FOREST MENSURATION

Managing Forest Ecosystems

Series Editors:

Klaus von Gadow

Georg-August-University,

Göttingen, Germany

Timo Pukkala

University of Joensuu,

Joensuu, Finland

and

Margarida Tomé

Instituto Superior de Agronomía,

Lisbon, Portugal

Aims & Scope:

Well-managed forests and woodlands are a renewable resource, producing essential raw material

with minimum waste and energy use. Rich in habitat and species diversity, forests may contribute to

increased ecosystem stability. They can absorb the effects of unwanted deposition and other

disturbances and protect neighbouring ecosystems by maintaining stable nutrient and energy cycles

and by preventing soil degradation and erosion. They provide much-needed recreation and their

continued existence contributes to stabilizing rural communities.

Forests are managed for timber production and species, habitat and process conservation. A subtle

shift from multiple-use management to ecosystems management is being observed and the new

ecological perspective of multi-functional forest management is based on the principles of ecosystem

diversity, stability and elasticity, and the dynamic equilibrium of primary and secondary production.

Making full use of new technology is one of the challenges facing forest management today.

Resource information must be obtained with a limited budget. This requires better timing of resource

assessment activities and improved use of multiple data sources. Sound ecosystems management,

like any other management activity, relies on effective forecasting and operational control.

The aim of the book series Managing Forest Ecosystems is to present state-of-the-art research

results relating to the practice of forest management. Contributions are solicited from prominent

authors. Each reference book, monograph or proceedings volume will be focused to deal with a

specific context. Typical issues of the series are: resource assessment techniques, evaluating

sustainability for even-aged and uneven-aged forests, multi-objective management, predicting forest

development, optimizing forest management, biodiversity management and monitoring, risk

assessment and economic analysis.

The titles published in this series are listed at the end of this volume.

Volume 13

and

Forest Mensuration

Somerset West, South Africa

Universit t G ttingen, Germanyä ö

By

Anthonie van Laar

Alparslan Akça

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Anthonie van Laar was born in the Netherlands in 1923. He studied forestscience at the University and Research Centre Wageningen between 1941 and1949. In 1958 he emigrated to South Africa and obtained his D.Sc. degree inForest Science at the University of Stellenbosch (1961), thereafter Dr.oec.pub(1973), and Dr.hab. (1979) at the University of Munich. His theses dealt withforest biometry and growth modeling. Since his retirement in 1988, van Laarcontinued his involvement in these subjects, more particularly in growth modelsfor Eucalyptus grandis.

Alparslan Akca (born 1936) studied forestry at the University of Istanbul,Turkey and at the University of Freiburg, Germany. He received his doctoratein forestry at the University of Freiburg on identification of land use classesand forest types by means of microdensitometer and discriminant analyses in1970, for his habilitations in photogrammetry and geodesy at the Universityof Istanbul in 1976, and in forest inventory and forest management at theUniversity of Gottingen in 1981. He was Professor for Forest Management,Forest Inventory, and Remote Sensing at the University of Gottingen. He retiredin 2001. His main research interest is in forest mensuration, forest inventory,and remote sensing in forestry.

PREFACE

Forest mensuration is one of the most fundamental disciplines within forestand related sciences. It deals with the measurement of trees and stands andthe analysis of the resultant information. During the early days of sustainedforest management simple measurement and estimation methods and with theanalysis of inventory and research data were available. The middle of last cen-tury, however, witnessed a worldwide increase in the need for more quanti-tative information about trees and stands. This generated the need for moresophisticated methods to obtain and analyze forest data. This development wasfollowed by a phenomenal explosion of information.

During the past decades there has been fruitful cooperation between theInstitute of Forest Inventory and Forest Growth, formerly “Institute of ForestManagement and Forest Yield Sciences” of the University of Gottingen,Germany and the Faculty of Forestry of the University of Stellenbosch, SouthAfrica. This book is one of the results of this fruitful cooperation between theseinstitutions.

The first edition of this book was published in 1997 by Cuvilliers inGottingen. It was completely revised and supplemented with a presentationand review of recently developed tools and methods to measure and analyzeforestry-related data. The purpose of this book is to provide information aboutthe subject for those readers who are involved in this category of quantitativemethods. Since the middle of last century the increased availability of personalcomputers, and software and the consensus that statistical methods are indis-pensable for estimating tree and stand parameters and for testing statisticalhypotheses, had a considerable impact on the progress of forest mensuration asa research and management tool.

vii

viii Preface

It has been written for forestry students and for practical foresters, and doesnot presuppose a more than elementary knowledge of mathematics and statis-tics. However, because of the notable influence of statistics on forest mensu-ration methods and techniques, and the crucial role of sampling techniques,it reviews and evaluates elementary statistical concepts. At the same time itwas thought to be imperative to discuss spatial structure and diversity and toadd a chapter on conventional and digital remote sensing. Numerous practicalexamples have been included in this edition. They are thought to be particul-arly useful for university and college students. Additional information aboutspecific topics has been added for the benefit of the advanced reader and waswritten in italics.

Forest mensuration as a scientific tool originated in Europe and has alwaysplayed an important role in the practice of forest management. In additionto conventional terrestrial methods, quantitative remote-sensing techniquesform an integral part of the forest mensuration curriculum at the Universityof Gottingen. Countries in the southern hemisphere can derive considerablebenefits from this longtime experience. Conversely, short-rotation plantationforestry in the southern hemisphere and more particularly in South Africa actedas a stimulus for the application of statistics and other quantitative methods asa decision-making tool.

The Authors are greatly indebted to Mrs. J. Wirkner for proofreading theEnglish text of the first edition, Mrs. S. Rudiger for the technical drawings andgraphs, and Mr. W. Tambour for his illustrations. We would furthermore like tosincerely thank Mr. Th. Beisch and Mr. H. Heydecke for their constant supportin electronic data-processing matters, as well as Mrs. F.J.A. Allwright, whoassisted with the correspondence and EDP in Stellenbosch.

The authors gratefully acknowledge the inclusion of research data fromvarious sources, more particularly from the Faculties of Forestry at theUniversities of Gottingen and Stellenbosch respectively and those obtainedfrom Professor Prodan in Freiburg and Dr. Forrest in Australia could be incor-porated into the current edition. They gratefully acknowledge permissionfrom Professor Dr. H.C.H. Kramer and Professor Dr. A. Akca, the authors of“Waldmesskunde,” to include numerous drawings from this book. And finallyour sincere thanks are due to Springer and Editors of the book series who madethe publication of this edition possible.

Anthonie van LaarStellenbosch, South AfricaAlparslan AkcaGottingen, Germany

May 2007

SUMMARY

The management of forests and tree plantations requires a quantitative esti-mate of the current and future volume and biomass of timber and by-products,at national, regional, and local levels. Such information is also needed forforest-policy decisions. Forest and forestry research requires a great deal ofadditional information, for example about the density of forests and stands,diversity, spatial distribution of trees within stands, the size and size distribu-tion of trees within stands, and the expected growth of trees and stands. For-est mensuration is the discipline which deals with these topics. During recentyears, considerable progress has been made to develop methods for measuringtree and stand characteristics, but also in instrumentation and in the statisticalanalysis of forest mensurational information. Sampling, based on inferentialstatistics, plays a dominant role in forest mensuration and forest inventory, pri-marily because of the high cost of collecting and processing field data. Mod-ern sampling methods make it possible to find an optimum sampling strategywhich produces sufficiently accurate estimates at the lowest cost. Quantitativeinformation is primarily obtained by ground surveys, aerial photographs, andsatellite imagery are increasingly applied to obtain basic information about thespatial distribution of forests, possibly also to classify these forests accordingto specified categories and to supplement ground surveys.

In this book the authors, who are emeritus professors at the Universities ofStellenbosch and Gottingen in South Africa and Germany respectively, summa-rize and review currently used forest mensuration and forest inventory methods.A large number of worked examples have been added, primarily for the benefitof undergraduate and postgraduate students.

ix

CONTENTS

PREFACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

Chapter 1: INTRODUCTION . . . . . . . . . . . . . . . . . . . . 1

Chapter 2: STATISTICAL PREREQUISITES . . . . . . . . . . . 51 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . 52 SCALES AND UNITS OF MEASUREMENT . . . . . . . . 6

2.1 Scales of measurement . . . . . . . . . . . . . . . . . 62.2 Units of measurement . . . . . . . . . . . . . . . . . 7

3 GRAPHICAL PRESENTATION OF DATA . . . . . . . . . . 74 DESCRIPTIVE STATISTICS . . . . . . . . . . . . . . . . . 95 PROBABILITY DISTRIBUTIONS . . . . . . . . . . . . . . 13

5.1 Normal distribution . . . . . . . . . . . . . . . . . . . 135.2 Binomial distribution . . . . . . . . . . . . . . . . . . 155.3 Poisson distribution . . . . . . . . . . . . . . . . . . . 165.4 Distribution of χ2 . . . . . . . . . . . . . . . . . . . 175.5 Distribution of student’s t . . . . . . . . . . . . . . . 18

6 ESTIMATION . . . . . . . . . . . . . . . . . . . . . . . . . 206.1 Bias, precision and accuracy . . . . . . . . . . . . . . 206.2 Estimators . . . . . . . . . . . . . . . . . . . . . . . 226.3 Estimating accuracy . . . . . . . . . . . . . . . . . . 23

7 REGRESSION AND CORRELATION ANALYSIS . . . . . . 257.1 Simple linear regression . . . . . . . . . . . . . . . . 257.2 Correlation analysis . . . . . . . . . . . . . . . . . . 267.3 Multiple regression analysis . . . . . . . . . . . . . . 267.4 Nonlinear regression . . . . . . . . . . . . . . . . . . 27

8 MOVING AVERAGE . . . . . . . . . . . . . . . . . . . . . 27

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xii Contents

9 SMOOTHENING BY FITTING EQUATIONS . . . . . . . . 2810 FREEHAND FITTING . . . . . . . . . . . . . . . . . . . . 29

Chapter 3: INSTRUMENTS . . . . . . . . . . . . . . . . . . . . . 331 DIAMETER-MEASURING INSTRUMENTS . . . . . . . . . 33

1.1 Calipers . . . . . . . . . . . . . . . . . . . . . . . . . 331.2 Biltmore stick . . . . . . . . . . . . . . . . . . . . . . 351.3 Diameter tapes . . . . . . . . . . . . . . . . . . . . . 361.4 Permanent diameter tapes . . . . . . . . . . . . . . . 361.5 Wheeler’s pentaprism . . . . . . . . . . . . . . . . . . 371.6 Finnish parabolic caliper . . . . . . . . . . . . . . . . 381.7 Barr and stroud optical dendrometer . . . . . . . . . . 39

2 RELASCOPES AND PRISMS . . . . . . . . . . . . . . . . 392.1 Angle gauges . . . . . . . . . . . . . . . . . . . . . . 392.2 Kramer’s dendrometer . . . . . . . . . . . . . . . . . 392.3 Bitterlich’s mirror relascope . . . . . . . . . . . . . . 402.4 Wide-scale mirror relascope . . . . . . . . . . . . . . 432.5 Bitterlich’s telerelascope . . . . . . . . . . . . . . . . 442.6 Prisms . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3 TREE HEIGHT . . . . . . . . . . . . . . . . . . . . . . . . 473.1 Telescopic poles . . . . . . . . . . . . . . . . . . . . 473.2 Hypsometers . . . . . . . . . . . . . . . . . . . . . . 473.3 Hypsometer according to the trigonometric principle . 473.4 Baseline slope correction . . . . . . . . . . . . . . . . 513.5 Hypsometer according to the geometrical principle . . 54

4 BLUME–LEISS RANGE-TRACER DRUM . . . . . . . . . . 565 TREE CROWN AND FOLIAGE . . . . . . . . . . . . . . . 576 SHORT-TERM RADIAL GROWTH RESPONSES . . . . . . 577 INCREMENT CORES . . . . . . . . . . . . . . . . . . . . . 588 BARK THICKNESS . . . . . . . . . . . . . . . . . . . . . . 599 RECENT DEVELOPMENTS IN INSTRUMENTATION . . . 59

9.1 Laser dendrometer LEDHA . . . . . . . . . . . . . . 609.2 Criterion 400 laser dendrometer . . . . . . . . . . . . 629.3 Digital hypsometer forestor vertex . . . . . . . . . . . 62

Chapter 4: SINGLE-TREE MEASUREMENTS . . . . . . . . . . 631 MEASUREMENTS ON STANDING TREES . . . . . . . . . 63

1.1 Age . . . . . . . . . . . . . . . . . . . . . . . . . . . 631.2 Diameter and basal area . . . . . . . . . . . . . . . . 641.3 Tree height . . . . . . . . . . . . . . . . . . . . . . . 671.4 Stem form . . . . . . . . . . . . . . . . . . . . . . . 68

Contents xiii

1.5 Tree volume . . . . . . . . . . . . . . . . . . . . . . 751.6 Bark thickness . . . . . . . . . . . . . . . . . . . . . 781.7 Tree crown and foliage . . . . . . . . . . . . . . . . . 801.8 Leaf surface area, leaf weight and sapwood area . . . . 83

2 VOLUME, LOG CLASSES AND WEIGHTOF FELLED TREES . . . . . . . . . . . . . . . . . . . . . . 852.1 Volume . . . . . . . . . . . . . . . . . . . . . . . . . 862.2 Log rules, grades and classes . . . . . . . . . . . . . . 892.3 Weight . . . . . . . . . . . . . . . . . . . . . . . . . 91

Chapter 5: MEASUREMENT OF STANDS . . . . . . . . . . . . . 951 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . 952 AGE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 953 MEAN DIAMETER . . . . . . . . . . . . . . . . . . . . . . 96

3.1 Arithmetic mean diameter . . . . . . . . . . . . . . . 963.2 Quadratic mean diameter . . . . . . . . . . . . . . . . 973.3 Basal area central diameter . . . . . . . . . . . . . . . 98

4 DIAMETER DISTRIBUTIONS . . . . . . . . . . . . . . . . 994.1 Weibull distribution . . . . . . . . . . . . . . . . . . . 1014.2 Beta distribution . . . . . . . . . . . . . . . . . . . . 1064.3 Gamma distribution . . . . . . . . . . . . . . . . . . 1074.4 Johnson’s SB distribution . . . . . . . . . . . . . . . . 1094.5 Decreasing distributions . . . . . . . . . . . . . . . . 1114.6 Other approaches . . . . . . . . . . . . . . . . . . . . 113

5 STAND TABLES . . . . . . . . . . . . . . . . . . . . . . . 1145.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 1145.2 Parameter prediction and parameter recovery . . . . . 114

6 STAND HEIGHT . . . . . . . . . . . . . . . . . . . . . . . 1166.1 Mean height . . . . . . . . . . . . . . . . . . . . . . 1166.2 Top height . . . . . . . . . . . . . . . . . . . . . . . 1186.3 Fitting height curves . . . . . . . . . . . . . . . . . . 1216.4 Precision of height estimates . . . . . . . . . . . . . . 1246.5 The standardized height curve . . . . . . . . . . . . . 125

7 STAND VOLUME . . . . . . . . . . . . . . . . . . . . . . . 1277.1 Standard tree volume tables and functions . . . . . . . 1277.2 Volume estimation with form height

and volume series . . . . . . . . . . . . . . . . . . . . 1287.3 Stand volume tables and functions . . . . . . . . . . . 1297.4 Estimation with yield tables . . . . . . . . . . . . . . 1307.5 Volume estimation from felled sample trees . . . . . . 1317.6 Critical height sampling . . . . . . . . . . . . . . . . 133

xiv Contents

8 SPATIAL DISTRIBUTION OF TREES . . . . . . . . . . . . 1348.1 Tests of randomness . . . . . . . . . . . . . . . . . . 1348.2 Spatial structure . . . . . . . . . . . . . . . . . . . . 1378.3 Structural diversity . . . . . . . . . . . . . . . . . . . 139

9 STAND DENSITY . . . . . . . . . . . . . . . . . . . . . . . 1409.1 Area-related indices . . . . . . . . . . . . . . . . . . 1409.2 Distance-related indices . . . . . . . . . . . . . . . . 146

Chapter 6: TAPER TABLES AND FUNCTIONS . . . . . . . . . . 1491 TAPER TABLES . . . . . . . . . . . . . . . . . . . . . . . . 1492 STEM PROFILE MODELS . . . . . . . . . . . . . . . . . . 151

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 1512.2 Taper functions . . . . . . . . . . . . . . . . . . . . . 1522.3 Polynomials and segmented polynomials . . . . . . . 156

Chapter 7: TREE VOLUME TABLES AND EQUATIONS . . . . 1631 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . 1632 VOLUME EQUATIONS WITH ONE PREDICTOR

VARIABLE . . . . . . . . . . . . . . . . . . . . . . . . . . 1652.1 Simple tariff functions . . . . . . . . . . . . . . . . . 1652.2 The incorporation of height into the tariff function . . 167

3 EQUATIONS WITH TWO PREDICTOR VARIABLES . . . . 1703.1 Graphic methods . . . . . . . . . . . . . . . . . . . . 1703.2 Regression equations . . . . . . . . . . . . . . . . . . 171

4 EQUATIONS WITH MORE THAN TWO PREDICTORVARIABLES . . . . . . . . . . . . . . . . . . . . . . . . . . 175

5 MERCHANTABLE VOLUME . . . . . . . . . . . . . . . . 177

Chapter 8: TREE AND STAND BIOMASS . . . . . . . . . . . . . 1831 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . 1832 BIOMASS COMPONENTS . . . . . . . . . . . . . . . . . . 184

2.1 Branches . . . . . . . . . . . . . . . . . . . . . . . . 1842.2 Foliage . . . . . . . . . . . . . . . . . . . . . . . . . 1852.3 Stemwood weight . . . . . . . . . . . . . . . . . . . . 1852.4 Bark weight . . . . . . . . . . . . . . . . . . . . . . . 1862.5 Root weight . . . . . . . . . . . . . . . . . . . . . . . 1862.6 Pooling of data . . . . . . . . . . . . . . . . . . . . . 1872.7 Randomized branch and importance sampling . . . . . 187

3 TREE-LEVEL REGRESSION MODELS . . . . . . . . . . . 1884 ADDITIVITY OF BIOMASS COMPONENTS . . . . . . . . 192

4.1 Stem biomass . . . . . . . . . . . . . . . . . . . . . . 1934.2 Branch and needle biomass . . . . . . . . . . . . . . . 1944.3 Root biomass . . . . . . . . . . . . . . . . . . . . . . 194

Contents xv

5 DUMMY-VARIABLES FOR TREE SPECIES . . . . . . . . 1966 RATIO ESTIMATORS AND CLUSTER SAMPLING . . . . 198

Chapter 9: GROWTH AND YIELD . . . . . . . . . . . . . . . . . 2011 DEFINITIONS . . . . . . . . . . . . . . . . . . . . . . . . . 2012 THE GROWTH OF SINGLE TREES . . . . . . . . . . . . . 202

2.1 Growth parameters . . . . . . . . . . . . . . . . . . . 2022.2 Stem analysis . . . . . . . . . . . . . . . . . . . . . . 2032.3 Single-tree models . . . . . . . . . . . . . . . . . . . 209

3 SITE CLASS AND SITE INDEX . . . . . . . . . . . . . . . 2133.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 2133.2 Site index curves . . . . . . . . . . . . . . . . . . . . 2153.3 Site index equations . . . . . . . . . . . . . . . . . . 217

4 THE GROWTH OF STANDS . . . . . . . . . . . . . . . . . 2224.1 Estimating stand growth based on actual

measurements . . . . . . . . . . . . . . . . . . . . . . 2224.2 Stand table projection . . . . . . . . . . . . . . . . . 2264.3 Recent developments . . . . . . . . . . . . . . . . . . 227

Chapter 10: SAMPLING FOR FOREST INVENTORIES . . . . . . 2291 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . 229

1.1 Sampling units . . . . . . . . . . . . . . . . . . . . . 2322 PLOT SAMPLING . . . . . . . . . . . . . . . . . . . . . . . 232

2.1 Plot shape . . . . . . . . . . . . . . . . . . . . . . . . 2322.2 Plot size . . . . . . . . . . . . . . . . . . . . . . . . . 2332.3 Plots on stand boundaries . . . . . . . . . . . . . . . 2382.4 Slope Correction . . . . . . . . . . . . . . . . . . . . 240

3 POINT SAMPLING . . . . . . . . . . . . . . . . . . . . . . 2413.1 Basic principles . . . . . . . . . . . . . . . . . . . . . 2413.2 Choice of basal area factor . . . . . . . . . . . . . . . 2463.3 Nonsampling errors . . . . . . . . . . . . . . . . . . . 2463.4 Efficiency of point sampling . . . . . . . . . . . . . . 249

4 SIMPLE RANDOM SAMPLING . . . . . . . . . . . . . . . 2514.1 Sample size . . . . . . . . . . . . . . . . . . . . . . . 256

5 ERROR PROPAGATION . . . . . . . . . . . . . . . . . . . 2596 STRATIFIED RANDOM SAMPLING . . . . . . . . . . . . 262

6.1 Basic principles . . . . . . . . . . . . . . . . . . . . . 2626.2 Allocation methods . . . . . . . . . . . . . . . . . . . 2666.3 Poststratification . . . . . . . . . . . . . . . . . . . . 2686.4 Block Sampling . . . . . . . . . . . . . . . . . . . . . 269

7 REGRESSION AND RATIO ESTIMATORS . . . . . . . . . 2717.1 Regression estimators . . . . . . . . . . . . . . . . . 2717.2 Ratio estimators . . . . . . . . . . . . . . . . . . . . 273

xvi Contents

8 DOUBLE SAMPLING (TWO-PHASE SAMPLING) . . . . . 2768.1 Double sampling for regression estimators . . . . . . . 2768.2 Double sampling for stratification . . . . . . . . . . . 2808.3 Double sampling for ratio estimators . . . . . . . . . . 281

9 CLUSTER SAMPLING . . . . . . . . . . . . . . . . . . . . 2829.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . 2829.2 Estimators . . . . . . . . . . . . . . . . . . . . . . . 284

10 MULTISTAGE SAMPLING . . . . . . . . . . . . . . . . . . 28710.1 Two-stage sampling . . . . . . . . . . . . . . . . . . 28710.2 Three-stage sampling . . . . . . . . . . . . . . . . . . 290

11 STRIP SAMPLING . . . . . . . . . . . . . . . . . . . . . . 29212 SAMPLING WITH UNEQUAL SELECTION

PROBABILITIES . . . . . . . . . . . . . . . . . . . . . . . 29512.1 List sampling (PPS sampling) . . . . . . . . . . . . . 29512.2 3P sampling . . . . . . . . . . . . . . . . . . . . . . . 298

13 SYSTEMATIC SAMPLING . . . . . . . . . . . . . . . . . . 30014 SAMPLING PROPORTIONS . . . . . . . . . . . . . . . . . 303

14.1 Simple random sampling . . . . . . . . . . . . . . . . 30314.2 Cluster sampling . . . . . . . . . . . . . . . . . . . . 305

15 ESTIMATING CHANGES . . . . . . . . . . . . . . . . . . 30815.1 Independent and matched sampling . . . . . . . . . . 30815.2 Sampling with partial replacement (SPR) . . . . . . . 309

16 LINE INTERSECT SAMPLING . . . . . . . . . . . . . . . 314

Chapter 11: REMOTE SENSING IN FORESTMENSURATION . . . . . . . . . . . . . . . . . . . . . 317

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . 3172 FUNDAMENTALS OF AERIAL PHOTOGRAPHY . . . . . 3183 DENDROMETRIC DATA . . . . . . . . . . . . . . . . . . . 323

3.1 Tree height . . . . . . . . . . . . . . . . . . . . . . . 3233.2 Number of tree crowns . . . . . . . . . . . . . . . . . 3323.3 Crown closure . . . . . . . . . . . . . . . . . . . . . 3343.4 Crown dimensions . . . . . . . . . . . . . . . . . . . 3363.5 Age . . . . . . . . . . . . . . . . . . . . . . . . . . . 3393.6 Profile of the stand’s growing space . . . . . . . . . . 339

4 ESTIMATION OF STAND VOLUME . . . . . . . . . . . . . 3415 ESTIMATION OF VOLUME INCREMENT . . . . . . . . . 344

APPENDIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . 353

INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377