2. when = 1—became negative, the logarithm was not defined.Chapter 3. Methods 823. whenever the form exponent of the Kozak’s (1988) function (c) was less than one, theresult of was undefined.In order to correct this problem, predicted values for were constrained between zeroand one using a logistic function as follows:1 + exp[fi] (3.55)where is the predicted relative height; and f is a function of the independent variablesselected.In order to find out the most important independent variables for the above model, theRSQUARE procedure in SAS (SAS Institute, Inc. 1985) was used to construct a linearfunction for fi using the following 11 independent variables: Hj,--—.]i—, h(t), x fiji, x H)2 x Two models were selected:1. f1(h, H) = b0 + b1h +b2h + + - + b5H + b6 ln +b7(h X fI) + b8 Xx H2 +b9(h x pit))2. f1(h, H) = xwhere b0 to b9 are the coefficients to be estimated.The two linear models for f, were substituted into Equation 3.55. The following twologistic models resulted.= 1/[1 + exp(bo + b1h + b21i +— + +b5fI +b6lnH +b7(h x ) +b8(h x )2 +b9(h; x H))] (3.56)Z = 1/[1 + exp(bo + b1h + b2h +-+ + b5 in fi +b6(h x H) +b7(h x H)2 + bs(h x ]))] (3.57)Chapter 3. Methods 83These two functions were fitted with the DUD option in the NUN procedure of SAS (SASInstitute, Inc. 1985) to data generated using the measured heights froni the stern analysisfit data for which the values were between 0 and 1. The coefficient estimates from theRSQUARE procedure for fi were used as starting values for the logistic nonlinear models.Model selection was based on the fit (R2, SEE) and prediction statistics (FT. SEE, RMSE,Bias, and MAD).3.3.4 Variables Selected for Estimating the Form ExponentIn order to find the variables correlated with c, graphical analysis was carried out. Plotsof against the variables found to be correlated with age, QD50 and transformations of these variables) were constructed. In addition, cjj values were plotted againstall the variables used by Kozak (1988) in the form-exponent.Using the knowledge gained from objective one and these graphs, all the variables foundto be correlated with r, and transformations of these variables, were used to find the bestmodel for predicting Cjj. The variables selected were: SI, Age, QD50 D,ratio, and Not all transformations of the variables used by Kozak (1988)were used. The variable ln(2 + 0.001) was dropped, because ln(Z) is undefined at thetree base; Kozak added an arbitrary constant of 0.001 to obtain a value at the tree base.The variable exp(Zi) was only marginally correlated with and also was dropped beforethe analysis. The RSQUARE procedure in SAS (SAS Institute, Inc. 1985, p. 711—724) wasused to find the best subset model based on remaining variables.The two best subset models for predicting based oii minimum MSE, highest R2 andR, and low C were:1+b5QDok+b6br± (3.58)IIit gej II ijtChapter 3. Methods 84andêijt = +b4A’ (3.59)it J 2t zjtwhere ij1 is the predicted form exponent; i is the tree number; j is the section numberon tree i: t is measurement period; and b0 to b8 are coefficients to be estimated. For bothequations, when = 0 at ground level, a small constant such as the one used by Kozak(1988) would he added to to become + 0.001 so that the models are defined.3.3.5 Dynamic Taper Model Assembly and FittingThe two selected models for predicting cj and the predicted values for Z, andwere incorporated into the Kozak (1988) taper function (Equation 3.44) by substitutingthese for the variable-exponent (c), D, and H rspectively. For comparison purposes,Kozak’s (1988) taper function was fit using Z, and Dil as independent variables. Thisdynamic version of Kozak’s (1988) model was included in order to compare the new dynamictaper models with a dynamic taper model obtained simply by using predicted height anddbh. The static model was included since it represented the best predictions possible usingmeasured height and dbh that is standard for current inventory. These four taper models(three dynamic and one static) are presented below.1. Kozak’s (1988) static taper model (IViodel 1).— a1 DbiZ.1+b2ln(Zjjf+0.001)+b3\/+b42t l-zjt — a0 X it X a X2. Kozak’s (1988) dynamic taper model (Model 2) (variables used: b, &t, and,‘ b1Z+b2 +0.001)+ba\/ +b4e3t+b5— j-ai 1vZ3— a0 it a2 x3. New dynamic taper model (Model 3A).a1 i/bl h1+O.OO1—aox it xa x 1\/Chapter 3. Methods 854. New dynamic taper model (Model 3B).— r— bo+bj:f+b2v +bl1+b4_+ .oQDook+b in 1i0+b7 001 +b8d — 1a D1 v 2‘ijt — a0 awhere a0 to a9 and bo to b10 are coefficients to be estimated; and q is given as ForModels 2, 3A and 3B, when hj will equal DI, since = = 1.0. Atthis poiit, is equal to a0 x D’ x a, which corresponds to predicted Therefore,at this point, the predicted form exponent has no impact on the since (M)ciit for anyexponent equals 1.0. The predicted value of will only be affected by the measured orpredicted and measured or predicted since HI is some fixed proportion of Thestatic model (Model 1) has the same properties, except that the measured values for dbhand height are the inputs instead of the predicted values.These taper functions are intrinsically nonlinear in the parameters, if they are assumedto have additive error terms. However, if it is assumed that they have multiplicative errorterms, as Kozak (1988) did, then the taper models will be iitrinsically linear, because theycan he expressed in linear forms by logarithmic transformation. Therefore, the four taperfunctions can he fit using either OLS or NLS, depending on the assumptions made aboutthe error terms.During the model fitting process, the above dynamic taper equations were assumed tohave either (1) additive error terms and were fitted using NLS or (2) multiplicative errorterms and were transformed using natural logarithms and fitted using OLS. This would resultin eight sets of fit statistics, two sets for each taper model. However, before the actual modelfitting process began, the two new dynamic taper functions (3A and 3B) were transformedand fitted using OLS (REG procedure in SAS Institute, Inc. 1985) based on the stern analysismodel development data set. They were then evaluated based on the validation stem analysisdata set. The fit and prediction statistics used for evaluation included R2, SEE, PRESS, Fl,SEE, RMSE, Bias, and MAD. The model with the better prediction statistics was selectedand labelled as Model 3.C’hapter 3. Methods 86In addition to these three models, a test using the coefficients of Model 1 (static), hutwith predicted height and dbh was included. Commonly, growth and yield models use anexisting taper function and simply input predicted height and clbh (e.g., Arney, 1985). Thestatic model with measured height and dbh was labelled Model la and the static model withinputted predicted height and dbh was labelled Model lb.Therefore, eight taper models were examined. Four models were dynamic (two linear andtwo nonlinear), two were static (one linear and the other nonlinear), and two were the staticmodels with inputs of predicted height and dbh. For clearer identification of the models,“N” was added to the model number to indicate a nonlinear fit and “L” was added to themodel number to indicate a linear fit.Using the above taper models, the following comparisons were therefore possible:1. Model la with Models 2 and 3. The static model (Model la) would be expected toperform better than Models 2 and 3 because it is based on measured height and dbh.Model la is the standard model used for current inventory taper predictions.2. Model lb with Models 2 and 3. Model lb is the standard for use in growth andyield predictions of tree taper. Model lb should perform better than Model 2 if thecurrent procedure used in growth and yield is accurate. However, if Model 2 performsbetter than Model ib, then the current procedure used in growth and yield should bequestioned. Models 2 and 3 are expected to perform better than Model lb becausethese dynamic taper functions were fitted as a system.3. Model 2 with Model 3. Model 3 is expected to perform better than Model 2 becauseModel 3 includes new variables which were found be correlated with tree form, andModel 3 was developed specifically for modelling taper over time.Chapter 3. Methods 873.3.5.1 Optimization of qAs stated previously, q (the percent height at which the join point ocdllrs) was found to varybetween 22 and 27 percent. An attempt was made to find the best value which minimizedthe variation around the taper function. During model fitting using NLS, the DUD iterativemethod in NUN was used with the OLS coefficient estimates as the starting values. Model3N was fit by trying to optimize the coefficient estimates as well as the value of q. The qvalue was limited to between 0.00 and 0.35 and incremented by 0.005. The procedure tookover 45 minutes of CPU time to converge. The optimum value obtained was very small(q=0.0003). This value was too small to make any impact on the model; therefore, two moreNLS runs were made for model 3N. One run was based on q=0.00 and the other on q=0.25.The results showed no difference for the coefficients using q=0.00 versus the optimum valueq=0.0003. However, for q=0.25, there were some differences. Optimization of q was triedfor Kozak’s (1988) taper model (model 2N) and the results showed that scaling was notnecessary.For all further analyses using models 3L and 3N. no scaling was assumed (i.e.. q = 0.0).However, for models laL, laN, lbL, lhN. 2L, and 2I the scaling factor of q=0.25 wasmaintained, to agree with Kozak (1988).3.3.5.2 Fitting the Taper Models using OLS and NLSAfter transforming the data using the natural logarithm, models 1 to 3 were fit using OLS.Based on the findings from the optimization process for q, model 3L was fit using no scalingfactor (q=0.0). All taper functions were fit using the REG procedure in SAS (SAS Institute, Inc. 1985. p. 655—709) using the stern analysis fit data. The resulting coefficientestimates were used in the evaluation of the OLS fitting method. These coefficient estimatesare unbiased and consistent, but are inefficient (Kmenta 1971) because of the error termcharacteristics outlined in Section 2.4. Graphical analyses of the residuals were performedchapter 3. Methods 88to check for lack of fit.After fitting the three taper models using OLS, the estimated coefficients were used asstarting values in the NLS fitting. NLS was applied to the three models using the procedureNUN with the DUD method in SAS (SAS Institute, Inc. 1985, p. 586) on the stem analysisfit data. The initial coefficient estimates were varied slightly and the model refit until thecoefficients estimates stabilized. This process ensured getting the global minimum valuesfor the sum of squared residuals. The resulting coefficient estimates are consistent andasymptotically normal, but might not be efficient (Judge et al. 1985, p. 198-2O1). Graphicalanalyses of residuals were performed to check lack of fit.3.3.5.3 Fitting the Taper Models using FGLSThe stem analysis data used for dynamic taper model fitting was characterised as havingresiduals which were non-iid. If the dynamic taper models were fitted using OLS or NLS,the coefficient estimates would be unbiased and consistent, but would not be BLUE and theusual estimates of the variances for the coefficients would be biased and inconsistent.Knowing that the OLS and NLS fitting methods for estimating parameters would resultin biased estimates of the variances for the coefficient estimates, the use of FGLS and FGNLSwas considered in order to carry out statistical inferences. The FGLS and FGNLS methodsare slightly different, but the general procedure in both methods is the same and only onemethod, the FGLS for linear models, will be described in detail.Assuming that there are no differences in parameters (regression coefficients) for thedifferent measurement periods, sections and trees, the FGLS parameter estimator is themost appropriate method. Each of the transformed dynamic taper models can be writtenas:jt = /3 + /31Xj + !32X -- +/3loXijno + (3.60)Chapter 3. Methods 89or simplified asK= /3o + Xtk/Jk + (3.61)k= 1where is the natural logarithmic transformation of djj; XJtk is the kt independentvariable, k = 1, 2, , K; K is the number of independent variables used in the model; i isthe tree ilumber; j is the section number; t is the measurement period; dj to /3o are thecoefficients to be estimated from the data; and is the logarithmic error term associatedwith the predicted 1n(dj) for the sample tree i, section j, and time period t. In matrix formthe above equation becomes:Y=Xd+€ (3.62)where Y is the column vector of dependent variable (in d); X is a matrix for the Kindependent variables dependimg on the model (these variables included predicted D andthe product of mM and the selected variables for c); /3 is the column vector of modelparameters to be estimated; and € is the column vector of ramnlom error termsThe vector Y and the matrix X appear as:Chapter 3. Iiethods 901 X111Ki1T, 1 X11T21 X1212 X19111 X1-3 X1MT2 X1M1T3K= Y211 X = 1 X9111 X2112 X2111.1 X22 X2T3K1 X2fT1 X21iT X2MT1KNMT3 1 XNMT1 XNMT32 XNMT3KFor the analysis, all the cl measurements on all sample trees of the model development dataset for all time periods were pooled together (5916 x 1). Different trees were assumed to havebeen randomly selected in the field and, therefore, were assunied to he independent of oneanother. However, these trees caine from different plots in different areas, they had differentcrown sizes and growth rates and, therefore, were assumed to have differing variances. Thematrix (X) for the independent variables is a x K matrix (5916 x 11). /F isa K x 1 (11 x 1) column vector shown below:eli’eli2andeNMT..where is an Z x 1 (5916 x 1) vector.Chapter3. Methods 91To fit the three taper functions using FGLS, the components of would be obtained byfirst calculating the ,ô1j’s (Equation 2.26), the components of followed by estimating(Equation 2.38). Finally, cz would be substituted into Equation 2.40 to get /3F• This fittingprocess would follow the following steps:1. The first stage would begin by pooling all the observations and running OLS usingEquation 3.60. Since the OLS coefficient estimates (b) are unbiased and consistent(Kmenta 1971), these would be used to estimate the residuals (et) for each tree asfollows:=— jt (3.63)where Yj is the predicted value using OLS.2. Using these residuals, a consistent estimate of Pu, assunhing first order autocorrelation,would he calculated for each section using Equation 2.26.However, the following problems were recognized in estimating the Pij ‘S.(a) For all trees, the last section had oniy one serial measurement. Therefore, calculation of j was not possible. Since it contained only one measurement, correlationwas assumed to be perfect and would be assigned a value of one.(b) For every tree, there were some sections with only two serial measurements. Thismeant that the calculated values may be unreliable.(c) Using 1ojj = 1 for the last sections for each tree meant these measurements wouldnot be transformed. As a result, these sections would be lost.To solve these problems, the possible solutions would have been:(a) If the time series measures are very few, it may be preferable to impose therestriction of a common p (Baltagi 1986), even if it is violated in the population,chapter 3. Methods 92in order to improve the small sample performance of the FGLS estimator (Greene1990, p.474). This means that a single p value for each tree (pj) would be used.From the equation for calculation of p (see Equation 12.32 of Kmenta 1971, p.512), sections with only one periodic measllremeut would not be used to estimatep using Equation 2.26. As a result, the estimated value for tree i will be biased.(h) Use the econometric methods for fitting panel data, i.e., use varying coefficientmodels (see Judge et al. 1985, p 415—425, for disscusion of these models). However,as already noted in Section 2.4, such models are not practical for taper modelling.(c) Another option would be to pool the sections together for all trees and try to findmodels that predict the variation in p for individual trees or p for each sectionalong the stem as functions of tree, stand, and site variables as:= function(tree, stand, and site variables)orfunction(tree, stand and site variables)Pooling sections together over trees to calculate a common pj for each tree, assugggested by Baltagi (1986), or pooling all observations for all sections for alltrees together for calculating a common pj, are possible. However, these treeswere sampled from different forests in different areas, meaning that they haveexperienced different growing conditions even for trees of the same age, and also,some trees were at different growth stages at the different locations. It is verytenuous to assume that all trees experienced the same growing conditions at different ages. Therefore, pooling the data together in this manner might result inpoor estimates. Also, based on Gertner’s (1985) study, serial correlation mightnot have been a serious problem for the nonlinear models, because the remeasurement period was five years. Gertner found the relative efficiency of NLS toFGNLS to increase with the measurement interval. For a measurement intervalChapter 3. Methods 93of five years, the relative efficiency of NLS compared to FGNLS was 99 percent;Gertner said that NLS is almost equivalent to FGNLS for measurement intervalsof five or more years.3. Since ô1 is a consistent estimator of Pij (Kmenta 1971), the original data would betransformed using this estimate and Equations 2.27 to 2.30.4. The resulting estimated residuals (ujj) (Equation 2.31) would he serially uncorrelatedand would he used to estimate sectional variances and covariances using Equations2.33 to 2.36. However, following problems were identified at this stage.(a) For all sections with one serial measurement, the variance could not estimated.Also, for sections with only two serial measurement, the estimated variances basedon two observations could be unreliable.(b) For a given tree, sections had unequal numbers of seria.l measurements (i.e., jdecreases from sections near the ground to one serial measurement for the lastsection). From Equations 2.33 and 2.36 for calculating covariances, some observations would not be utilized (from larger sections with more observations).Therefore, the estimated covariances between such sections could he unreliable.Some possible solutions to these problems include:(a) Pooling the data over sections for all trees and then calculating common estimatesfor variances and covariances. However, these trees were samples from differentforests in different areas, and the variances among sections vary considerably.(b) Pooling the data for all sections of each tree and then calculating a single variancefor each tree. This choice also does not make much sense, because it is apparentfrom graphs of versus that the variances for the different sections of a treedecrease as increases.Chapter 3. Methods 94(c) Another option would be to pooi the data together for each section over all treesand try to find models that predict the variation in cr along the stem as functionsof tree, stand, and site variables:= function(tree, stand, and site variables).The resulting coefficient estimates using F’GLS would be asymptotically efficient if aconsistent estimate of could he obtained. Since the fitted model would involve the useof a large quantity of data, the asymptotic properties of FGLS (Kmenta 1971) would alsoapply. This means that the coefficient estimates and their variances would be better thanthose estimated using OLS and NLS. However, with the problems noted in obtaining thevariance-covariance matrix, O, estimated coefficients could be greatly biased and even lessefficient than OLS or NLS estimates.Therefore, FGLS was not used. If the model to be developed is to be used for prediction,as the dynamic taper model would be, then it is more important to have unbiased coefficientestimates. OLS and NLS fit provide such estimates.3.3.6 Dynamic Taper Model Testing and EvaluationModel validation involves determining the quality of model predictions. As used by manymodellers, model validation refers to the process of assessing, in some sense, the degree ofagreement between the model and the real system being modelled (Reynolds et al. 1981).Snee (1977) and Neter et al. (1985) suggested the following ways of validating a model:1. Examining the model’s performance on the model-building data set or self-validation.This indicates how good the model is for predicting within the confines of the data set.2. Comparing the model predictions and coefficients with theoretical expections, earlierempirical results, and simulation results. This is not possible if there is no priorinformation available.Chapter 3. Methods 953. Collecting new data to check the model and its predictive ability. This would be thebest choice, but it may not be possible or may be very expensive, making it infeasible.4. Using part of the data collected as an independent data set to check the model’s prediction (data splitting) or the techniques of cross-validation. jacknifing, and bootstraping(Gong 1986).The validation technique most used by taper modellers is to split the data into twosets of varying proportions. Newnham (1988, 1992) and Kozak (1988) used one-half oftheir data for validation, Perez et al. (1990) and Byrne and Reed (1986) used one-third,while Max and Biirkhart (1976) used one-fourth. The data for validation can either besystematically selected (Newnham 1988, 1992) or randomly selected (Kozak 1988; Byrneand Reed 1986 Perez et al. 1990; Max and Burkhart 1976). As stated previously, the PSPand the stem analysis data were split randomly into two sets: the model development (70%)and the validation or prediction data sets (30%). Only a small proportion of data (30%)was retained for model validation because with only 135 trees, a bigger proportion had tobe used for model fitting.The process of model selection should he based on both practical and statistical considerations. For practical and theoretical purposes, a taper model should he easy to apply,coefficient estimates should be easily derived, the variables used should be easy to measure,and the model should be able to track the real system being modelled. In this case, themodel selected should be able to take into account the variation in tree form over time fromground to tree tip.Model selection for this research involved comparing t.he measured and calculatedtotal tree volume with and V, respectively, as generated by each fitted taperfunction. The accuracy and precision of the predictions for each taper equation dependedon how well the fitted taper equation was able to track the tree profile. Unfortunately, thefitted taper models could not be integrated to an exact form therefore, numerical integrationChapter 3. IViethods 96was used for volume prediction.Comparison of the four taper models for each of the two fitting techniques (NLS andOLS) was based on fit and prediction statistics. The fit statistics used were R2, SEE, andPRESS (for linear models only), while the prediction statistics used were Bias, RMSE, MAD,SEE, and FT. It should be noted that the predicted values used in calculating the predictionstatistics were based on rather than in which was ilSed in the fitting process, in orderto facilitate comparison of the linear and nonlinear models and to better relate to the realsystem being modelled.When a model is logarithmically transformed, its predicted values will systematicallyunderestimate the actual values (Baskerville 1972; Flewelling and Pienaar 1981). Therefore,the logarithmically transformed taper models had to he corrected for this bias. According toFlewelling and Pienaar (1981), if the number of observations minus the number of coefficients(DF) to he estimated is greater or equal to 30, and the variance estimate (MSE) for the errorterm of the logarithmic model is less than 0.5, the bias will be less than one percent. Forthis research, the smallest DF was 5907 and the largest MSE was 0.066. With such largeDFs and small error variances, the biases would be very small. Nevertheless, Flewelling andPiennar’s (1981) correction factor (CF) of exp(MSE) was used for the OLS fitted models.This CF value was applied by calculating the predicted value of as follows:= exp(MSE) x exp(ln(dt)) (3.64)All the fit and prediction statistics identified above were used in selecting and evaluatingthe developed dynamic taper model, because no single criterion is best. A model can performbetter using one measure and poorer using another; therefore, using any one criterion alonemight not give a true picture of the model.Graphical analysis also was used as part of model evaluation process. Graphs of Bias infor all trees and for three dbh classes along the stem were plotted. As well, graphs forChapter 3. Methods 97predicted and observed tree 4t over time were plotted for the large dbh trees. Predictedtotal volumes for each tree were determined by numerical integration of the taper functions.The observed (actual) estimates of the volume for each tree were calculated using Smalian’sformula (Husch et al. 1982, p. 101-403). The purpose of these graphical analyses was todetermine which regions along the stem were more biased for a given tree or group of trees.Chapter 4ResultsThis chapter is divided into three sections. Section 4.1 presents the results of investigatingthe influence of tree, stand, and site factors on the variation of tree form and taper over time(study objective one) .Also, individual tree taper and form variation with age at differentstem positions and with age and height above ground are presented as three-dimensionalplots. Section 4.2 presents the selected models for predicting total tree height, dbh, relativeheight, and the form exponent (c), along with the fit and prediction statistics (study objectivetwo). Also, the dynamic taper models developed are given. Section 4.3 presents the resultsof assessing the dynamic taper models for their prediction abilities for and total treevolume.4.1 Variation in Tree Form and TaperExaminations of the changes in the form exponent (r) of Equation 2.1 with age, dbh, height,D/H ratio, CL, CR., QD50 and SI were performed using both final felling and periodicmeasurements for the stem analysis data. Since the relationship between r and these variableswas expected to vary over the tree stern, three positions on the tree stem were chosen (0.111,0.5H, and 0.8H).4.1.1 The Variation of Tree Form and Taper with Tree, Stand, and Site FactorsTree form is known to vary with some tree, stand, and site variables. In this subsection, theresults of investigating the relationships between the form exponent (r) and total tree height,dbh, D/H ratio, AGE, SI, QD50 CL and CR are presented. Graphs of these relationships98Chapter 4. Results 99were based on the stem analysis data composed of trees measured intervals of five years (887tree-measures), except for CL and CR, which were based on the final felling data (135 trees).4.1.1.1 Total Tree HeightAt the base of the tree, r increased with increase in total height (Figure 4.4A). The variabilityin r is greater over the height range than the variability of r over the dbh range (Figllre 4.4B).At O.5H or O.8H, the value of r was constant throughout the height range (Figures 4.SA and4.6A). However, the relationship of r with height may be confounded by changes in dhh andstand density as the trees grow.4.1.1.2 Diameter at Breast HeightAs trees illcreased in dbh, the valile of r increased (Figures 4.4B, 4.5B, and 4.6B). The mostnotable changes in the shape coefficient (r) occurred at the base of the tree (Figure 4.4B).Dbh is an important factor in the prediction model for r, particularly for tree butt sections.4.1.1.3 Dbh to Height RatioThe trend of the form exponent (r) with D/H ratio indicated a steep slope at the base of thetree (Figure 4.4C). The value of r also rose with D/H ratio at the middle and top positions ofthe tree stem, but the slope was less steep (Figures 4.5C and 4.6C). The use of D/H ratio asa variable to indicate changes in r may prove more important for predicting form exponentchanges than use of dbh and height separately.4.1.1.4 Age at Breast HeightFor a young tree, no real shape differences were apparent with change in time for any pointon the tree stem (Figures 4.4D, 4.5D, and 4.6D). However, for older trees shape changeswere evident at the base and top of the tree (Figures 4.4D and 4.6D). At the tree base, theChapter 4. Results131IIL0.801OTIL TE K9GHT (m)‘a4[IAa:a’i.e11 14 17SITE t’DEX (m)aa a— a a__ aa•,_•• a-.1 :.- --— a :: 41 % a a• a—a•—a -aFigure 4.4: Form exponent (r) by (A) Height (m), (B) Dbh (cm), (C) D/H ratio (cm/ni),(D) Age (years) at breast height, (E) SI (m), and (F) QD50 (cm) at 0.111 above ground.B100Aa • —a • a aa•a • j• •—‘— a— aii.( ‘‘-‘‘a a4322.4i.eCaa a— a.—q.’1 a> —— — —%•_ aa• ——a—lb aaa0%07 o 1:7 19DUJET9 TO HBGHT R4110 (cn.kn)4’Q.50BFAST H9GHT DETER (au)4.80 B4 aaaa aa a—aq•— aa.i a•a.C 10 3, , 0BEAST HBGHT AEEaIaII— a—•— :i‘‘Iii’’Ia—i.!1 I.:I—aI’ ll’—’..III.4322.41.8Fa aaa a aI :1Laa• a”I—Ia;iiI!I I—I•II aI II’IIIIII’I! I I •_ a• — •a—al:.!10 15QUADRATIC MEAN DIAJ4ETER (au)Chapter 4. Resultsa ••_.•..- a-p.I —•a.0:7 01 11 I’s I.?0T10 PAT daMn)101Figure 45: Form exponent (r) by (A) Height (m), (B) Dbh (cm), (C) D/H ratio (cm/rn).(D) Age (years) at breast height, (E) SI (m), and (F) QD50 (cm) at 0.5H above ground.AS41I42a2.4•...a... I.•— a.— —I.___I010 io10TflTHB47a’C1 10 15Ar€OHT OLaT(S01Chapter 4. Results 102a32a’B4aa •2.4 —a___--Ba-—I’—.-Ioa I’SB*ST HBGHT DLITER (an)4.80432.4 -aSS B—a1..:——a.0.80 10Figure 4.6: Form exponent (r) by (A) Height (m), (B) Dbh (cm), (C) D/H ratio (cm/rn),(D) Age (years) at breast height, (E) ST (m), and (F) QD50 (cm) at 0.SH above ground.A4V.2.4 —I’S0.80TOTATFEE HEIGHT (m)CBa-,iII_I —N—. .•••a— a BI.—-I S0b.3 0.5 0’7 0’9 11 1:3 i:5 17 2DITO HEIGHT PATIO ()Ez 50BAST H8GHTE(31I.1_I: I‘iIIliiiIIiiI!i11F4312.4• a—•. •1.8 I!11111111111H III’•Z 0.8 isQUADPATIC ME1d DI.aNET (an)I II 14 17WEWDEX(rn)Chapter 4. Results 103form exponent changed from a conical shape (r of 2) to a neiloid shape (r of 3) and to highervalues (up to about 5), representing concave shapes. At the tree top, there seemed to besome trend from a paraboloid (r of 1) to a conical shape.The large variability of r with age for all three positions, similar to that shown for height,indicates that the trends of individual trees were confounded by changes in other tree andstand variables. Although no obvious trend of r with age was shown in Figures 4.4D. 4.5D,and 4.6D, the age variable will likely be important once other variables such as dbh areincluded in the prediction equations.Six representative trees were selected and the form exponent or shape variation with agewas analysed for the same relative height positions (Figures 4.7 and C.18 to C.22 of AppendixC). From these individual tree trends, it can be seen that the form exponent (r) increasedwith tree age, particularly for the base section. The increase was more pronounced for largetrees (trees 16.2 and 46.1; Figures C.18 and C.22, respectively) than for small trees (trees25.2 and 43.2: Figures C.19 and C.21, respectively). For all the trees at young ages, theform exponent was almost the same at all relative heights; however, the r value at the baseincreased faster with increasing age, compared to the other relative heights which maintainedalmost the same value. This means that the trees maintained almost the same shape fromground level to tree top as they grew.At young ages, trees had similar shapes at the three positions (Figures 4.7 and C.18 toC.22) with an r value ranging from 1 to 1.5 (more or less paraboloid in shape). However, asthe trees grew, they tended to assume different shapes for the different positions; this mightbe confounded by changing stand density. At 0.1H, trees tended to become conic and thenneiliod if grown in less dense stands (Figure C.18, for tree 16.2; and Figure C.20, for tree42.1), while at 0.5H and 0.8H trees continued to be more or less parabolic.Chapter 4. Results 1044O.1H MEASURES3.5O.5H MEASURES3 O.8H MEASURESaIz 2.5wi.:0.509 14 19 24 29 34 39 44 49 54BREAST HEIGHT AGE (YEARS)Figure 4.7: Form exponent (r) by breast height age (years) for tree 13.2, dbh=15.6 cm.height=16.8 m, and AGE=54 years, QD50=9 cm, at three relative heights (0.111, 0.5H, and0.811) above the ground.4.1.1.5 Site IndexAlthough the relationship of r with SI was quite variable for all positions on the stem(Figures 4.4E, 4.5E and 4.6E), there appeared to be some increase in the form exponentwith increasing SI. This relationship was clouded by changes in dbh due to differences indensity, tree breast height age, and changes in the crown length (Figures 4.8A, 4.8C, and4.8E).4.1.1.6 Predicted Quadratic Mean Diameter at Age 50The relationship of r with QD50 was not very strong (Figures 4.4F, 45F, and 4,6F). Therewas a stronger relationship at 0.1 H than at 0.5H and 0.8H. It should he noted that the datacontained only dominant and codominant trees and this will have an impact on the results.Other measures of stand density. such as basal area per hectare, number of stems per ha.stand density index, and relative density, were examined, but none showed any strongerrelationship than QD50. Basal area per hectare does not necessarily reflect changes in treeChapter 4. Results 105size or average competition, which might have resulted in the lack of a strong relationshipwith r. Also, the number of sterns per hectare does not reflect the tree size. Stand densityindex and relative density showed stronger relationships with r than basal area per ha andnumber of sterns per ha, but weaker than QD30.4.1.1.7 Crown Length and Crown RatioFor dominant and codorninant trees, the relationship betweell r and crown length wasstronger at the tree base (0.1H) (Figure 4.8A) thall for 0.5H and 0.8H (Figure 4.8C and4.8E). As the tree crown increased in length, the base became more swollen, changing shapefrom a cone (r=2) to a neiloid (r=3) or even higher r values. The relationship of r withcrown ratio varied greatly at the base of the tree (Figures 4.8B, 4.8D, and 4.8F). A strongrelationship occurred at the middle and top of the tree, with an increase in crown ratio,corresponding to an increase in the form exponent. As the crown ratio increased, the shapeof the upper bole of the tree changed from a paraboloid shape (r=1) to a conical shape(r=2). Crown length appeared to be a good indicator of butt swell, whereas crown ratio wasa good indicator of stern change within the crown.4.1.2 Variation in Form and Taper within Individual Trees over TimeThe form and taper of individual trees varies with many factors, as shown above. In thissubsection, the results of examining the variation in tree shape and taper for specific relativeheights (h/H) over time are given. Also, 3-dimensional graphs of r versus age and heightabove ground are presented.4.1.2.1 Form and Taper Variation along the StemThe variation in form and taper along the stern for a particular tree over time is demonstratedin Figures 4.9 and 4.10 (see also Figllres D.23 to D.32 of Appendix D). Figure 4.9 for treeC’hapter 4. Results—————•.i.—i-I._: _ i.’gIt1 ‘——,_—._.. —_02 0.4 0.6NRA11O03Zd aa. —ai.e•aaa.——a-a0.L0 02 0.4 0.6 0.8R&11O4F31za——-—i.e,SS0.2 4 06D% R&T1OA106B— —— 4 —• a• .III.. —— ——•5• a--—IIII— I• —•.a_.._:—•— ——I.eS—S3.22.4I .C0._4 è 12 6RDWN LENGTh (m)C31V.2.4I.eSa ——S•a—— ——:-aa —j 12cLENGTh (m)E3.2V04 -US a a• is a_IllS aa— • —a.S4 8csw4 LENGTh (m) 12Figure 4.8: Form exponent (r) by crown length (m) and crown ratio at 0.111 (A and B), 0.H(C and D), and 0.8H (E and F) for the final felling data.Chapter 4. Results 10713.2 showed that at a young age (14 years), the tree had a similar shape (paracone) fromthe ground to the top. The shape began to change as the tree grew. At older ages (e.g., 54years), the tree tended to have different shapes at different relative heights, although theytended to maintain the same shape in the upper part of the stem (above approximately 60percent of the height). Similarly, tree 16.2 (Figure D.23), at 13 years, also appeared like aparacone in shape from base to top. As it grew, the lower parts began to change in shape,curving inwards (concave-shaped), and eventually at age 53 years, the tree had a differentshape at the base. In contrast, the upper parts remained unchanged in shape or tendedto curve outwards (parabolic or conical). This is characteristic of the trees growing in lowdensity stands (trees 16.2 and 46.1; Figures D.24 and D.32).From these figures, it can be seen that trees have roughly the same shape from ground totop at very young ages (parabolic or conic. r of 1 or 2). As age increases, the shape at thebase changes for larger dbh trees to a neiloid (r of 3) or even larger r values (Figure D.23and Figure D.31). Smaller dbh trees tended to maintain almost the same shape (r between1 and 2) throughout their lifetime (Figures D.25 and D.29).4.1.2.2 Form Variation with Tree Age and HeightTree shape changes with age at a given height above ground are presented in Figures 4.11and E.33 to E.37 (Appendix E). From these three-dimensional plots, it is shown that atyoung ages, the tree shape (form) was relatively constant over the height of tree (see alsoFigures 4.4D, 4.5D, 4.6D, and Appendix E), with an r of less than 1.8. As the tree grew inheight and increased in age, differences in form started to appear and differentiation betweenthe base and upper stern parts began. However, for small dbh trees growing in dense stands(e.g., Figures E.34 and E.36), no major ridges and valleys occurred compared to large dbhtrees growing in more open stands (Figures E.33 and E.37).Chapter 4. Results 108ID54 YEARSx44 YEARS4 A34 YEARSI— 24 YEARSz14YEARS00*XAcj00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8RELATIVE HEIGHT ABOVE GROUND (hA-I)Figure 4.9: Form expoient (r) by relative height for tree 13.2. dbh=15.6 cm, height=16.8m, age=54 years. and QD50=9 cm, for different measurement periods.Chapter 4. Results 1091.2?IE c54 YEARSZ cex *XD 44YEARSO ACt IX 34YEARS0.824 YEARS> *oCD A0.6 AD< 14YEARSA3UJ0.4 EAXwA0.2uJIU0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9RELATIVE HEIGHT ABOVE GROUND (h/H)Figure 4.10: Relative diameter by relative height for tree 13.2, dbh= 15.6 cm height16.8 m.age54 years, and QD50=9 cm. for different measurement periods.Chapter 4. Results 110Figure 4.11: Form exponent (r) by height (m) above ground and breast height age (years) fortree 13.2, dbh15.6 cm, height=16.8 m, age=54 years, and QD509cm. for all measurementperiods.4$#1.9Chapter 4. Results 1114.2 Dynamic Taper Model BuildingThis section presents the results of the dynamic taper modelling process and evaluation(objective two). The models for predicting height, dbh and relative height are presentedin Subsections 4.2.1, 4.2.2 and 4.2.3, respectively. This is followed by presentation of thedynamic taper models in Subsection 4.2.4. Results of the evaluation of the dynamic tapermodels are presented in Section 4.3.4.2.1 Height Prediction ModelThe total tree height prediction models that were fit and tested included two linear modelsand one nonlinear model (Equations 3.47 to 3.49). For all three models, plotting the residualsagainst the predicted height, and against each of the independent variables, indicated noobvious vio1atiois of the assumptions of lillear and nonlinear least squares. However, it isknown that the PSP data used to fit these models are dependent, because the same tree wasmeasured more than once (possible presence of serial correlation).Because of this depeidence, the models could not be evaluated based on the t and Fstatistics, since the usual estimated variances associated with the parameter estimates usingOLS are biased (Kmenta 1971). Although the data were not independent, the predictionabilities of the models were not affected, because the coefficient estimates are unbiased andconsistent. Therefore, the three models were evaluated only for their prediction abilitiesbased on the fit and prediction statistics.The linear models had almost the same fit statistics (R2, SEE, and PRESS, Table 4.6),with the values for Equation 3.47 being slightly lower than for Equation 3.48. The predictionstatistics are also better for Equation 3.47. except for the mean bias. Therefore, Equation3.47 was deemed to be better than Equation 3.48 in terms of both fit and prediction.Equation 3.47 also had slightly better fit statistics (higher R2 aild lower SEE) thanEquation 3.49 (Table 4.6). However, for the prediction statistics, which are important forChapter 4. Results 112Table 4.6: Fit and prediction statistics for the linear and nonlinear height (m) predictionmodels using PSP data.Fit (n=1334 trees) Prediction (n=574 trees)Eqilation R2 SEE PRESS FT SEE Bias RMSE MADLinear3.47 .9337 1.0236 1405.0585 .9396 1.0184 0.0524 1.0131 0.79533.48 .9331 1.0283 1416.8288 .9391 1.0221 0.0096 1.0176 0.8050Nonlinear3.49 .9324 1.0329— .9394 1.0178 0.0694 1.0124 0.8052a2 is the coefficient of multiple determination; SEE is the standard error of estimate in rn; PRESS isthe predicted sum of squares residuals (Draper and Smith 1981, p. 325—327); Fl is the fit index; SEE is theestimated standard error of estimate in m; Bias is the mean bias in m; RMSE is root mean squared error inm; and MAD is the mean absolute bias in m.comparing models fitted using different techniques, the models had similar FT values of about0.939, with the nonlinear model having a slightly lower SEE than the linear model. Sincethe two models had a similar Fl, the model with the lower SEE and RMSE would be moreaccurate. However, Equation 3.47 was slightly less biased (Bias 0.05 m) compared to thenonlinear model (Bias = 0.07 m). Both models had almost the same MAD values. The plotsof the residuals against the predicted height for the fit data showed no particular problemsfor either model. Thus, the linear and the nonlinear models could not easily be separatedbased on their fit and prediction statistics; instead, other measures or factors had to be takeninto account.Other factors considered when selecting the height prediction model included:1. The number of parameters to estimate. The nonlinear model requires fewer variables tobe measured (only two) and fewer parameters to be estimated (only three). Whereas,the linear model requires an additional variable (stand density) to be measured whichChapter 4. Results 113could prove to be costly. Also, the linear model has six parameters to be estimated.2. The growth of trees is a nonlinear process and the growth in height of trees is a functionof available nutrients and sunlight for a given site, species, and age. For a given speciesand site, this nonlinear process is a function of age. Therefore, a linear function wouldbe a mere approximation of this growth process and such a function would he usefulmostly for predictions within the range of the fit data. In terms of forecasting (i.e.,making predictions beyond the fit data range), nonlinear models are expected to workwell and are known to be more flexible than linear models (Payandeh 1983). Giventhe fact that the calibration data range is very small, choosing a more flexible modelwas considered to be beneficial.Based on the above information and the prediction statistics in Table 5.6, the nonlinearmodel (the conditioned logistic function by Goudie (1984. referenced in Goudie et al. 1990))was selected. The model selected was:—1.0 + exp[8.5368 — 0.9000ln(50)— 1.4900ln(SIk — 1.3)]Hkmo — 1.3 + SIk — 1.3) 1.0 + exp[8.5368 — 0.9000 lfl(AGEkm) — 1.4900 ln(SIk — 1.3)](4.65)It should be noted that this model was based on the 1334 dominant and codominant treesin PSP model development data set only.The selected model was recalibrated using the final felling stem analysis data (135 trees).The resulting fit showed that some coefficients changed values, but the changes were small,and no coefficient changed signs. The final form of the model was:H — 1 3+SI —1 3 1.0 + exp[8.2660 — 1.1414ln(50)) — l.l743ln(SIk —1.3)] 466I 1.0 + exp[8.2660 — 1.1414lu(Age)— 1.1743ln(SIk — 1.3)]Chapter 4. Results 114Table 4.7: Fit and prediction statistics for the dbh (cm) prediction models using PSP data.Equation Fit (n=1334 trees) Prediction (n=574 trees)Number R2 SEE PRESS FT SEE Bias RMSE MAD3.51 .9123 1.8417 4552.4135 .8783 2.2279 0.1207 2.2162 2.04033.52 .9114 1.8525 4616.8883 .8828 2.1902 0.1073 2.1692 2.03723.53 .9216 1.7433 4084.3257 .8844 2.1771 0.0927 2.1600 2.0350a2 is the coefficient of multiple determination; SEE is the standard error of estimate in cm; PRESS isthe predicted sum of squares residuals (Draper and Smith 1981, p. 325--327); FT is the fit index; SEE isthe estimated standard error of estimate in cm; Bias is the mean bias in cm; RMSE is the root mean squarebiases or residuals (cm); and MAD is the mean absolute bias in cm.4.2.2 Prediction Model for Diameter at Breast HeightThree models for predicting dbh (see Section 3.3.3) were fit using the PSP data set andtested using the validation PSP data set. Comparison of the prediction models was basedon the fit and prediction statistics.Equation 3.53 had the best fit and prediction statistics (highest R2 and Fl, lowest SEE,SEE, PRESS, Bias, and MAD, Table 4.7) compared to Equations 3.51 and 3.52. Therefore,it was selected for use in the dynamic taper model development. However, it is apparentfrom the fit and prediction statistics that none of the three models predicted dbh very well.The dbh prediction model selected was:Dkmo = 4.6810 + 0.42lOHkmo— 0.0072Hmo + O.O993SIk + 0.0381SI— 0.2333QD5ok+0.0054Qok + 0.0035(QD5ok x AGEkrn) + 0.138OAGEkm (4.67)As was the case for height prediction, the above model was recalibrated using the finalfelling stem analysis data. The resulting model with new coefficient estimates was:Chapter 4. Results 115= —1.7567+ 1.2433H — o.o268H; + O.4662SIk — O.O197SI— O.31O4QD50k+O.O137QDOk + O.O128(QD.ok x Age) — O.O76OAge1 (4.68)4.2.3 Relative Height Prediction ModelFor prediction of relative height, the two prediction models (see Section 3.3.4) were fit usingNLS for the model development stem analysis data set and tested on the validation data.The t or F tests were not used because the observations in the stem analysis data set weredependent. Therefore, comparison of relative height prediction models was based on the fitand prediction statistics.Equation 3.56 was better than Equation 3.57 in terms of prediction statistics (lowestSEE and highest Fl, Table 4.8) and in terms of bias (lowest Bias and MAD). The plotof the residuals against predicted relative height showed no obvious lack of fit. Therefore,Equation 3.56 was selected as the model for predicting relative heights. The fitted logisticfunction used for prediction of relative heights was:= 1/[1 + exp(—(3.128023 + O.338347h — 2.O67O61h1+ O.O9242 + O.3122756 +3 t3O.179957H—2.528776 in— O.O17853(h x fIji) + O.OOO179(h x—O.OO6469(h x 1)))] (4.69)4.2.4 Dynamic Taper Function SelectionUsing logarithmic transformations and an OLS fit, Model 3B (with 11 estimated coefficients)fitted the data better than Model 3A (with nine estimated coefficients) (Table 4.9). However,Model 3A had better prediction statistics (higher Fl, lower SEE, Bias, RMSE, and MAD)than Model 3B. Since the dynamic taper function will be used mainly for predictive purposes,Chapter 4. Results 116Table 4.8: The fit and prediction statistics for the relative height prediction logistic models.Fita PredictionEquation R2b SEE FT SEE Bias RMSE MAD3.56 0.9904 0.0295 0.9829 0.0398 0.0067 0.0397 0.02303.57 0.9880 0.0331 0.9795 0.0437 0.0078 0.0436 0.0261aFjt statistics are based 011 5916 sectional measures and the prediction statistics are based on 2668sectional measures.bR2 is the coefficient of multiple determination; SEE is the standard error of estimate; PRESS is thepredicted sum of squares residuals (Draper and Smith 1981, p. 325—327); Fl is the fit index; SEE is theestimated standard error of estimate in cm; Bias is the mean bias; RMSE is root mean squared error; andMAD is the mean absolute bias.Model 3A was selected as more appropriate than Model 3B and will hereafter be referredto as Model 3.When Model 3 was fitted using NLS (Section 3.3.5.1) allowing the coefficient q to vary,it was found that this coefficient was not necessary and it was eliminated. Therefore, Model3 became:b1+b2/+b3 ----+b7h•±O.OOl= ao x x [i—(4.70)This model no longer has the same properties as Kozak’s (1988) model (Equation 2.15),since it is not conditioned to pass through the Kozak’s join point of However, themodel is conditioned in such a way that when = then = 0, and when = 0,then a0D”, which is the predicted diameter inside bark at tree base.4.2.5 Taper Models FittedThe selected taper models (dynamic Models 2 and 3 and static Model la) were fitted usingOLS and NLS as described in Section 3.3.5. Since the fitting techniques resulted in differentChapter 4. Results 117Table 4.9: Fit and prediction statistics for the two dynamic taper functions for diameterinside bark (d) (cm) using OLS.Model Fita PredictionType R21’ SEE PRESS FT SEE Bias RMSE MAD3A .9068(.9722) .2554(.7807) 385.3950 .9733 .7538 .0754 .7524 .55633B .9152(.9657) .2436(.8672) 354.4663 .9674 .8343 .0840 .7324 .6067aT1e fit statistics were based on 5916 sectional measures while the prediction statistics were based on2668 sectional measures.The fit statistics are based on in and values in brackets were estimated usingthe antilogarithm of ln(c1t).bR2 is the coefficient of multiple determination; SEE is the standard error of estimate in cm; PRESS isthe predicted sum of squares residuals (Draper and Smith 1981, p. 325—327); Fl is the fit index; SEE is theestimated standard error of estimate in cm; Bias is the mean bias in cm; RMSE is root mean squared errorin cm; and MAD is the mean absolute bias in cm.coefficient estimates (Table 4.10), it likely that they have different predictive abilities forboth diameter inside bark over the tree stein and total tree volume.4.3 Taper Function EvaluationUsing the coefficient estimates in Table 4.10. t.he three taper models were evaluated forand Vj predictions. Both linear and nonlinear fits were evaluated.4.3.1 Model Evaluation for Predicting Diameter Along the Tree StemThe OLS fitted models (Models laL, 2L, and 3L) were corrected for underprediction since alogarithmic transformation was used (Equation 3.64). A correction factor (CF) of 1.00647,1.03283, and 1.03212 were used for Models laL, 2L, and 3L respectively.From the fit and prediction statistics (Table 4.11), it can be seen that the static tapermodels (Models laL and laN) outperformed all the other taper models in all respects, asexpected, since measured dbh and height were used. They had both the best fit and the bestChapter 4. Results 118Table 4.10: Coefficient estimates for the static and two dynamic taper models fitted usingOLS and NLS.aCoefficient Kozak (1988) Static Kozak (1988) Dynamic New Dynamic ModelEstimates laL laN 2L 2N 3L 3Na0 1.60636 1.38712 1.47846 1.33651 0.70635 0.93242a1 0.66631 0.77700 0.64360 0.77898 1.07554 0.99682a9 1.01998 1.00916 1.03060 1.01086——b1 0.35906 0.00999 -1.54544 -0.83186 0.85991 0.38479b2 -0.04343 0.07509 0.44487 0.23072 -0.23125 0.13417b3 0.22627 -0.97381 -4.69504 -2.58667 0.05795 0.041.51b4 -0.02953 0.52304 2.62135 1.49741 -0.75920 -0.14300b5 0.23267 0.30002 0.09211 0.13466 0.00025 0.00015b6— —— 0.87850 0.44204b7— ——— -0.58807 -0.44441am the table, L represents linear models; N representsnonlinear models; and ‘a” represents modelsassessed using measured height and dbh. It should be noted that Models lbL and lbS have the samecoefficient estimates as Models laL and laX respectively.Chapter 4. Results 119prediction statistics. When the same fitted equations (Models lbL and lbN) were assessedby inputting predicted height and dbh, they performed poorer than the dynamic tapermodels (Models 2 and 3), particularly in the nonlinear form (poorer prediction statistics).Comparing the dynamic taper functions based on the fit statistics, Model 3L was a betterdynamic taper function for fitting the data for diameter inside bark than Model 2L (higherR2, lower SEE and PRESS). Also, Model 3N was a better model than Model 2N (lower R2and SEE). However, based on the prediction statistics, IViodel 2L was a better function (allprediction statistics are lower) than model 3L. For the functions fitted using NLS, Model 2Nwas more biased (higher Bias and MAD), but more precise (lower SEE and RMSE) thanModel 3N, even though the FT values were equal. The NLS fitted models were better thanOLS fitted models (lower prediction statistics). However, based on these statistics, a clearchoice between Models 2 and 3 cannot be made.Since the statistics in Table 4.11 represent the overall fit and prediction of the models andfitting methods, they do not indicate which model is more or less biased along the stem. Inorder to effectively assess the taper functions and the two fitting methods (OLS and NLS),the mean biases in diameter prediction were assessed for a given percent height above groundusing the validation stem analysis data (Table 4.12). Such analyses are very important forselecting a good taper function.The static taper models (Models laL and laN) had lower biases than the other models(Table 4.12) with Model laN having the lowest biases along the stem. However, for the staticmodel assessed using predicted height and dhh (Model lb) was more biased when OLS- andNLS-fitted than Model 2. For section Model lb had especially high biases. For thedynamic taper functions, the NLS-fitted functions had lower biases for the stem sectionsnear the ground, than the OLS-fitted models, but very high biases for upper stem sections,particularly for the last section (O.9H). Kozak’s (1988) taper model altered using predicteddbh, height, and relative height (Model 2L), had very low biases compared to the dynamictaper model (Model 3L), except at the 90 percent of height. Model 2L had the lowest biasesChapter 4. Results 120Table 4.11: Fit and prediction statistics for the three taper functions for diameter insidebark (d) (cm) using OLS and NLS.aIViodel Fitb PredictionNumber R2c SEE PRESS FT SEE Bias RSME MADlaL .9816 .1136 76.5707 .9890 .4830 -0.1307 .4825 .3557laN .9917 .4270— .9900 .4606 0.0693 .4601 .3141lbL .9816 .1136 76.5707 .9788 .6712 .0118 .6701 .4749lbN .9917 .4270— .9778 .6869 .0022 .6858 .48752L .9049 .2580 395.230 .9758 .7243 .0363 .7232 .53202N .9805 .6531 .9812 .6372 .0216 .6362 .45103L .9097 .2514 376.415 .9751 .7340 .0382 .7327 ..54823N .9809 .6468 — .9812 .6383 -.0047 .6372 .4462am the table, L represents linear models; N represents nonlinear models; “a” represent the static modelsassessed using measured height and dbh; and “b” represents static models assessed using predicted heightand dbh.bThe fit statistics were based on 5916 sectional measures and the prediction statistics were based on 2668sectional measures. These fit statistics, R2, SEE, and PRESS for the linear models (Models 1L, 2L, and 3L)are based on the natural logarithm transformed ln values.CR2 is the coefficient of multiple determination; SEE is the standard error of estimate in cm; PRESS isthe predicted sum of squares residuals (Draper and Smith 1981, p. 325—327); Fl is the fit index; SEE is theestimated standard error of estimate in cm; Bias is the mean bias in cm; RMSE is root mean squared errorin cm; and MAD is the mean absolute bias in cm.Chapter 4. Results 121Table 4.12: Mean biases for all taper functions fitted with OLS and NLS for diameter insidebark (cm).aPercent No. of Mean bias in sectional diameter inside bark (cm)Height Sections Kozak (1988) New ModelAbove Static Dynamic DynamicGround laL laN lbL lbN 2L 2N 3L 3N<5.0% 364 -0.01 -0.06 -0.30 -0.30 0.04 -0.02 0.37 0.0710.0% 268 0.09 0.08 -0.01 0.01 -0.00 0.11 0.38 0.1120.0% 218 0.03 0.10 -0.01 0.08 0.06 0.19 0.09 0.0330.0% 201 0.01 0.07 -0.03 0.04 0.05 0.13 -0.10 -0.0240.0% 187 0.05 0.06 0.04 0.04 -0.01 0.07 -0.16 -0.0150.0% 183 0.13 0.07 0.06 -0.01 -0.11 -0.03 -0.09 0.0460.0% 221 0.21 0.09 0.29 0.19 0.02 0.05 0.28 0.2070.0% 243 0.17 0.04 0.13 0.02 -0.01 -0.13 0.41 0.0780.0% 322 0.16 0.08 -0.06 -0.13 0.08 -0.23 0.44 -0.1390.0% 428 0.05 0.06 0.76 -0.75 -0.36 -0.79 -0.25 -0.93100.0% 33 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00am the table, L represents linear models; N represents nonlinear models; “a” represents the static modelsassessed using measured height and dbh; and “b” represents static models assessed using predicted heightand dbh.followed by Models 3N and 2N; while Models 3L, lbN, and lbL had the highest biases,particularly for 0.1Further analyses were needed in order to judge the taper models and the fitting techniquesfor different tree sizes, beca.use it is the large size trees that are important for a. forestmanager. The data were further divided into three dbh classes to check the models’ predictionabilities for different dbh classes. The three dbh classes were: (1) 0.1 to 10.0 cm, (2) 10.1 to20.0 cm, and (3) greater than 20.0 cm. Models lb, 2, and 3 were more biased than Kozak’s(1988) static model (Models la) (Figure 4.12). Generally, all models predicted for lowersections (sections less than 10 m above ground) with little bias (Bias of less than or equalChapter 4. Results 122to 0.3 cm). For upper sections, all models were biased with a maximum of 1.5 cm at 19.3m above ground. However, only 4 trees were available in the validation data set for thisheight, therefore this could be the reason why the Bias is high. This upper section biasesoccurred particularly for the OLS-fitted dynamic models (Models 2L and 3L). The NLSfitted functions were less biased for all sections (maximum Bias of 0.7 cm for Model lbN for19.3 m above ground). It is impossible to determine the predictive abilities for by thetaper functions for different tree sizes from Figure 4.12, since trees of all sizes are included.Breaking the data into diameter classes was neccessary to determine which function was abetter predictor of for large dbh classes which contain more volume.For small trees (dbh class 1), Kozak’s (1988) static taper functions (Model laL and laN)had low biases (Bias less than + 0.2 cm) compared to the dynamic taper functions whichhad higher biases (Bias less than ± 0.4 cni) for sections less than 10 m above ground (Figure4.13). Also, Models lhL and lbN had lower biases compared to Models 2 and 3, except forthe last section (14.3 m above ground) for OLS-fitted models. For NLS-fitted models, allthe models had similar biases (i.e., they were all less biased, Bias < 0.2), except for the lastsection. However, as tree size increased, the differences in biases increased.For medium trees (dbh class 2), Kozak’s (1988) static taper function (Model laN) stillhad the lowest Bias (Figure 4.14). However, Model laL was not different from Model 2 (bothwere equally biased, but in opposite directions) for sections less than 10 iii above ground.Model lbL was had the highest Bias. Models 2L and 3L were the best linear models exceptfor the section at 0.6 m above ground where they were poorer than model laL. For the NLSfitted taper functions, the dynamic taper functions showed lower biases than Model laNup to 13.3 m above ground; for the upper sections (above 13.3 m), all models were equallybiased. At the butt (0.3 m and 0.6 m above ground), Model lbN was more biased thanModels 2N. This indicates that the use of predicted height and dbh without refitting themodel results ill higher biases. Both Models 2N and 3N had lower biases than the OLS-fittedfunctions (Bias of + 0.5 cm for sections less than 10 m above ground and Bias < 1.0 cmChapter 4. Results 12:3E()-o00a)>0____________a)________ _________a)______S(U(I)(U(Ua).OLS2‘--c_Model la-1• -Model lb--Model 2-0--2 ModeI3—3— I I I I I I I I I I I I I I0.3 1.3 3.3 5.3 7.3 9.3 11.3 13.3 15.3 17.3 19.3Height Above Ground (m)3.NLS Modella-Model lbrModel2-0-Model 3:E0000Ii)>0a)a)S(U(I)(U(Ua)_.) 1 . I I I I - . I I •I I I I I0.3 1.3 3.3 5.3 7.3 9.3 11.3 13.3 15.3 17.3 19.3Height Above Ground (m)Figure 4.12: Mean biases for diameter inside bark (cm) prediction at various heights aboveground for trees of all sizes (275 trees). The section lengths between 0.3 iii and 13 rn are0.6 m above ground and the other unmarked section lengths are a metre apart.Chapter 4. ResultsEC.)-o0(5G)>0£1)2Cu0C’)CumCua)EC)000a)>0a)a)SCu0CoCuCua)0.3 1.3 3.3 5.3 7:3 9.3 11.3 13.3Height Above Ground (m)124Figure 4.13: Mean biases for diameter inside bark (cm) prediction at various heights aboveground for trees of dbh class 1 (dbh < 10.0 cm for 180 trees). The section lengths between0.3 m and 1.3 m are 0.6 m above ground and the ohter unmarked section lengths are a metreapart.;3 —OLS Modella--Model lb-)IE2 Model2•E3.Model 31*..:-2—3 I I I I I I I0.3 1.3 3.3 5.3 7.3 9.3 11.3 13.3Height Above Ground (m)3=NLS Modella--Model lb2 Model 2€3.Model 31E!f&JRi._ S S—.__-C—1--2-3Chapter 4. Results 125for sections more than 10 m above ground). However, for sections more than 10 m aboveground Models laN, 2N, and 3N seemed to be more biased than the OLS-fitted functionsexcept for Model lbL which had equally high biases.All the taper models were most biased when predicting d1 for large dbh trees (dbh class3, Figure 4.15). Of the OLS-fitted models. Model laL was more biased for sections less than9 m above ground than the other models except Model 2L, but it was less biased for sectionsmore than 9 m above ground. Model lbL had lower biases than all the other models forsections less thaii 9 m above ground, but it was the worst for sections more than 9 m aboveground. Model 3L performed better than Model 2L for both lower and upper sections, butnot for the middle sections. Apart from a few middle sections, Model lbL was more biasedthan Model 3L. Overall, Model 2L performed the poorest.For the NLS-fitted models, Model 2N had slightly lower biases than Model laN forsections below 12 m above ground; however, for the upper sections (sections more than 12 rnabove ground) Model laN had the lowest biases of all the NLS models. Model lbN performedconsistently poorer than Model 2N for all sections along the stem, confirming the idea thatthe use of predicted height and dhh without refitting the model will result in more biasedmodels, while Model 3N was consistently less biased than IVIodel 2N for all sections aboveground. Model lbN gave had the poorest predictions.Comparing the OLS-fitted and the NLS-fitted taper models for sections less than 10 mabove ground, the NLS-fitted models were less biased (Bias < 0.5 cm compared to Bias >-2.5 cm). For upper sections (sections above 10 m), Models laL and laN had the lowestbiases. Kozak’s (1988) taper models refitted using OLS, whether static or dynamic (ModelslaL, lbL, and 2L), were characteristically biased for lower sections, particularly the firstthree sections (Figures 4.14 and 4.15). These models overpredicted for lower sectionsand underpredicted it for upper sections. For overall bias, however, these overpredictionsand underpredictions for cancel each other out, and Model 2L appears less biased thanModel 2N (Table 4.12).Chapter 4. Results 126--OLS Model 1 aModel lb2-Model 2.E.ModeI3-o421ID..?•.tiLbCl)FaJ(ClID-2-3 I I I I I I I I I I0.3 1.3 3.3 5.3 7.3 9.3 11.3 13.3 15.3 17.3Height Above Ground (m)3.NLS ModellaModel lb2 -ModeI2ModeI3C420. -1-C(Cla)-2-—3— I I0.3 1.3 3.3 5.3 7.3 9.3 1t3 13.3 153 17.3Height Above Ground (m)Figure 4.14: Mean biases for diameter inside bark (cm) prediction at various heights aboveground for trees of dbh class 2 (10.0 < dbh < 20.0 cm for 91 trees). The section lengthsbetween 0.3 m and 1.3 m are 0.6 m above ground and the other unmarked lengths are ametre apart.Chapter 4. Results 127QIC’:‘2-• . Cfr Model laModel lbModel2‘S-—-• ,er Model3-- ‘S—3 I I I I I I I I I I I I0.3 1.3 3.3 5.3 7.3 9.3 11.3 13.3 15.3 17.3 19.3Height Above Ground (m)‘2NLS/ -‘.•‘2 ,‘C ‘—910‘L50ci,EModella• ---1 ModelibModeI2ci,II)Model3-2—S., I I I I I I I I I I I I I I I0.3 1.3 3.3 5.3 7.3 9.3 11.3 13.3 15.3 17.3 19.3Height Above Ground (m)Figure 4.15: Mean biases for diameter inside bark (cm) prediction at various heights aboveground for trees of dbh class 3 (dbh > 20.0 cm for 4 trees). The section lengths between 0.3m and 1.3 m are 0.6 m above ground and the other unmarked section lengths are a metreapart.Chapter 4. Results 128Overall, large dbh trees are the most important because they contain more volume. Also,lower sections contain a large proportion of the total tree volume. Therfore, a taper modelthat has good predictions for lower sections for large dbh trees would be the best. Based onthese results, Model 3N qualifies as the best model.In order to select the best dynamic taper function, selection criteria cannot be based onthe prediction of alone. Trees are important for their volume, and a taper function thatis less biased for volume prediction (particularly for the lower sections which contain themajority of the total tree volume) is more desirable. Therefore, the dynamic taper functionswere further assessed for predicting total tree volume.4.3.2 Evaluation of the Taper Functions for Total Tree Volume PredictionKozak’s (1988) refitted static taper models (Models laL and lbL) had the best overallprediction statistics for volume (Table 4.13). The OLS-fitted functiois had higher FT valuesthan their counterpart NLS-fitted functions, except for Model 2L. Kozak’s dynamic taperfunction fitted with OLS (Model 2L) was the most biased for all dbh classes, except for treeswith dbh less than 8.0 cm where it was equal to all other models. For small dbh classes(dbh 8.0 cm), all the models had very low biases, and were not distiguishable. As dbhincreased, the OLS-fitted models became more biased than the NLS-fitted models, exceptfor the last dbh class, where all models had high biases. Model la is better than Models 2and ib, particularly for OLS fitting. Model lb had lower biases than Model 2, except forlarge dbh classes (18.1—22.0 cm). Model 3 is better than Model 2, particularly for the OLSfitting. Model lhN had the poorest predictions for the largest dbh class.All models had larger biases for the four largest trees (20.l--22.0 cm). All the OLS-fittedmodels had particularly high biases for both the lower and upper sections, with Model 2Lhaving the highest. Since large dhh trees contain more volume than small trees, a taperfunction that has lower biases for such trees is desirable. It should he pointed out that meanbiases for all trees can be influenced by large biases for a few individual trees.Chapter 4. Results 129Table 4.13: Mean biases for volume (m3) prediction using all taper functions for differentdiameter classes.aNo. Mean biases in volume (cubic metres)Dbh of Kozak (1988) New ModelClass Trees Static Dynamic DynamiclaL laN lbL lbN 2L 2N 3L 3N0.1- 2.0 8 0.0002 0.0003 0.0002 0.0002 0.0002 0.0003 0.0003 0.00022.1- 4.0 38 0.0001 0.0003 0.0000 0.0002 0.0003 0.0003 0.0005 0.00024.1- 6.0 43 0.0001 0.0001 -0.0005 -0.0004 0.0002 -0.0002 0.0007 -0.0006.1- 8.0 39 -0.0001 -0.0003 -0.0008 -0.0009 0.0005 -0.0005 0.0011 -0.00028.1-10.0 52 0.0009 0.0002 0.0002 -0.0005 0.0017 0.0002 0.0022 0.000510.1-12.0 29 0.0015 0.0001 0.0012 -0.0000 0.0027 0.0009 0.0021 0.000812.1-14.0 24 0.0017 0.0004 0.0025 0.0014 0.0021 0.0020 0.0025 0.001614.1-16.0 21 0.0032 0.0025 0.0007 0.0003 -0.0039 0.0004 -0.0028 -0.000916.1-18.0 11 0.0020 0.0034 -0.0001 0.0017 -0.0126 0.0007 -0.0038 0.000518.1-20.0 6 0.0075 0.0143 0.0054 0.0130 -0.0238 0.0094 0.0010 0.011320.1-22.0 4 -0.0009 0.0137 0.0106 0.0256 -0.0390 0.0196 0.0080 0.024522.1+ 0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000Total 275 0.0010 0.0010 0.0005 0.0006 -0.0010 0.0008 0.0010 0.0009Fl 0.9971 0.99.53 0.9963 0.9926 0.9856 0.9947 0.9950 0.9939a11 the table, L represents linear models; N represents nonlinear models; “a” represents the static modelsassessed using measured height and dbh: b” represents static models assessed using predicted height anddbh; FT is the fit index; and No. of trees represent tree-measures.Chapter 4. Results 130All models had poor predictions for older trees (51—57 years old) (Table 4.14). Model laLhad the smallest Bias while Model lbN had the highest. Model lbL had lower biases thanModel 2L, but Model lbN had higher biases than Model 2N. Model 3 had lower biases thanmodel 2, particularly Model 3L which was the best dynamic model for volume prediction.The biases associated with the last three age classes were exceptionally high. Therefore,further assessment was necessary for the taper models.When predicted volumes for large dbh trees were plotted against age (Appendix F), thedifferences in volume predictions by the static and dynamic taper models for the two fittingmethods became more visible. These figures show that the static model was superior tothe dynamic models and the NLS-fitted models were better than the OLS-fitted models.For the small tree (tree 25.1, Figure F.40), all taper functions (static and dynamic) hadsimilar predictions, except Model lb which consistently underestimated tree volume for allages. The static taper models (Models laL and laN) had more accurate volume predictionsfor the big trees (Figures F.38, F.39, and F.41 for trees 14.1, 17.1 and 41.1, respectively)as expected. Of the OLS-fitted models, Model 2 had the poorest volume predictions; itconsistently overestimated tree volume. Model lb consistently underestimated tree volumeas tree aged. Models laL, lbL, and 2L showed poorer volume predictions for tree 41.2than Model 3L. For the NLS-fitted taper functions, all models slightly underestimated treevolume, with Model lbN having the highest underestimates. Model 2L had the highestbiases, overestimating volume for the larger trees. However, at young ages (trees 14.1 and17.1, Figures F.38 and F.39, respectively), all models had similar predictions for tree volilme.From these results, it is apparent that Models 2L performed the poorest, followed by ModelslbL and lbN. The best model was Model la followed by Model 3 fitted by either OLS orNLS.Chapter 4. Results 131Table 4.14: Mean biases in volume (rn3) prediction for all taper functions by 3-year ageclasses.aMean biases in volume_(cubic_metres)Age No. of Kozak (1988) New ModelClass Trees Static Dynamic DynamiclaL laN lbL lbN 2L 2N 3L 3N1- 3 18 0.0003 0.0004 0.0005 0.0007 0.0007 0.0007 0.0005 0.00044- 6 16 0.0002 0.0004 0.0001 0.0003 0.0004 0.0004 0.0007 0.00037- 9 28 0.0005 0.0006 0.0005 0.0005 0.0009 0.0006 0.0015 0.000610-12 22 0.0002 0.0002 -0.0001 -0.0001 0.0005 0.0001 0.0017 0.000513-15 23 -0.0002 -0.0005 0.0004 0.0002 0.0008 0.0005 0.0026 0.000916-18 28 -0.0004 -0.0005 -0.0003 -0.0004 -0.0005 0.0000 0.0023 0.000819-21 15 -0.0000 -0.0002 -0.0005 -0.0006 -0.0014 -0.0003 0.0028 0.001122-24 22 0.0004 -0.0001 -0.0003 -0.0007 0.0005 -0.0002 0.0015 0.000125-27 16 0.0012 0.0006 0.0003 -0.0003 0.0010 0.0003 0.0017 0.000428-30 12 0.0003 -0.0005 -0.0008 -0.0015 0.0006 -0.0008 0.0005 -0.000831-33 18 0.0023 0.0014 0.0012 0.0005 0.0011 0.0011 0.0017 0.001034-36 7 0.0067 0.0069 0.0034 0.0040 -0.0021 0.0037 0.0030 0.004137-39 14 0.0018 0.0006 0.0001 -0.0009 0.0002 -0.0001 -0.0006 -0.000740-42 9 0.0058 0.0063 0.0037 0.0046 -0.0033 0.0044 0.0016 0.004343-45 7 -0.0002 -0.0008 -0.0003 -0.0005 -0.0040 -0.0003 -0.0040 -0.002046-48 12 0.0029 0.0039 0.0007 0.0023 -0.0098 0.0017 -0.0048 0.000549-51 4 0.0025 0.0117 0.0087 0.0185 -0.0252 0.0143 0.0014 0.015052-54 3 0.0031 0.0027 -0.0034 -0.0033 -0.0124 -0.0029 -0.0170 -0.008255-57 1 0.0051 0.0221 0.0269 0.0444 -0.0303 0.0368 0.0118 0.036458+ 0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000aJ the table, L represents linear rnodels N represents nonlinear models: “a” represent the static modelsassessed using measured height and dbh “h” represents static models assessed using predicted height anddbh; and No. of trees represent tree-measures.Chapter 5DiscussionThis chapter is divided into two sections. First, the variation of tree form and taper fordominant and codominant lodgepole pine trees is discussed. Individual tree taper and formvariation with age at different heights along the stem is also discussed. Second, the dynamictaper functions that were fit are discussed along with their predictive abilities for diameterinside bark (d1) and total tree volume.5.1 Variation in Tree Form and TaperUnderstanding the variation in tree shape and taper is very important. Since trees with thesame dbh and height can have a different form and taper, they can have different volumes.For example, a parabolic shaped tree will have more volume than a cone shaped tree withthe same dbh and height. Also, implications for management options exist (i.e., pruning orthinning to get different tree shapes).Tree form and taper were found to increase with increases in height, dbh, D/H ratio, CLand age, particularly for lower sections, and with an increase in CR for middle and uppersections. SI showed no clear relationship with tree form and taper changes. For QD50,the relationship was strong for the tree base, hut relatively poor for the middle and uppersections.Change in tree shape as tree height increases likely is only associated with changes inother factors that are related to height, as height on its own should not affect changes iiitree shape. However, large diameter trees tend to be tall in most cases, because trees put onincrement in height and diameter together (Sections 2.1.1 and 2.1.2). This is probably the132Chapter 5. Discussion 133major factor affecting tree shape changes. As trees grow, they increase in diameter along thestem, including dhh. Trees change shape because diameter increment is not uniform alongthe whole tree stem, a factor confirmed by Clyde (1986) and Larson (196.5). This was shownin this research by increases in the shape coefficient (r) for the base and crown sections ofthe tree.The variation in tree shape with height and dbh was indicated by a high linear correlationbetween r and the D/H ratio for all trees. Tree shape increased with an increase in the D/Hratio. An increase in the D/H ratio, which was used as a measure of taper by IKozak (1988)and Marshall (1991), seems to be an indicator of tree shape changes, particularly for the buttsections of the trees. An increase in the D/H ratio indicates that the tree will be puttingon more radial growth than terminal growth in a relative sense. As a result, the tree willchange shape. Therefore, the use of the D/H ratio as a variable to indicate changes in treeshape is justifiable. IKozak (1988) and Newnham (1988, 1992) included D/H ratio in theirmodels to account for butt swell.For young trees, no real shape differences over the stem were apparent, irrespective ofwhere the trees were growing. This seems to suggest that at young ages, factors that affecttree shape and taper changes have not yet begun to act, and that the small shape differencesthat may occur are only indicative of genetic variability, agreeing with Larson (1963) andNewnham (1965). However, as trees grow, different sections start changing into differentshapes. At the base, tree shape changes from a parabolic shape towards a neiloid; at thetop, tree shape seems to change more slowly from parabolic to conic. Young trees had a formexponent ranging from 0.8 to 1.5, which agrees with Forsiund’s idea of a paracone for trees atyoung age. Also, a look at variation in form for individual trees with age at different heightsshowed that tree form coefficient tended to increase over tinie. Different parts of the treesincreased in r at different rates, which gave trees different shapes for different sections. Forold trees, shape changes were particularly evident at the base and top (within the crown).This was witnessed by the presence of humps for lower sections of the trees at older ages onChapter 5. Discussion 134the 3-D graphs, compared to valleys for upper sections; no humps and valleys were presentfor the same trees at youig ages.One possible reason why trees have uniform shapes at young ages could be that trees areputting their growth efforts into height to achieve good crown positions for sunlight ratherthan on radial increment. Also, all trees have similar crown size at this age, and they havenot yet differentiated into crown classes. As age increases, growth occurs unequally alongthe stem, resulting in different shapes at various points along the stem. Radial growth isundoubtedly correlated with age, but it is not caused by age; rather, growth is caused bythe accumulation of photosynthetic products. This reasoning agrees with the existing theoryabout tree shape (Figure 2.1), and the literature review by Larson (1963, 1965) and Gray(1956).Site index was found to have no clear relationship with the shape or taper of trees. Treeheight growth is related to site productivity. Therefore, site index was expected to affectchange in shape of trees, as pointed out by Larson (1963). Trees growing on good sites aresupposed to add on more height growth than trees in growing on poor sites. Also, treesgrowing on good sites will put on more diameter growth along the stem than trees growingon poor sites. As a result. they will have different shapes for different sections of the tree.The lack of a clear relationship between r and site index for this study might have beencaused by confounding due to differences in density and tree age, as well as changes in thecrown length. No relationship of tree shape and taper with site index also was found byNewnham (1965) and Burkhart and Walton (1985).The relationship of r with QD50 was not very strong; however, there was a better linearrelationship at O.1H than at O.5H and O.8H. This means that form and taper of trees, particularly for butt sections, increases with QD50. High QD50 values are associated with lowerstand densities for a given stand age. These results agree with the findings of Larson (1963),Gray (1956), Newnham (1988) and Valinger (1992) that tree form and taper decreases withincreasing stand density. As noted by Smithers (1961), trees that grow in less dense standsChapter 5. Discussion 135(e.g., tree 46.1) were almost conical, and trees in very dense stands (e.g., tree 25.2) werewhiplike and hardly had any difference in shape and taper from the ground to the top. Treesput on different amounts of diameter increment along the stern depending on where theyare growing (Clyde 1986). If a tree is growing in a dense stand, it will put on less diametergrowth compared to a tree growing in less dense stand, because the amount of shared resources (water, mineral nutrients and sunlight) will he more limiting for a tree growing in adense stand. Trees growing in dense stands will have less butt swell also, because they willbe shelting each other from wind and providing support to each other. As a result, treesgrowing in a dense stand will be less tapered than trees growing in less dense stands.Crown length, like dhh, was found to be highly related to tree shape, hut mainly forthe lower tree sections . Among individual trees, trees with longer crowns (e.g., tree 46.1with a CR. of 0.81) had bigger shape changes from the top to the base than trees with shortcrowns (e.g., tree 25.2 with a CR of 0.46). When tree crowns increased in length, the treebase became more swollen and changed in shape from a cone to a neiloid and even moreconvex shapes. These results agree with Larson (1963, 1965) and the literature review inSection 2.2 of this thesis, in that crown length is very important in determining tree shapeand might be a strong contributor to tree butt swell. Gray (1956) showed that trees becomemore cylindrical in form with an increase in stand density and a decrease in crown length.For this study, an increase in stand density and a decrease in crown length resulted in similarresults to Gray’s, but trees tended to have a parabolic form. Trees growing in less densestands had large crowns (e.g., trees 16.2 and 46.1) compared to trees growing in very densestands (e.g., tree 25.2).The relationship between tree shape (r) and tree crown ratio varied greatly at the baseof the tree. However, a strong relationship existed between r aiid the middle (0.5H) and top(0.8H) sections of the trees. An increase in crown ratio corresponded to an increase in theform exponent (change in shape). As crown ratio increased, the shape of the upper bole ofthe tree changed from a parabolic to a conical shape.Chapter 5. Discussion 136Thus, crown length seems to be a good indicator of changes at the base of the tree (buttswell), whereas crown ratio is a good indicator of stem change within the crown. Therefore,a model with D/H and crown ratio should be able to capture the variations in taper andform in both the lower and upper ends of the tree stem.Prom the results of this study and the literature review, it appears that the main factorscontrolling tree form and taper are stand density and tree crown size. Stand density controlsthe amount of canopy a given tree will possess and, as a result, it controls the form and taperof the trees. It was observed that older trees with small crowns growing in dense stands (e.g.,tree 25.2: age of 48 years, QD50 of 6 cm or 8175 trees per ha and a crown ratio of 0.48)do not necessarily have differentiated shapes for different sections along the stem. However,young trees with large crowns and growing in less dense stands (e.g., tree 46.1: age of 20years, QD50 of 22 cm or 425 trees per ha and a crown ratio of 0.81), have drastic changesin shape along the stem. Age does not appear to be the controlling factor for tree shapechanges, since no obvious trend of r with age was found for all trees. However, age shouldbe an important variable in a prediction model for diameter inside bark along the tree stembecause other factors, that do affect tree form and taper changes (e.g., crown size and standdensity), change over time.Stand treatments such as thinning, pruning and fertilization, will affect tree shape andtaper by altering both tree crown size and stand density. For example, thinning will reducestand density and, in the process, will allow individual trees more space to expand theircrowns. If stands are heavily thinned, trees will grow like open-grown trees with long crowns,which means that they will have high taper and more conical shapes. However, when treesare pruned, crown sizes are reduced for a given height, which should be similar to increasingthe stand density. As a result, pruning will decrease taper and makes trees appear moreparabolic in shape. Therefore, thinning trees heavily would not be appropriate if the aim isto produce less tapered trees with high volume recovery after sawing, unless heavy thiningis accompanied with heavy pruning to reduce the crown size. Fertilizing may increase treeChapter 5. Discussion 137growth vigour; the tree puts on more branches and increases crown size, resulting in increaseddiameter growth along the stem. However, a fertilized tree will also increase in height growth.With increased height and diameter growth, the resulting tree may not show a large changein taper because the D/H and CR ratios might not have changed. This could be the reasonGordon and Graham (1986), Tepper et al. (1968), and Thomson and Barclay (1984) foundtaper to be oniy slightly affected by heavy fertilization.Because the crowns of dominant trees extend above the general level of the canopy andreceive light from above and the sides, they are better able to put on more diameter growthand increase in taper, change shape, and have butt swell. Crowns of codominant trees formthe general level of the stand canopy and receive sunlight from above only; therefore, they puton less diameter growth than dominant trees, resulting in less tapered and more uniformlyshaped (paracone shaped) trees with little butt swell. These dominant and codominanttrees will have higher CR and D/H ratios than intermediate and suppressed trees. Crownsof intermediate trees are within the general canopy, competing with crowns of other trees,making the crowns smaller and receiving minimum sunlight from above. Diameter growthalong the stem is much lower thai for dominant and codominant trees. This means thatintermediate trees would be expected to have essentially uniform shapes from ground to treetop, minimum taper, and little or 110 butt swell. Suppressed trees, which are completelydeprived of direct sunlight, would have very small crowns and diameter growth along thestem would be minimum because they would be putting all their energy into height growthto gain access to direct sunlight. This would result in trees with virtually no taper (CR andD/H ratios very small) and the shape would he uniformly cylindrical from the ground to thetop (MacDonald and Forslund 1986; Clyde 1986; Gray 1956).The form and taper of trees also varies with species (Clyde 1986; Koch 1987), dependingon whether the species is shade tolerent or iitolerent. For lodgepole pine, a shade intolerentspecies, diameter growth would quickly reach a maximum and then decline rapidly, comparedto shade tolerent species such as spruce (i.e, it would have a low D/H ratio) (Clyde 1986 andChapter 5. Discussion 138Assmann 1970). This could he due to the species’ inability to niaintain high levels of growthwith increased competition. As a result, shade intolerent species would have relatively simpletaper and would have less differentiated shapes (Clyde 1986; Koch 1987). In contrast, shadetolerent species would maintain diameter growth much longer, even when shaded by adjacenttree crowns. As a result, these species would have more taper (high D/H ratios) than shadeintolerent species and would be more differentiated into different shapes from ground to topthan shade intolerent species.In summary, it was found that tree shape and taper change along the stem at one time,and over time, change with changes in tree and stand variables. The tree and stand variableswhich were important included D/H ratio, CL, CR, and QD50. AGE was found to beimportant also, but its importance could be due to other factors which change over time.It also was found that trees have a simple shape (i.e., they are all parabolic in shape fromground to top) at young ages. However, as they increase in size, different portions of thestem change into different shapes, because of unequal growth in diameter along the stem.Thus, to model taper over time, a process that will either capture or mimic the changes instand density, crown size and D/H ratios as the trees grow should be involved.5.2 The Dynamic Taper FunctionsModelling taper over time should provide a taper function that is able to predict accurateand total tree volume over time. Such a taper function could be incorporated into atree growth model that will be relatively simple to understand, easy to calibrate, have wideapplication, and be less cumbersome to handle. The dynamic taper modelling process usedin this study involved predicting height and dbh over time. predicting relative height, findingthe correct model form for the form exponeilt, and, finally, putting these models togther intoa dynamic model. This model was fit using two common fitting techniques, OLS and NLS.The new dynamic taper model that was developed (Model 3) took into account the factchapter 5. Discussion 139that tree shapes and taper change continuously along the entire length of the tree at one time,and over time at the same position, by incorporating models which predict dbh, height, andform changes over time. These change in tree shape and taper along the stem were expressedas functions of tree and stand variables which change over time. As a result, the dynamictaper function was able to predict, with reasonable accuracy, the change in diameter insidebark along the stern bole and in total tree volume over time.Kozak’s (1988) static taper function (Model la), had better and volume predictionthan the same function converted into a dynamic model (Model 2). This was expected, because Model la is based on measured dbh and height. This means that very good predictionmodels for height and diameter would improve the predictive abilities of any fitted dynamictaper function. The most probable reason why Model 2 was slightly more biased than Modella for large dhh classes is that the dbh prediction model was poor. Model 2 had better predictions (lower biases) for and than Model lb as expected. Model lb represents theprocess used in growth and yield modelling. This means that the process currently used ingrowth and yield modelling of using an existing fitted taper function and inputting predictedheight and dbh for projecting stand growth and yield and filture inventory can result in biased projections. The new technique of fitting the system as a dynamic process improvedthe predictions for and V and would provide better estimates for future growth andyield and inventory projections.Model 3 was marginally superior to Model 2 in predicting aiid obviously superiorin predicting volume. The poorer performance of Model 2 could he attributed to the twovariables, ln(Z + 0.001) and exp(Z) in Model 2 that were excluded from Model 3. Thevariable ln(Z + 0.001) was dropped, because ln(Zt) is undefined at the tree base; Kozakadded an arbitrary constant of 0.001 to obtain a value at the tree base. The addition ofthis constant does not affect the predictive abilities of taper function, but it does shift thewhole function. The variable exp(Zi) was only marginally correlated with and mightinflate small values. Also. Model 3 had additional variables, AGE and QD50, added to theChapter 5. Discussion 140ct prediction model.The fact that differences between Model 2 and 3 were small was surprising because treeform and taper were found to be correlated with QD50, particularly for lower tree sections.However, it could he attributed to the limited taper variability of the pine data used, or tothe fact that the inclusion of the D/H in the Model 2 might have already accounted formost of the variation in taper. Also. since height was a function of age and site index anddbh was a function of height, age and density, these variables (site index, age, and QD50)were already accounted for in the taper model to some extent. One possible conclusion isthat the tree form exponent (i.e., the shapes of tres) might not be changing as fast as thetrees grow. This would mean that shape change is very inargiiial after trees reach a certainage. Another possibility is that the form exponent. which is a function of crown size anddistance from crown, might already be well modelled and trying to add any more variableswill not make any further improvements. It is possible that additional improvements couldbe made by including tree and stand variables when predicting the base (the model for basediameter) of Model 3.The NLS-fitted dynamic taper models were less biased than the associated OLS-fittedtaper models. The differences in predictions for the different fitting methods were mostapparent for the static Kozak (1988) taper model (Models la) and the dynamic version ofthis model (Model 2). The associated OLS-fitted taper models were poorer in predictingboth and V than the same models fitted using NLS.The dynamic taper models (Models 2 and 3) had predictions for df and volume whichwere sllrprisingly similar to the predictions of Model la. This good performance of thedynamic taper models, particularly Model 3, could be attributed to two reasons:1. The data set used (the stem analysis data) was composed of only dominant and codominant, trees. These data had less variation than if suppressed and intermediate treeswere included. Also, the age range of the data was narrow (1-61 years). Therefore,Chapter 5. Discussion 141the form and taper variation among trees might have been small, resulting in a moreaccurate dynamic taper function.2. Lodgepole pine is known to have relatively simple taper (Koch 1987; Clyde 1986). As aresult, a very complicated model was not required for this species, compared to speciestha.t have large taper such as Douglas-fir, western red cedar, a:nd spruces.Even though Model 3 showed surprising good predictions for both and I/h, there areareas that could be improved. Such areas include:1. The height and dbh prediction models were selected using a large data set (PSP data),but fit using a very small data set (final felling stem analysis data). Therefore, thesemodels might have coefficients which are unreliable for use beyond the range of thedata used.2. According to the literature on tree growth, dbh growth depends on density and crownsize, and the form and taper of trees is just a reflection of the crown status of trees.Therefore, crown length or crown ratio could have been important variables in thedynamic taper model; these variables were not available for time periods previous tothat of the time of felling. However, the D/H ratio is known to be highly correlatedwith crown length (Burkhart and Walton 1985, Newnham 1988). Since the D/H ratiowas included in the model for predicting taper over time, it can he assumed that it didaccount somewhat for lack of a crown size measure. Also, the D/H ratio is known tobe a good measure of taper (Kozak 1988, Marshall 1991).3. The dynamic taper models developed were based on only dominant and codominanttrees, which are less affected by stand density than intermediate and suppressed trees.These models should not be used for data that include intermediate and suppressedtrees without re-calibration, as they may produce biased results. However, the useChapter 5. Discussion 142of QD50, and the D/H ratio might somewhat account for the variation due to crownposition.4. The dynamic taper models could be improved when fitted using NLS by incorporatingfunctions for height and dhh predictions for optimization as well. However, this coulddelay or completely prevent the fitting algorithm from reaching optimum solutions.These problems are not all difficult to solve. For example, the height and dbh modelscould be refitted based on a data set with a wider range. The problem of lack of crown sizemeasurements cannot easily be solved unless there exists data that have crown size measuredover time (PSP data with upper stem measurements). Such data also could improve the dbhmodel because crown diameter was found to be important for dbh prediction (Sprinz andBurkhart 1987). The use of dominant and codominant trees is not a big problem since thiscan be solved by calibrating the model with a data set composed of all crown classes. Also,the presence of QD50 and the D/H ratio might account for crown class variations.Based on Clyde’s (1986) findings, a dynamic taper modelling approach to growth modelling seemed to be an infeasible idea. However, from the results of this study, dynamic tapermodelling does appear feasible. Reasons for this difference could include the following.1. The static taper models used by Clyde assumed that tree form did not change alongthe stem over time. Therefore, her dynamic taper models would not account for thechanges in stem form over time. Such a deficiency might have caused the poor results.2. The prediction models for height and dbh used by Clyde were functions of age alone.She never included other tree, stand, and site variables such as site index for heightand dbh prediction and stand density and height for dbh prediction that were foundto be important in this study.Chapter 6Conclusions and RecommendationsForest management has become more intensive and reached a stage where acquiring the bestinformation, in the most efficient way, is very important. Much of the necessary informationis provided by forest inventories and growth and yield projections. Both forest inventoriesand growth and yield projections require accurate and current information for both tree andstand volume. Understanding and modelling taper changes over time will help improve thequality of the information provided.In this study, several aspects of variation in form and taper of trees along the stem overtime were examined for dominant and codominant lodgepole pine trees from the interiorof British Columbia. The objectives of this research were twofold: (1) to investigate thevariation in tree form and taper over time as affected by changes in stand, tree, and sitecharacteristics; and (2) to use the knowledge gained from objective one to develop a dynamic taper function. The goal of developing this dynamic taper function was to be able toaccurately predict the tree shape and taper cha.nges over time given certain stand conditions.The first objective of this study has been achieved. The impact of various factors ontree shape and taper changes along the stem at one time and over time was studied. Itwas found that lodgepole pine trees have a simple shape (parabolic) from ground to top,a statement made by Forslund (1982) without confirmation. At the same time, the resultsdemonstrated that taper and form changes over time were different at different heights alongthe stem. The pattern of variation in tree taper and form was different for different ages,crown sizes, and stand densities. Trees growing in less dense stands were more tapered ata given height than trees growing in more dense stands. Since stand density and crown143Chapter 6. Conclusions and Recommendations 144size influence the distribution of diameter growth along the stem, it is possible for a forestmanager to manipulate stand density and tree crown length in order to produce trees withdesired characteristics.Using the knowledge gained from objective one. the second objective of this study wasachieved because two dynamic taper functions were developed which was able to accuratelypredict diameter along stem and tree volume over time. These dynamic taper models werenot fit using a technique that would account for the non-iid characteristics of the stem analysisdata used because of a number of problems involved with using such fitting techniques.Nevertheless, the dynamic taper models gave precise predictions for along the stem,especially for lower sections which contain more volume, and for total tree volume, whenfitted using either OLS and NLS. These models took into account the fact that the geometricshapes of the tree stem changes continuously along its entire length and over time at thesame position. One of the dynamic taper models (Model 3) expressed change in tree sha.pealong the stem as a function of tree and stand variables which change over time.Model 3 has the following advantages:1. It gives consistently accurate estimates of diameter along the stem over time, particularly for lower sections, and total tree volume over time. This model comparesfavourably with Kozaks (1988) static taper model (Model la), used for current inventorv. It is also an improvement over the static model (Model ib) with inputs ofpredicted height and dbh, presently used for growth and yield.2. Since it includes age and predicted quadratic mean diameter at age 50 (a stand densitymeasure) in its form exponent model, this model is good for tracking tree shape andtaper changes over time. This could be the reason that it was found to be less biasedfor d1 and V than Model 2.3. Like Kozak’s (1988) model (Model la), it ta.kes into account the continuous variation inform along the stem (a factor that was confirmed by objective one), and thus requiresChapter 6. Conclusions and Recommendations 145a single taper function. In addition, tree form changes over time were modelled.4. It is easy to fit using the standard fitting methods (OLS and NLS).However, Model 3 has some problems.1. It had poorer predictions than desired for upper stern diameter, particularly for thetwo sections before the tip.2. It is not easily integrated to obtain volume (merchantable or total) directly. Therefore,potentially time consuming numerical methods have to be used. Also, it is not possibleto transpose and obtain the height directly, as with Max and Burkhart’s (1976) taperfunction. This means height has to be obtained by iterative methods. However, thisis becoming less of a problem with the ever increasing computering power available.Apart from the few problems noted above, the developed dynamic taper model was ableto take into account form variation along t.he stem over time. The future of the dynamictaper modelling philosophy looks brighter than was portrayed by Clyde (1986).Probably the greatest promise in the dynamic approach to modelling tree taper is thefact that it allows visualization of how tree shape and taper will vary given a set of standconditions. This would allow a forest manager to simulate the manipulation of the standdensity and crown length through thinning and pruning in order to achieve specific treeshape and volume objectives.Also, this study is a contribution to increasing to body of information on growth modellingin the following ways.1. A detailed study was conducted on how tree shape and taper changes over time fordominant and codominant lodgepole pine. This increased understanding of the basicchanges in tree form and taper over time and how various factors (site quality, standdensity, crown length, etc) affect tree form and taper.Chapter 6. Conclusions and Recommendations 1462. The dynamic taper functions developed, which incorporated site, stand, and tree information. can be used to estimate current and future volume and taper of individualtrees. These models could he incorporated easily into an individual tree growth modelfor volume predictioitBy incorporating tree, stand, and site factors into a simple variable-exponent taper modelby Kozak (1988), a dynamic taper function was developed. This dynamic taper function wasable to track the behaviour of very complex tree shape and taper changes over time withreasonable accuracy. The dynamic taper modelling approach will he a useful tool in forestmanagement because dynamic taper models (e.g., Model 3) can be developed that will enableforest managers to simulate stand development in order to achieve specific objectives.This study has provided the basis for further investigation into the variation of tree shapeand taper over time and dynamic taper modelling for B.C. forests. However, some complications were encountered. These led to the following suggestions for further investigationsinto the process of dynamic taper modelling.1. There is a possibility of applying FGLS or FGNLS for fitting dynamic taper functions,if the modeller is willing to make simplifications concerning serial correlation and heteroskedasticity by pooling data. The error variances and covariances, and the measuresof serial correlation, could be assumed to be functions of tree, stand, and site variables,in order to remove the problem of undefined values.2. Dynamic taper models could he improved by including a crown size measure, becausecrown length and crown ratio were found to be highly correlated with tree shape. Thiswould require upper stern measures of trees in PSPs along with crown size over time,rather than stem analysis data for fitting the model. Alternatively, crown size couldbe monitored for trees on PSPs and then these trees could he felled at a later date forstem analysis.Chapter 6. Conclusions and Recommendations 1473. Although the new dynamic taper model (Model 3) had good predictions for andvolume, some improvements could be achieved if the prediction model for base diameter(0.Dal) was improved by adding some tree, stand, and site variables.4. Dynamic taper models could be extended to other species and crown classes (i.e. include intermediate and suppressed trees). These models may not he as good as thosedeveloped for this thesis, since lodgepole pine has relatively little taper, compared tospecies like Douglas-fir, western red cedar, and spruces.5. Additional work could be done to examine the effect of other stand density measures,and how specific silvicultural treatments such as thinning and pruning directly affecttaper and form changes along the stein over time. The dynamic taper functions couldthen he re-examined and modified if necessary.6. The accuracy of any dynamic taper model depends on the precision of the modelsemployed for predicting height and dbh. Another model for dbh prediction could beselected and fitted, preferably a nonlinear model. since the dhh model used for for thisstudy was not as accurate as desired.The current procedure used in growth and yield of inputting predicted height and dbhinto an existing taper function was found to result in biased predictions for and V.Therefore, the dynamic taper modelling approach should be used because it was found toprovide more accurate predictions for and V than the method currently used.This research opens a new approach to estimating volume for growth and yield models andprojecting future forest inventories. The dynamic taper modelling approach could replacethe process currently used. Such a modelling approach would enable forest managers tosimulate the effects of various silvicultural operations on tree shapes over time. 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Some anatomical responses of loblolly pine to soil-waterdeficiencies. Tappi 44: 179—184.Smith, J.H.G. 1965. Biological principles to guide estimation of stand volumes. Photogram.Eng. 31: 87—90, 95.Smith, J.H.G. 1980. Influences of spacing on radial growth and percentage latewood ofDouglas fir, western hemlock, and western red cedar. Can. J. For. Res. 10:169—175.Smithers, L.A. 1961. Lodgepole pine in Alberta. Bulletin 127. Ottawa, Ontario, Canada,Department of Northern Affairs and National Resources, and Department ofForestry. 153 p.Snee, R.D. 1977. Validation of regression models: methods and examples. Technometrics19: 415—428.Sprinz, P.T. and H.E. Burkhart. 1987. Relationships between tree crown, stem, and standcharacteristics in unthinned loblolly pine plantations. Can. J. For. Res. 17:534—538.Spurr, S.H. 1952. Forest inventory. Ronald Press Co., New York. 476 p.Stage, A.R. 1973. Prognosis model for stand development. USDA For. Serv. Res. Pap.INT—137. 32 p.Stark, R.W., and J.A. Cook. 1957. The effects of defoliation by the lodgepole needle miner.For. Sci. 3: 377—395.Sterba, H. 1980. Stern durves—a review of the literature. For. Abst. 41: 141—145.Sweda, T. 1988. A theoretical stem taper curve (II). J. Jap. For. Soc. 70: 199—205.Tepper, H.B., H.E. Wilcox, and F.A. Valentine. 1968. The effect of fertilization and depthto ground water on wood deposition in red pine. For. Sci. 14: 2—6.Thomas, M., S.L. Ranson, and J.A. Richardson. 1973. Plant physiology. 5th Ed. LongrnanGroup Ltd., London. 1061 p.Literature Cited 158Thomson, A.J. and H.J. Barclay. 1984. Effects of thinning and urea fertilization on thedistribution of area increment along the hole Douglas-fir at Shawnigan Lake.British Columbia. Can. J. For. Res. 14: 879—884.Thomson, A.J. and G.A. Van Sickle. 1980. Estimation of tree growth losses caused by pestactivity. Can. J. For. Res. 10: 176—182.Valinger, E. 1992. Effects of wind sway on stein form and crown development of scots pine(Finns sylvestris L.). Aust. For. 55: 15—21.Walters, D.K. and D.W. Hann. 1986. Taper equations for six conifer species in SouthwestOregon. Oreg. State Univ. For. Res. Lab. Res. Bull. No. 56. 42 p.Wilson, B.F. 1984. The growing tree. University of Massachusetts Press, Amhurst. 138 p.Wykoff, W.R. 1984. Introduction to prognosis model—Version 5.0. USDA For. Serv. Gen.Tech. Rep. INT—193. p. 44—52.Tykoff, W.R. 1986. Supplement to the user’s guide for the stand prognosis model version5.0. USDA For. Serv. Gen. Tech. Rep. INT—208. 36 p.Wykoff, W.R., N.L. Crookstoll. and A.R. Stage. 1982. User’s guide to the stand Prognosismodel. USDA For. Serv. Geu. Tech. Rep. INT—133. 112 p.Appendix AGlossary of Variables UsedH or height is total tree height in metres.D or dbh is diameter inside bark at breast height in centimetres.d is diameter in centimetres inside bark at height ii above ground.h is height above ground in metres.BH is breast height (1.3 metres above ground).V is total volume in cubic metres.Age is tree age at breast height in years.CL is crown length in metres.CR is crown ratio or crown length to total tree height ratio.SI is site index or height of site trees (average height of dominant and codominant trees) inmetres at age 50 years.Q D is quadratic mean diameter in centimetres.QD50 is quadratic mean diameter at age 50 years in centimetres.SDI is the stand density index.RD is the relative density.SPH is the number of stems per hectare.BA is basal area in square metres per hectare.AGE is the mean breast height age per plot in years.159Appendix BQuadratic Mean Diameter Curves160Appendix B. Quadratic Mean Diameter CurvesI403020100Figure B.16:0zZzC.)C.0 50 100 150 200BREAST HEIGHT AGE (years)Quadratic mean diameter (cm) for site index 15 m by breast height age.161iS0040ov •iS•30•0 —0 50 100 150 200BREAST HEIGHT AGE (years)Figure B.17: Quadratic mean diameter (cm) for site index 10 m by breast height age.Appendix CTree Form Variation with Tree Breast Height Age along the Stem162Appendix C. Form Variation with Tree Age 16:34.O.1H MEASURESO.5H MEASURESHMEASURE1.50.508 13 18 23 28 33 38 43 48 53BREAST HEIGHT AGE (YEARS)Figure C.18: Form exponent (r) by breast height age (years) for tree 2 in plot 16 (16.2),dbh=22.2 cm, height=21.1 m, Age=53 years, and QD50=18 cm at three relative heights(O.1H, O.5H, and O.8H) above the ground.AO.1H MEASURES3.5O.5H MEASURES3 O.8H MEASURESI—z10.501 6 1’l 16 21 26 1 36 41 46BREAST HEIGHT AGE (YEARS)Figure C.19: Relative diameter by breast height age (years) for tree 2 in plot (25.2), clbh=8.3cm, height=9.3 in, Age=46 years. and QD50=6 cm at three relative heights (0.IH. 0.5H, and0.811) above the ground.Appendix C. Form Variation with Tree Age 164q0.1K MEASURES35 -*-0.5H MEASURES0.8H MEASURESI—z 2.5wz021.5LL 1—--———-W10.56 11 16 21BREAST HEIGHT AGE (YEARS)Figure C.20: Form exponent (r) by breast height age (years) for tree 1 in plot 42 (42.1),dbh=13.9 cm. height=13.O m, Age=21 years, and QD50=18 cm at three relative heights(O.1H, O.5H. and O.8H) above the ground.0.1H MEASURES3.50.5K MEASURES3 0.8H MEASURESI—z 2.5wz02- 2><1.5LL0.504 14BREAST HEIGHT AGE (YEARS)Figure C.21: Form exponent (r) by breast height age (years) for tree 2 in plot 43 (43.2).dbh=5.5 cm, height=5.1 m. Age=14 years, and QD50=12 cm at three relative heights (O.IH.0.511, and 0.811) above the ground.Appendix C. Form Variation with Ttee Age 165A*O.1H MEASURES3.5O.5H MEASURES3 O.8H MEASURESz 2.5150•LJ10.5010 15 20BREAST HEIGHT AGE (YEARS)Figure C.22: Form exponent (r) by breast height age (years) for tree 1 in plot 46 (46.1),dbh=17.O cm, height=9.7 m, Age=20 years, and QD50=22 cm at three relative heights (O.1H,0.511, and 0.8H) above the ground.Appendix DTree Form and Taper Variation along the Stem Against Relative Tree Height166Appendix D. Form and Taper Variation along the Stem 167wHw4:wFigure D.24: Relative diameter by relative height for tree 16.2, clbh22.2 cm, height=21.1m, Age=53 years, and QD5018 cm for different measurement periods.053 YEARS0 N43 YEARS4N A33 YEARSA1) xN 23 YEARSI—.13YEARSw 3 zzQ A><W00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9RELATIVE HEIGHT ABOVE GROUND (IvH)Figure D.23: Form exponent (r) by relative height for tree 16.2, dbh=22.2 cm, height=21.im, Age=53 years, and QD50=18 cm for different measurement periods.053 YEARSN43 YEARSaA33 YEARSK23 YEARS13 YEARS00Ac0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9RELATIVE HEIGHT ABOVE GROUND (h/H)Appendix D. Form and Taper Variation along the Stem 1685046 YEARSN36 YEARS4 A.26 YEARSx16YEARSF— w6YEARSz0xw0NA ciciNND0.1 0.2 0.3 d.4 0.5 0.6 0.7 0.8 0.9RELATIVE HEIGHT ABOVE GROUND (M-I)Figure D.25: Form exponent (r) by relative height for tree 25.2. dbh=8.3 cm, height=9.3 m.Age46 years, and QD50=6 cm for different measurement periods.1.4046YEARSNO 1.2 ‘ 36YEARS0 A 26 YEARS1• DNA Z 16YEARSN A 6 YEARS0 X0.6 0NA,AA cix0w>I— 0.2-uJ0 0.1 0:2 0:3 0.4 0.5 0.6 0.7 0.8 0.9RELATIVE HEIGHT ABOVE GROUND (h/H)Figure D.26: Relative diameter by relative height for tree 25.2, dbh=8.3 cm. height=9.3 m.Age=46 years, and QD50=6 cm for different measurement periods.Appendix D. Form and Taper Variation along the Stem 169r021 YEARSN16 YEARS411 YEARSK6 YEARSz0cL02O K K 0Ii- D, W 0D NTM10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1RELATIVE HEIGHT ABOVE GROUND (h4-I)Figure D.27: Form exponent (r) by relative height for tree 42.1, dbh=13.9 cm, height=13.0rn, Age=21 years, and QD50=18 cm for different measurement periods.— 1.4•K 021 YEARS4f. NL..J I. 16YEARSZ AD Ao A 11YEARsDNA K K0 0 6YEAJRSLU>0.80.6LU 0NA00.4 TMLU 0>0LUTM0 I I I0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9RELATIVE HEIGHT ABOVE GROUND (h/H)Figure D.28: Relative diameter by relative height for tree 42.1, dbhl3.9 cm, height= 13.0in. Age21 years, and QD5018 cm for different measurement periods.Appendix D. Form and Taper Variation along the Stem 170C14 YEARS9 YEARS44 YEARSw3z0ci.><LUC°N o100 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9RELATIVE HEIGHT ABOVE GROUND (h/i-i)Figure D.29: Form exponent (r) by relative height for tree 43.2, dbh=5.5 cm, height=5.i m.Age=14 years, and QD50=12 cm for different measurement periods.1.6A 014YEAPSA 9’F4SA0 12 4YEARS00W.o A< flQ.0w AI— 0Ui 0.60wi’. C>A0.2Ui0 I I I I I0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9RELATIVE HEIGHT ABOVE GROUND (h/H)Figure D.30: Relative diameter by relative height for tree 43.2, dbh=5.5 cm, height.5.1 m.Age=14 years, and QD50=12 cm for different measurement periods.Appendix D. Form and Taper Variation along the Stem 171020 YEARSN15 YEARS410 YEARSx5 YEARS!: :. NN N A0 0.1 0.2 0.3 d.4 0.5 0.6 0.7 0.8 0.9 1RELATIVE HEIGHT ABOVE GROUND (M-i)Figure D.31: Form exponent (r) by relative height for tree 46.1, dbh=17.O cm. height9.7m, Age=20 years, and QD50=22 cm for different measurement periods.1.602OYEAPSXO I5YEARSZ z AO 1.2 A 1OYEARS5(t5ON A K0.M0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9RELATIVE HEIGHT ABOVE GROUND (WH)Figure D.32: Relative diameter by relative height for tree 46.1, dbh= 17.0 cm. height=9.7 m.Age20 years. and QD50=22 cm for different measurement periods.Appendix ETree Form Variation with Height Above Ground and Breast Height Age172Appendix E. Form Variation with Height and Age 173Figure E.33: Form exponent (r) by height (m) above ground and breast height age (years) fortree 16.2, dbh22.2 cm, height=21.1 m, Age=53 years, and QD50=18 cmfor all measurementperiods.Figure E.34: Form exponent (r) by height (m) above ground and breast height age (years) fortree 25.2, c1hhS.3 cm, height9.3 m, Age46 years, and QD06cm for all measurementen o d s.LBIJ0IAppendix F. Form Variation with Height and Age 174IFigure E.35: Form exponent (r) by height (m) above ground and breast height age (years) fortree 42.1, dbh= 13.9 cm, height= 13.0 m, Age=21 years, and QD50=18 cm for all measurementperiods.Figure E.36: Form exponent (r) by height (m) above ground and breast height age (years) fortree 43.2. dbh=5.5 cm, height=5.1 m, Age14 years, and QD5012 cm for all measurementperiods.IJ0LB12.81$S—Appendix E. Form Variation with Height and Age 175IFigure E.37: Form exponent (r) by height (rn) above ground and breast height age (years) fortree 46.1, dbh17.O cm, height=9.7 m, Age2O years, and QD50=22 cm for all measurementperiods.9$IsCAppendix FVolume Predictions per Tree for all Taper Functions176Appendix F. Tree Volume Prediction 1770.450/OLS/0.4—ACTUAL/0.35 Model Ia/Modelib / f0.3 -D- /Model2LJJ /025 Model3• /0 /> I.LU /02 /H—0l5• /0.10.0502 7 12 17 22 27 32 37 42 47 52BREAST HEIGHT AGE (YEARS)0.4NLS0.35ACTUAL03 ModellaModellb-0-LU 0.25 Model 2D Model30.2 /LULUH0.150.10.052 7 12 17 22 27 32 37 42 47 5’2BREAST HEIGHT AGE (YEARS)Figure F.38: Total tree volume (iii) by breast height age (years) for tree 14.1 (clbh=21.9cm, height2O.9 rn, QD50=18 cm. and SI=20 m).Appendix F. Tree Volume Prediction i 780.4OLS/0.35-.-/ (ACTUAL //0.3 Modella/Modellb / “--0.25 Model2 /I’—D—J ModeI3 //0.2ww4’I.0.150.10.050 I I I I I I7 12 17 22 27 32 37 42 47 52 57BREAST HEIGHT AGE (YEARS)0.4NLS0.35ACTUAL/0.3 io&laModel lbw025• Model2D0.2Model3uJwI—0.150.10.050 I I I I7 12 17 22 27 32 37 42 47 52 57BREAST HEIGHT AGE (YEARS)Figure F.39: Total tree volume (m3) by breast height age (years) for tree 17.1 (dhh=2 1.2cm, height=20.6 m, QD50=16, and SI=18).Appendix F. Tree Volume Prediction 179C,,wDFigure F.40: Total tree volume (in3) by breast height age (years) for tree 2.1 (dbli=8.1 cm.3 8 13 18 23 28 33 38 43 48BREAST HEIGHT AGE (YEARS)height=9.4 m, QD50=6 cm. and SI1O m).Appendix F. Tree Volume PredictionuJc)wD180Figure P.41: Total tree volume (m3) by breast height age (years) for tree 41.2 (dbh=20.icm, heightz10.4 iii, QD50=25 ciii. and Sb21 iii).8 13BREAST HEIGHT AGE (YEARS)

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