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An application of the average diameter method of preparing growth tables for lodgepole pine Jewesson, Roy Stanley 1954

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AN APPLICATION OF THE AVERAGE DIAMETER METHOD OF PREPARING GROWTH TABLES FOR •LODGEPOLE PINE by ROT STANLEY JEWESSON •A THESIS SUBMITTED IN PARTIAL FULFILMENT OF .THE REQUIREMENTS FOR THE DEGREE OF MASTER OF FORESTRY in .The Faculty of Graduate Studies We accept this thesis as conforming to the standard required from candidates for tbe .degree of MASTER OF FORESTRY .Members of the Faculty of Forestry .The University of British Columbia April, 195H. ABSTRACT In the Foothills Region of Alberta lodgepole pine (Pinus contorta var. l a t i f o l i a ) frequently develops even-canopied, undifferentiated stands, the growth of which tends to be controlled by the density of stocking as well as by site and age. In order to consider a l l these factors, a method for the construction of tables to be used in the prediction of growth of lodgepole pine has been developed. A stand density factor based on basal area per acre and the average stand diameter i s introduced and by means of the relationship of these two variables with age a series of basal area growth tables are devised. Similarly, by means of the relationships which stand age and the ratios of total cubic volume and number of trees per acre to basal area per acre, bear to average stand diameter, two more sets of growth tables have been constructed to show the manner in which these ratios progress with age. The site percent concept of describing site i s discussed. A table showing the average stand diameter developed by a stand of given age and number of trees per acre on the regional average site was constructed, and the technique of measuring site quality of any individual area by expressing the actual stand average diameter as a per-centage of the tabular value i s described. The validi t y of using the site percent method i s proven s t a t i s t i c a l l y and the method of using this measurement to introduce a site correction into the growth tables is shown. Site variation adds but a small refinement to the prediction of growth for this species, possibly because of an actual small variation in the quality of forest land upon which the species w i l l develop in the Foothills region, or possibly because of a bias in the sample. The techniques discussed are based upon the data from 79 sample plots in the Foothills region of Alberta. While i t i s believed that the method of deriving the tables and their use i s well substantiated by the discussions given, no claim i s made for the accuracy of the tables themselves. The data has weaknesses in both the upper and lower classes of age and average diameter and the curves f i t t e d therefrom have suffered accordingly. ACKNOWLEDGMENT The author expresses his appreciation to the Department of Resources and Development, Forestry Branch, Calgary, Alberta, for permission to use the data from which the techniques and tables described herein were dev-eloped. These data must be considered confidential unti l such time as the Forestry Branch releases them for publication. CONTENTS Page Introduction 1 The Sample 6 The Analysis Method of Compilation 9 Age 10 Stand Density 12 Site Percent Concept lit The Importance of Site and Stand Density 25 Application of Site to the Basic Data 30 Volume 3^ Number of Trees P^' Application of the Growth Tables ILLUSTRATIONS Page Figure • 1 2 3 5 6 7 * 9 10 11 12 13 1^  Map showing sampling area. Red dots indicate location of one or more sample plots... 7 Relationship between total height and age of seedlings. ... 11 Relationship between stand age and average stand diameter for a l l the data. ... 13 (a) Relationship between basal area per acre and average stand diameter for the average site and density and (b) between the coef-ficient of variation of basal area and average stand diameter. ••• 15 Basal area growth curves for average site. ... 17 Relationship between average stand diameter and number of trees per acre by age classes' for regional average site. ... 20 Probit values i l l u s t r a t i n g distribution of site percents in the basic data. ... 23 Illustrating the lack of linear correlation between site percent and age. ... 24-Relationship between standard deviation of basal area and stand age. ... 27 Relationship between residuals from the basal area growth curve and site percent. ... 30 Illustrating the lack of linear correlation between density classes and site percent. ... 33 Relationship between (a) total cubic volume-basal area ratio (R) and average stand dia-meter and (b) coefficient of variation of this ratio and average stand diameter. ... 36 Ratio R growth curves for an average site. ... 37 Illustrating the lack of linear correlation between standard deviation of the ratio R and age. ... 38* Page Figure 15 16" 17 Relationship between the absolute residuals of the ratio R and site percent. The relationship between number of trees per acre and average stand diameter. Ratio K growth curves for an average site. 3 9 41 * 3 Table I n H I 17 7 Distribution of the basic data. Stand age by average stand diameter for the average site. Site corrections - Basal area. Site corrections - Ratio R. Site corrections - Ratio K 12 3 1 to 2^ Plate I I I I I I IV Lodgepole pine, 8*0 years, site H, D.C. 17. follows page Lodgepole pine, g>0 years, site U , D.C. I I I . follows Plate I Lodgepole pine, 105 years, site I I , D.C. I I I . follows Plate I I Lodgepole pine, 225 years, site I I I , D.C. I I I . • follows Plate I I I APPENDICES Appendix I Standard volume table f o r lodgepole pine. Volume i n t o t a l cubic f e e t . II Average stand diameters attained by stands of lodgepole pine on the regional average s i t e * I I I Basal area growth t a b l e . Total basal area per acre by age, s i t e and r a t i o classes. IV Ratio of t o t a l cubic volume to basal area per acre by age, s i t e and r a t i o classes. V Ratio of number of trees per acre to basal area per acre by age, s i t e and r a t i o classes. AN APPLICATION OP THE AVERAGE DIAMETER METHOD OF PREPARING GROWTH TABLES FOR LODGEPOLE PINE Introduction The Foothills Region of Alberta (Boreal, B 19)(11) presents a rolling and often broken terrain between 3000 and 5000 feet above sea level, climbing sharply in the west to the Rocky Mountains and sloping eastwards to the Plains regions. Generally, the topography may be described as a series of undulating ridges created by glacial move-ment and deposition, running from west to east. The valleys so formed contain the streams of the Eastern Slope water-shed, which form the headwaters of the Saskatchewan and 2 Athabasca River systems. The soils are usually deep and gravelly, sandy or s i l t y loams, normally well drained but frequently marshy. The tops and steeper sides of the ridges are usually of a dry, rocky or deep, sandy structure. The native rock i s predominantly limestone. The predominant forest tree species of the region i s the subclimactic lodgepole pine (Pinus contorta var. l a t i f o l i a Engelm.) which i s sustained in more or less pure stands as a result of periodic fi r e s which have prevented the white spruce (Picea glauca (Moench) Voss) climax type from gaining dominance. The pine usually occurs in pure stands but may often be found in mixtures with white or black spruce (Picea mariana (M i l l . ) ) , poplar (Populus  tremuloides Michx.), larch (Larix l a r i c i n a (DuRoi) K. Koch) or alpine f i r (Abies lasiocarpa var. lasiocarpa (Hook) Nutt.). It grows on a wide range of site conditions - from the rocky or sandy ridges to the near-muskegs found in the valley bottoms - but prefers deep, sandy, well drained soils on level or south exposures. On such sites i t thrives and reaches i t s optimum growth. While lodgepole pine apparently has a wide t o l -erance to-site conditions, i t i s much less tolerant to light competition. When i t occurs in pure stands density of stocking becomes a decisive factor in determining i t s growth in diameter, height and volume. Characteristically, and perhaps genetically, the species shows very l i t t l e d i f f e r -entiation in the development of individuals within pure 3 stands. Very frequently i t appears that no one stem i s able to grow at a greater rate - particularly in height-than i t s neighbors. This feature, together with i t s intolerance to light competition generally results in the formation of even-canopied stands. When an extremely high density of stocking occurs in young stands they become stagnated and remain for many years in a near dormant condition. This occurs less frequently in the Foothills Region than at the higher elevations of the Subalpine Region (11). The problem of devising growth tables for lodge-pole pine i s therefore complicated by the necessity of including a factor to consider the density of the stands, in addition to the usual variables of site and age. The usual concept of considering that volume growth w i l l vary directly as the percentage relationship of the basal area of a given stand to that of a "normal" stand cannot be accepted where there are such indications of irregularity. The inclusion of an extra independent variable suggests the need for a much larger sample than would ordinarJLy be required. By developing growth tables in the manner to be described, however, the problem can be solved by an easily handled group of empirical data. The methods used in this study are, with some exceptions, extensions and modifications of principles and techniques developed by several well known authors. The 4 use of average stand diameter as the main independent variable in analyzing the growth of even-aged stands of lodgepole pine was f i r s t used by Barnes in 1931 (21). He showed that the number of trees per acre i s more closely related to the mean stand diameter than to age and site. This principle has been adapted to the use of basal area and average stand diameter as a measure of stand density. Used in conjunction in this manner these two factors are synonymous with number of trees per acre but give a more restricted definition of the density factor (3)» Because basal area i s subject to a smaller range of variation i t is more convenient to handle in mensurational techniques. With density of stocking thus defined the relation of basal area to average stand diameter i s plotted for the average conditions represented by the data. The dispersion of individual plots about this mean curve shows the manner in which a l l stands represented by the sample are related to the average condition. The technique of using this dispersion to break the main curve into a family of curves representing the growth trends of stands of various densities i s described by Osborne and Schumacher (16) and Duerr and Gevorkiantz (3) and a modification of i t i s used here. A general curve of stand age on average stand dia-meter i s constructed and the basal area curves are trans-formed into growth curves by eliminating the common variable of average diameter and plotting basal area directly on age. 5 The expression of growth in terms of average diameter has generally been found to exclude the necessity for the inclusion of any measure of site variation. Bruce (12) and Barnes (21), however, introduce a correction factor based upon the average height of the trees of average diameter to give a refined estimate of volumes per acre. Duerr and Gevorkiantz (3) use the dispersion of their "basic data about the curve of average diameter and age to i l l u s t r a t e the variation in stand diameters which may be attributable to varying site conditions. The site percent concept of measuring site in stands of lodge-pole pine as developed by Parker (17) has been used in this study and site, where referred to, i s to be considered in this manner. The value of the addition of a site variable i s assessed and compared to the usefulness of the density factor. It was found that, while both variables are of significant value, the inclusion of site (measured as site percent) i s of relatively small importance compared to density ( measured in terms of basal area and average diameter). For refined predictions of growth, however, the v a r i a b i l i t y Introduced by site differ-ences i s accounted for by means of the standard units method described by Bruce and Schumacher ( l ) . By means of the tech-niques outlined thus far a series of basal area growth curves are devised for stands of varying densities on different sites (Appendix III). For a stand of a given density and site percent there i s a definite relation between the average stand diameter, basal area per acre, number of trees per acre and any expression of volume per acre/ The ratio of volume per acre to basal area per acre w i l l be constant for a given set of conditions of site, density and average diameter, but w i l l change as the latter increases with age (3)« A set of curves describing the development of the ratio of total cubic volume per acre to basal area has been produced (Appendix IV") using tech-niques similar to those used in constructing the basal area growth curves. Similarly a set of curves representing the ratio of number of trees per acre to basal area has been devised (Appendix V ) . By means of the information contained in Appendices III, I V and V i t i s possible to predict future basal area, cubic volume and number of trees per acre for any stand of lodgepole pine (within the region sampled) for which the present age, average diameter, number of trees, basal area and volume per acre are known. The data used in this study, were collected in the f i r s t f i e l d season of a project which was to take two or three years to complete. They are thus only partially complete for the region they purport to represent. This weakness i s reflected in the shapes and f i t t i n g s of some of the curves illustrated in this work. Basically, these curves were fit t e d by the freehand method and balanced in two or more groups (4). In some cases, however, where i t was apparent that the data would show an improbable relationship the curve was adjusted to a more likely f i t . The Sample Eighty-five sample plots, varying in area from one-tenth to one-half acre, were examined in pure lodgepole pine stands in the Foothills Region of Alberta between the 7 Figure 1. Map showing sampling area. Red dots indicate location of one or more sample plots. Red Deer and Athabasca Rivers. These samples were selected to represent as wide a variety of site and density conditions as possible in as many age classes as could be found. No attempt was made to randomize or systematize the location of the plots, but the sample i s without the bias of an attempt to choose samples to f i t a predetermined standard such as the "normality" concept. Essentially the only consciously assigned parameter was that of stand purity; only stands containing 75$ or over (by basal area) of lodgepole pine were used in the analysis. For failure to f i t this standard, six of the original eighty-five plots were discarded. The remaining seventy-nine plots were distributed by age and average stand diameter as shown in Table I. No stands less than 35 years of age and very few over 120 years were sampled, with the result that any calculations concerned with very young or old stands were based upon estimated good curve f i t t i n g s . Nor were the plots well distributed over the Foothills Region, but concentrated in i t s southern portion. Table I Distribution of the Basic Data Av. Diam. (in.) Age Class 30 50 70 90 110 130 Total Number of Plots 1 1 1 2 a — — — - - a 3 2 - 2 1 - — 5 4 - 3 4 9 2 - ia 5 — 3 3 a 3 — 17 6 - 2 — 6 — — a 7 - - - a 5 - 13 8 - - . - 3 - - 3 9 - - - - -. 1 1 10 — — — — - 1 1 11 - — — — — _ — 12 - - - - - 3 3 13 - - - - - 1 1 Total i i 6" 9 35 10 6 79 Plct-te H Z Lodge: pol<z pin<z2Z5y<za.rs, sitalLTQQIII (To f o / l o w P/o.t«It?> 9 The data used in this study are therefore not adequate to form the basis of growth tables which may be considered representative of lodgepole pine i n the region described. They w i l l , however, serve to i l l u s t r a t e the method of constructing such tables. The Analysis Method of Compilation A l l plot data were assembled on a per-acre basis. The t a l l y sheets for each plot were summarized and the following values calculated: a) Number of trees per acre. b) Basal area per acre (trees 0.6" d.b.h. and over). c) Average stand diameter (by basal area). d) Total stand age. e) Total cubic foot volume per acre. f) Ratio of total cubic volume to basal area (e * b). g) Ratio of total number of trees to basal area (a * b). While the plots were selected in essentially pure stands of lodgepole pine,a certain number of trees of other species - white spruce, poplar, etc. - occurred in some stands. If the total basal area of these other species exceeded 25% of the total basal area per acre the plots were discarded. Of the remaining plots, those containing species other than pine were compiled according to the number of "effective" stems. This classification considers that trees 10 of speoies other than pine occupying a place in the stand w i l l affect the growth of the stand as would an equivalent number of pines of the same size. It i s frequently found that these other speeies w i l l be present in two population distributions, one of a diameter range similar to the main pine stand and another of smaller diameters occuring as a definite understory. Where this condition was encountered only the f i r s t portion was included in the "effective" stem classification. The understory, usually composed of a large number of small stems could not seriously affect the growth of the main stand and, i f included, gave a much distorted average stand diameter and basal area. Cubic foot volumes were calculated according to the table of volumes shown in Appendix I. Compilations for each plot were based on individual height-diameter curves. Age The age of each plot was determined from borings taken at a point one foot above the ground from sufficient trees to provide a ring count constant to within two or three years. The number of years required for a seedling to reach one foot in height - seven years in a l l cases - was added to the age shown by the core. Seedling height-age data were colleoted in the Sundre, Rooky Mountain House and Entrance areas. A simple analysis of the variance about the means of each group showed the data to be from homogenous pop-ulations so they were a l l grouped together. The relation-11 A 0 £ 4- C 8 .10 12 14- IG /6 20 Zl 'Totcl Ago- (y^o.rs) Figure 2. Relationship between total height and age of seedlings, ship of seedling height to age i s shown in Figure 2. The sample plot data were next grouped by one-inch average diameter classes and the mean average diameter and mean age of each class computed. These were plotted and a smooth curve drawn through the points (Figure 3)« The average stand ages according to the average stand diameter, as read from this curve, are shown in Table II. This curve expresses only the relative ages between stands of different diameters and, since they represent the average stand of the 12 Table I I Stand Age by Average Stand Diameter for the Average Site Av. Stand Age Dia. (in.) (years) Av. Stand Age Dia. (in.) (years) 1 33 2 43 1 i 1 11 7 96 6 106* 9 121 10 135 11 152 12 172 13 194 .sample they can only apply specifically where such a stand i s encountered. Otherwise they only show, for example, that i t takes on the average about 10 years for a stand to grow from 6 inches to 1 inches in average stand diameter. Stand Density It i s proposed that the basal area of a stand considered with i t s average stand diameter w i l l give a satis-factory measure of i t s density. This i s simply another way of referring to the number of trees per acre and may be inter-preted as being synonymous with the latter expression. A stand of a given average diameter having a high basal area per acre w i l l be considered, to have a higher density than one of equal diameter with a lower basal area. The general relation-ship between basal area and average stand diameter w i l l express the manner in which these two quantities develop simultaneously for the stand of average density. The dispersion of the individual plot data about this curve w i l l indicate the range of densities encountered in the sample and the manner in which this variation i s affected by changes 1 3 c^ (inches) Figure 3 « Relationship between stand age and aver-age stand diameter for a l l the data. in average diameter. The individual plot basal areas were averaged by diameter classes and the means plotted over average diameter to form a strong curve (Figure 4 (a)). Assuming that the f i e l d sample gives equal representation to a l l d e n s i t i e s , t h i s curve represents the trend of basal area with average diameter and consequently with age (Figure 3), under the fundamental assumption that the present stand of a given average diameter w i l l , a f t e r a period of undis-turbed growth, develop into a stand of a greater average diameter s i m i l a r to present representatives of the l a t t e r diameter c l a s s . This curve (Figure 4 (a)) represents the develop-ment of stands having an average stocking or density. It i s generally held that, as stands approach maturity those which are understocked or overstocked tend to approach an average or normal stocking. The rate at which t h i s occurs depends upon the o r i g i n a l density of the i n d i v i d u a l stand. This increase or decrease i n the r e l a t i o n of stocking to the average condition i s attributed to the e f f e c t s of density upon mortality and diameter growth. The dispersion of i n d i v i d u a l p l o t s about the average basal area curve may be measured i n terms of the standard deviations of basal area f o r each diameter class about the curved basal area f o r each p l o t . These, expressed as percentages since the mean basal area i s d i f f e r e n t f o r each c l a s s , were plotted on the mean diameter f o r each average diameter class (Figure 4 (b)). Thus pictured they indicate the trend of the data to become l e s s dispersed as the average stand diameter increased. There i s a greater range of density i n stands of 2 inch diameter {29%) than i n stands of 12 inches [12%), confirming the assumption that 15 0 Z A 6 6 / 0 / 2 /Average $tand D'uxmatar Cmc has ) Figure 4« (a) Relationship between basal area per acre and average stand diameter for the average site and density and (b) between the coefficient of variation of basal area and average stand diameter. the stands tend to approach the average condition with increasing age. These coefficients serve as a basis for the construction of additional curves of basal area on 16 average diameter f o r densities above and below the average. Four more such curves were constructed by choosing a r b i -t r a r i l y to displace the upper and lower curves by 50% of the average curve basal area at the 5 inch diameter c l a s s . A te s t Of the v a r i a t i o n within t h i s class indicates that 99% (3 standard deviations) of the stands with average diameters i n the class w i l l not have basal areas exceeding 230 square feet or l e s s than 76 square feet per acre, and thus w i l l f a l l within the 50% displacement. With the c o e f f i c i e n t of v a r i a t i o n at the 5 inch class as standard, the c o e f f i c i e n t of v a r i a t i o n of each other diameter class was transformed to a f a c t o r proportionately l a r g e r or smaller than the standard. This f a c t o r , applied to the 50% standard displace-ment, gave a percentage displacement appropriate to each class and dependent upon the r e l a t i v e s i z e of i t s c o e f f i c i e n t of v a r i a t i o n . These percentages, converted to basal areas, were added to or subtracted from the average curved basal area value f o r each inch c l a s s . The r e s u l t i n g values were plotted on average diameter to give the mean curve of the two extreme density classes (D.C. I and V). Two more curves, midway between the extremes and the average, calculated i n the same manner, give a t o t a l of f i v e density curves. This i s an adaptation of the method of constructing harmonized curves described by Osborne and Schumacher (16) and Duerr and Gevorkiantz (3). In accordance with the assumption that present conditions represent past and future conditions, these density curves state, f o r example, that a stand i n D.C. I 17 7£e n ™—~~~"—— - Y L — I — — t K 200 160 & H V ^/ B v. cL/20 d w < Of 8 0 0 CQ » i — • — • — • — * m / / PI T \ / » * J 1 / 1 rf / i / i 0 40 50 /20 /60 £ W fitcLnd Ago* (y<2(\rs) Figure 5* Basal area growth curves for average site. today w i l l remain in that class during i t s undisturbed existence and w i l l develop in basal area and average diameter according to the trend of the curve for that class. These density curves were next combined with age by eliminating the common factor of average diameter IS* (Figure 3)» giving a series of basal area growth curves (Figure 5) for five density classes on the average site of the sample. These are tabulated in Appendix III under Site II, By means of these tables i t i s possible to predict future basal area for stands on an average site i f the present basal area and age are known, and to deter-mine the growth in basal area for different periods of time. These relationships refer to stands growing upon the average regional s i t e . Under the premise that better sites w i l l produce stands of higher volume, basal area and average diameter for a given age and density than w i l l poor sites, i t may safely be considered that these average site growth curves must be lowered for poorer than average sites and raised for above average sites. Site Percent Concept It may often be demonstrated that on identical sites stands of pine having different densities of stocking w i l l develop consistently different growth indications. A dense stand w i l l , at a given age, have a shorter mean dominant height and a smaller average stand diameter than a less dense one. It i s considered, therefore, that the usual definition of site as related to dominant height at a standard age w i l l not be adequate for this species since stand density may effectively change such heights. As shown by Parker (17)» average stand diameter i s f a i r l y highly correlated with 1 dominant height and may be used alternatively as a site 1. Correlation Coefficient - 0.6*33 19 indicator. For efficiency in f i e l d work the average stand diameter i s more useful since i t i s normally calculated for nearly every cruise. It i s frequently d i f f i c u l t to d i f f e r -entiate clearly between crown classes in pure stands of lodgepole pine with the result that inaccuracies i n classification and height measurements occur, causing an error in the site index calculated from them. Average stand diameter, age and density of stocking are therefore considered to be the most effective and useful factors in determining a measure of site for pure lodgepole pine stands. Each plot of the basic data was assigned a site value according to the site percent method (17). This concept i s based on the premise that the average site of a region w i l l produce, in a given number of years and with a given number of trees per acre, stands of lodgepole pine with a certain average stand diameter; a poorer than average site w i l l , at the same age and with the same density, produce a lesser average stand diameter; a better than average site w i l l produce a greater average diameter. The relation between the diameter of the stand on the average site and that of the other stand, expressed as a percentage, gives an index value to the better or poorer s i t e . Thus:-x . 100 = Site Percent y where x i s the average diameter of a given stand, X i s the average diameter of a similar stand on the average s i t e . 20 Numbar- of Trazs p<trr Acre. Figure 6. Relationship between average stand diameter and number of trees per acre by age classes for regional average s i t e . This concept takes into consideration the variation which may occur in height and average diameter of stands of a given age on identical sites, but of different degrees of stocking. Such variations would normally indicate a variation in site quality where there i s none. They are compensated for in this method by the inverse effect of stand density. Thus the site percent value w i l l remain constant for a given site although the average stand diameter and density fluctuate. The basic data were grouped and treated in the following steps (17) * 1. By Number-of-tree classes; average stand diameter '21 was plotted on the number of trees per acre by number-of-tree classes giving one ind i c a t o r curve f o r a l l ages. 2. By Number-of-tree classes and Age classes; average stand diameter was plotted on the number of trees per acre by age classes, on semi-logarithmic paper. (Unharmonized)• 3. Same grouping as step 2; age was plotted on number of trees per acre by age classes to give a correction (4) f o r v a r i a t i o n i n age within an age c l a s s . (Unhar-monized). 4. By Age classes and Number-of-tree classes; average stand diameter was plotted on age by number-of-tree classes. (Harmonized). 5. By. Number-of-tree classes and Age classes as deter-mined from step 4; average stand diameter was plotted on number of trees per acre by age classes on semi-logarithmic paper. (Harmonized, Figure 6). For i n t e g r a l numbers of trees and ages, average stand diameters were read from Figure 6 and arranged i n a table of diffe r e n c e s . Extraordinary differences were smoothed out a r i t h m e t i c a l l y . Tabular values f o r the o r i g i n a l data were compared with actual values and the aggregate difference calculated. A general tendency f o r the tabular values to be lower than actual values was noted, causing an excessive aggregate difference. A multi-plying _•-. factor o f + 1.0273 was applied to r a i s e a l l tabular values, with the re s u l t that the aggregate difference was 22 reduced to 0.8 and the average deviation to 6.26$. A table of corrected values i s shown i n Appendix I I . This table gives the average stand diameter which w i l l be attained by a pure stand of lodgepole pine on an average s i t e , at a given age and density of stocking. Average s i t e i s the mean s i t e represented by the basic data. To determine the s i t e value of any area bearing a stand of pine: a) Determine the average stand diameter (x). b) Determine the stand age. c) Determine the number of trees per acre of,the stand. d) Interpolate a r i t h m e t i c a l l y i n the table of Appendix II f o r the same age and number of trees to determine the average stand diameter f o r an average s i t e (v.). e) Substitute x and y_ i n the equation -x # ioo = Si t e Percent y The r e s u l t i n g value represents the r a t i o of the s i t e i n question to the average s i t e . Thus s i t e percents of over 100 represent better than average s i t e s ; those l e s s than 100 are below average s i t e s . Each plot of the basic data was assigned a s i t e percent value. They ranged from s i t e percent 80 to 136 with a mean of 101.3, standard error of 1.03 and standard deviation of 9*2. The data were then grouped into f i v e - u n i t s i t e percent classes and each class assigned frequencies according to the occurrence of plots i n that c l a s s . Probit values f o r cumulative frequency percentages plotted over s i t e percent show a nearly straight l i n e r e l a t i o n s h i p from s i t e percent 2 3 s o > : y 4 p" > Si t e Percent Figure 7 « Probit values i l l u s t r a t i n g dis-tribution of site percents in the basic data, SO to 1 2 0 , but tend to drop off for higher sites (Figure 7 h Most of the distortion i s due to the presence of two very-high quality sites in a low age class. The balance of the distribution i s more or less normal. If the two high quality plots are thrown out the mean i s reduced to 1 0 0 and much of the distortion removed. It would, however, be impractical to exclude plots on this basis with such a small sample of the best sites. On the premise that the data may be considered to be normally distributed with respect to site and that i t represents the range of stand and site conditions to be found in the region, i t i s apparent that approximately 6$fo of a l l sites w i l l f a l l within 9 . 2 site percent units of the mean. The balance w i l l be more or less equally distributed on either side of the mean beyond these limi t s . It i s often 2 4 uo o X no M 9 6 -O a -0- -- - O- -10 90 3<3 HO ro 90 HQ I30 /JO '70 '9o Figure 8. Illustrating the lack of linear cor-relation between site percent and age. convenient to arrange site classifications by groups rather than by absolute numbers. A division of the data into three groups may be effected as follows, by setting as limits the positive and negative values of one standard deviation from the mean si t e : Group I - better than average sites. Group II - average sites. Group III - poorer than average sites. This w i l l place approximately two-thirds of the sites in the average (II) group. Such a division assumes that the majority of the data may be handled on the basis of an average si t e . Only those stands on exceptionally good or poor sites need be considered as being sufficiently influenced by site to warrant changes in treatment. The basic data were next arranged by age classes and a correlation analysis of the linear regression of site percent and stand age carried out. 25 Analysis of variance of site percent on age. Source D.F. Sum Squares M.S. V.R. Variation of site due to age 1 228.26 228.26 2.89 Residual variation 77 6071.89 78.85 Total variation 78 6300.15 The variance ratio being not significant at the 5$ level indicates that with a probability of better than 95$ there i s no correlation between site percent and age. This i s illustrated in Figure 8. The Importance of Site and Stand Density The average curve of basal area and age (Figure 5, D.C. I l l ) expresses the growth of stands of lodgepole pine in the region sampled for average conditions of site and density. Variations in the basal area of individual stands about this curve may be caused by differences of s i t e , density and age, or by errors of chance and sampling. The residual between the basal area of each plot of the basic data and that read from this average curve represents the amount by which that plot has been affected, in part at least, by conditions of site and density which are different from the average. The pattern of this variation as affected by changes in average diameter has already been used to determine the relationship of ar b i t r a r i l y chosen density classes to the average curve. There may also be included in this variation the effect of changes of site and age. The effect of age may be demonstrated by plotting 26* the basal area residuals, expressed as standard deviations for each age class, upon the average age of the class (Figure 9). This correlation may be confirmed by expressing the standard deviations as percentages (coefficients of var-iation) and again plotting them on age. A trend similar to that of Figure 9 i s obtained. To eliminate this effect of age the residuals from the basal area curve may be expressed as standard units -• the standard deviation at the plot age as shown in Figure 9 divided into the actual plot basal area residual (1). The relative importance of the effects of both site and density upon the variation in basal area may now be assessed. Assuming that a linear equation w i l l express the regression of standard units of basal area upon site and density an analysis of covariance was carried out. a) Analysis of covariance of standard units of basal area and site percent Source D.F. Sum Squares M.S. V.R. Variation of standard units attributable to site 1 21.090 21.09 20.85 Residual variation 77 78.033 1.01 Total variation 78 99.123 b) Analysis of covariance of standard units of density classes basal area and Source D.F. Sum Squares M.S. V.R. Variation of standard units attributable to density 1 46.750 46.75 68.75 Residual variation 77 52.373 0.68 Total variation 78 99.123 27 J>0 30 SO TO 90 HO / 3 0 /^ TP /fc? 190 £6and Ac^ f<2 Cia<\<r$) Figure 9. Relationship between standard deviation of basal area and stand age. In each case the si g n i f i c a n c e of the variance r a t i o indicates a strong c o r r e l a t i o n between the independent variables of s i t e percent or density classes and standard units of basal area. There i s , however, a strong p o s s i b i l i t y that i n each case of the single regression a part of the c o r r e l a t i o n effect i s due to the i n t e r a c t i o n of the other independent variable; that i s , that a part of the strong c o r r e l a t i o n between s i t e percent and standard units of basal area i s due to the i n t e r a c t i o n of the unfixed density classes. The p o s s i b i l i t y of such i n t e r a c t i o n was tested by means of a j o i n t regression of basal area on both s i t e and density. The l i n e a r equation y » 0.70878 x x + 0.02669 x 2 - 4.773 was f i t t e d to the data by the least squares method, where x^ represents observations i n density class u n i t s , x represents 28" observations in site percent and y equals standard units of basal area. The analysis of variance of this joint regression i s as follows: Source D.F. Sum Squares M.S. V.R. Variation due to regression 2 52-310 26.155 42.46 Residual Variation 76 46.813 .616 Total Variation 7* 99.123 Variation due to density (x^) l 46.750 46.750 75«89 Extra variation due to site (Xg ) l 5.560 5.560 9.03 Variation due to regression 2 52.310 Residual variation 76 .616 Total Variation 1* 99.123 The join* regression i s significant at the 5% level. The partial regression due to the effects of density alone i s shown to be very much more significant than that due to the addition of site as determined by the site percent method. Density alone accounts for a total sum squares of 47.75 out of the total due to regression of 52*31 while site accounts for only 5.56. This relation i s confirmed by the calculation of partial correlation coefficients for the three variables: r y x r x 2 = 0.632 - Ri T7H'T1 = °'326 = R 2 r X l x 2 - 0.349 - R These indicate that about 40$ of the variation in basal area i s accounted for by density differences, while only 10$ i s 29 attributable to s i t e . The significance of these coefficients may be measured by their variance ratios according to the equation: V.R. = N - m - 2 (R2) 1 - R2 where N equals the number of observations, m equals the number of variables eliminated, R equals the partial cor-relation coefficient, V tR i i s the variance ratio and (N - m -2) are the number of degrees of freedom. Partial Correlation V.R. Coefficient R l 50.667 •*2 9.022 With 1 and 76 degrees of freedom these variance ratios are highly significant and in the same order as those of the partial regressions. The results of these analyses indicate that the variation about the average growth curve of basal area i s significantly correlated with site (expressed as site percent) and stand density (measured in density classes). The in d i -vidual effect of density i s , however, relatively more important than that of s i t e . It may be considered, therefore, that for some purposes the addition of site percent as a variable to be considered in predicting the growth of stands of lodgepole pine w i l l constitute a refinement which i s l i k e l y to add comparatively l i t t l e accuracy to such predictions. •HO Figure 10. Relationship between residuals from the basal area growth curve and site percent. The Application of Site to the Basic Data If site i s to be considered, the basal area growth curve must be adjusted accordingly. Since the variation about this curve, expressed as standard units, is correlated with site percent i t i s possible to use this range of variation to s p l i t the basic curve into others representing poorer and better than average sites. The data were arranged by site percent classes. The relationship of standard units of basal area was plotted on site percent and a curve f i t t e d (Figure 10). Since i t has been shown that site percent w i l l have arela-tively small effect upon growth i t is not l i k e l y that small classes of site percent w i l l be of much importance. It is 31 proposed, therefore, to group the data into three classes of s i t e as previously discussed, se t t i n g the l i m i t s of the middle class as plus and minus one standard deviation from the mean s i t e percent of the basic data. Table I I I Site Corrections - Basal Area S i t e Group Si t e Quality Limits i n S i t e Percent Average Si t e Percent (adjusted) Standard Unit Correction I Excellent 110.1 - above 119.4 + 2.00 II Average 92.1 - 110.0 100.8 0 III Poor 92.0 - below 87.3 - 1.58 The average s i t e percent value was calculated f o r each s i t e group and the curved value of standard units found from Figure 10 f o r t h i s average. To construct the average basal area growth curve f o r s i t e I i t i s necessary to f i r s t read basal area values f o r even ten-year age i n t e r v a l s from Figure 5» D.C. I l l ; from Figure 9 read curved values of standard deviation f o r these same ages. To determine the amount by which the s i t e I curve must be raised above the average curve at each age, multiply the standard deviation at that age by the standard units correction f a c t o r f o r s i t e I (Table I I I ) , and add t h i s amount to the basal area value read from Figure 5 at the same age. With basal area values thus calculated f o r each age i t i s possible to con-struct the s i t e I growth curve. The s i t e I I I curve may be developed i n a s i m i l a r manner, deducting the appropriate 32 amount at each age from the average curve value of basal area to derive a growth curve below the average. Since s i t e I I represents the average condition no adjustment of the average curve i s necessary (standard unit correction of 0). Figure 5 therefore represents the growth curves f o r f i v e density classes i n s i t e I I . Assuming that the pattern of density classes i s s i m i l a r f o r each s i t e , the average curves f o r s i t e I and s i t e I I I may be treated i n the same manner as before to develop a sheaf of density curves f o r these s i t e s . These have been constructed and are tabulated i n Appendix I I I . These breakdowns are only j u s t i f i e d where two conditions are s a t i s f i e d : (1) v a r i a t i o n i n growth i s a t t r i b u t a b l e to the measures of both s i t e and density, and (2) s i t e and density are not correlated l i n e a r l y with age or each other. The f i r s t has already been discussed; the second condition i s i l l u s t r a t e d by Figures 8 and 11, and the following analysis of variance. Analysis of variance of density classes on age Source D.F. Sum Squares M.S. V.R. Var i a t i o n of density classes due to age 1 .0015 .0015 .0016 Residual v a r i a t i o n 77 70.8885 .920 Total v a r i a t i o n 78 70.8900 The variance r a t i o i s not s i g n i f i c a n t at the 5$ l e v e l i n d i c a t i n g that with a p r o b a b i l i t y of 95$ or better there i s no c o r r e l a t i o n between density classes and stand age. 3 3 S 3 c ! . i 2 ! l SO 90 /OO HO /ZO 130 140 Sit a. Par can't Figure 11. Illustrating the lack of linear correlation between density classes and site percent. An examination of the basic data and the table in Appendix III indicates that where site i s considered, most of the plots f a l l into the three middle density classes (II, III, IV) with the majority in density class III. When site i s ignored the plots tend to spread into a l l five density classes. This corroborates the previous discussion suggesting that the effects of site and density tend to overlap and consideration of the latter alone may, in many cases, be as effective as the use of both. Where a refined estimate i s needed, however, and where exception-al l y poor or good sites are encountered the site tables w i l l be useful. This apparent lack of variation with site may be due to a lack of sensitivity in the site percent method, a comparatively small range of site conditions in the region or an inadequate sample. While there may appear, in the ?4 f i e l d , to be a wide diversity of site conditions i t i s possible that in effect such a range does not exist - that lodgepole pine i s more or less restricted in the conditions of site on which i t w i l l develop. The small range of site percents in the sample tends to bear out this observation. Volume It i s possible, by means of the tables thus far constructed to predict future basal area and the growth i n basal area for various periods of time of stands whose present age, basal area per acre, number of trees per acre and average stand diameter are known. The next step i s to express this growth in terms of volume. This may be done by the use of the ratio between volume and basal area, working again through the common variables of stand diameter and age. In stands of small average diameter the ratio of cubic volume (or any other expression of volume) to basal area i s small. As the stands grow older they increase in average diameter and height. Basal area increases as a function of diameter increment alone, while volume increases depend upon both diameter and height growth. Thus the volume w i l l increase at a more rapid rate than basal area and the ratio between them w i l l increase proportionately with both age and average diameter. As the stand approaches maturity height growth tends to f a l l off while diameter increases continue, causing the rate of increase of the ratio of volume 35 to basal area to drop off again. The average diameter of a stand i s dependent upon the number of trees i t contains and the predominance of a certain class or group of classes of tree diameters. Thus a stand may have a large basal area and a low average diameter because of a high stocking of small trees; similarly a stand of the same basal area may have fewer, larger trees and a correspondingly high average diameter. The ratios of volume to basal area of these two stands at the same age at the beginning of a period of time w i l l be different from each other - they w i l l s t i l l be different at the end of the period. That i s to say, future ratio i s correlated with, but different from, present ratio and a single expression of the relationship of ratio with age w i l l not be adequate to include a l l the variations which may be caused by differences in stand diameter and basal area. It i s necessary to con-struct a series of such relationships which w i l l express the progress of any ratio with age. The ratio (R) between total cubic volume and basal area per acre for each plot was computed and a l l the ratios grouped by average diameter classes. The mean ratio (R) of each class was plotted on the mean average diameter, forming a strong curve (Figure 12 (a)). By reasoning similar to that used in constructing the basal area density curves, i t may be considered that the dispersion of the individual plot ratios (R) about this curve w i l l indicate the range of ratios; the manner in which this dispersion i s affected 36 d 0 0 >' 1 i o > 3 ^ , " o > 1 '* e-1 °/ a • 9 N \ \ N 4 > 2 4 6 3 10 12 /4 Figure 12. Relationship between (a) total cubic volume-basal area ratio (£) and average stand diameter and (b) coefficient of variation of this ratio and average stand diameter. by changes of average diameter w i l l indicate the trend of the ratios with increasing diameter and age. The coefficients of variation of the ratios (R) plotted by diameter classes (Figure 12 (b)) were used to construct a sheaf of ratio curves. This construction was based on a displacement of 1+0% at the 4 inch class between the average curve and the highest and lowest curves. This new sheaf of curves was next combined, as in the case of the density curves, with Figure 3 to eliminate average diameter and produce a series 37 d d I ft git*. R -DT nr. • . 1 — 1 IL I f I. — ^ 1 20 40 80 /CXP 160 Stand Ago (yaars) ZOO Figure 13. Ratio CR) growth curves for an average s i t e . of five ratio growth curves. These are represented in Figure 13 and are tabulated in Appendix IV under Site I I . They show, for the average site, the manner in which the ratio (R) of total cubic volume to basal area reacts to changes in age, by different ratio classes. Better than average sites may be expected to pro-duce higher volumes and proportionately higher ratios than average sites - the opposite holds for poor site s . The next 8 C o < §« g 2 3<3 3* 6 a' \ \ s / * - — — 6 O \ • ro /to iSo /50 £trand Aga (years) /TO /SO F i g u r e 14. I l l u s t r a t i n g t h e l a c k o f l i n e a r c o r r e l a t i o n b e t w e e n s t a n d a r d d e v i a t i o n o f t h e r a t i o ( £ ) a n d a g e . s t e p , t h e r e f o r e , was t o a d j u s t t h e a v e r a g e r a t i o c u r v e f o r t h e s e o t h e r s i t e s . F r o m t h e a v e r a g e c u r v e o f F i g u r e 13 (R c l a s s I I I ) v a l u e s o f R w e r e r e a d f o r e a c h p l o t a n d t h e r e s i d u a l s b e t w e e n t h e s e a n d t h e a c t u a l R v a l u e s w e r e c a l -c u l a t e d . T h e s e r e s i d u a l s , g r o u p e d b y a g e c l a s s e s , w e r e e x p r e s s e d a s s t a n d a r d d e v i a t i o n s a n d p l o t t e d o n a g e ( F i g u r e 14). No c o r r e l a t i o n b e t w e e n a g e a n d t h e s t a n d a r d d e v i a t i o n i s a p p a r e n t . T h i s i s c o n f i r m e d b y t h e f o l l o w i n g a n a l y s i s o f v a r i a n c e . A n a l y s i s o f v a r i a n c e o f s t a n d a r d d e v i a t i o n o n a g e , S o u r c e D . F . Sum S q u a r e s M . S . V . R . V a r i a t i o n o f s t a n d a r d d e v -i a t i o n a t t r i b u t a b l e t o a g e 1 0.16 0.16 0.68 R e s i d u a l v a r i a t i o n 4 9.40 2.35 T o t a l v a r i a t i o n 5 9.56 39 i 8-a o -y o 160 + 4> -20 2 / o / ! / ' / ; / / / / / / / / / / //! to -£ 7 i i / 39 X ' « ! ! /t / / f / / i i 80 90 100 UO IZO /3o 144 £ita Par cant Figure 15. Relationship between the absolute residuals of the ratio (Rj and site percent. Since the variance ratio is not significant at the 5$ level, i t is not l i k e l y that any relationship exists between age and the standard deviation of the ratio R. The absolute residuals of R may therefore be plotted directly on site percent by site percent classes as shown in Figure 15 > and the values of the absolute residuals determined from the curve for the average site percent of each class (Table IV). These values, added to or subtracted from the R value at each ten-year interval on the average curve (R class III) of Figure 13, give a point on the ratio growth curve for ^0 Table IV Site Corrections - Ratio (R) S i t e Av. Site Absolute Percent Residual of R I 119.4 + 3.95 II 100.8 0 I I I 6*7.3 - 2.80 each respective s i t e group. From these main curves f o r each s i t e i t i s possible, by means of the c o e f f i c i e n t of v a r i a t i o n method already described, to construct a s e r i e s of r a t i o growth curves f o r each s i t e . These, as constructed, are tabulated i n Appendix IV. From these tables i t i s possible to predict future r a t i o s and periodic changes i n r a t i o f o r any stand whose present t o t a l cubic foot volume per acre, basal area per acre and s i t e are known. Sim i l a r tables f o r other expressions of volume may be constructed i n a l i k e manner. Tests s i m i l a r to that i l l u s t r a t e d i n Figure 11 show there i s a very low or n i l c o r r e l a t i o n between r a t i o classes and the age of stands and between r a t i o classes and s i t e percentage. Number of Trees I t has been demonstrated that both basal area per acre and average stand diameter increase s t e a d i l y with increasing age. It may also be shown that the number of trees per acre w i l l decrease with increasing average diameter and age (Figure 16). The r a t i o of basal area to the number O 1 4 6 a \o ia i * ^ v a r a ^ a /Stand Dia.ma.t'ar (inch *s) Figure.16. The relationship between number of trees per acre and average stand diameter. of trees w i l l thus also decrease as the stands grow older. As in the case of the ratio (R) of basal area to volume i t i s possible, because of the variations in basal area and numbers of trees, to have two stands of the same age with different ratios of basal area to number of trees (K). Thus, again, i t i s not practical to have one expression of the development of K with age to represent a l l possible variations of stand diameter, basal area and stocking. The same method used to construct the tables of basal area to volume ratio (R) may be employed to derive growth tables for 2^ K. For each plot of the basic data the actual K was c a l -culated and an average curve was constructed, through the common variable of average stand diameter, of K on age. The c o e f f i c i e n t of v a r i a t i o n method was used to break t h i s curve into three r a t i o classes f o r the average s i t e (Figure 17). High s i t e s may be expected to produce higher basal area and thus lower r a t i o s (K) than lewer s i t e s with stands of the same number of trees and the same age» In order to adjust the curves of Figure 17 the same procedure was followed as before. Since the residuals between the actual pl o t values of K and the curved values, expressed as standard deviations, were found to be correlated with age they were transformed into standard units and the following corrections f o r s i t e derived as previously described. Table V Site Corrections - Ratio (K) Site Av. Site Standard Units Percent Correction 1 119.4 - 0.53 II 100.8 0 III 87.3 + 0.50 By applying these corrections to the middle curve of Figure 17 a series of average K curves f o r the three s i t e groups were constructed. These, tabulated, appear i n Appendix V. They express the development of the r a t i o between basal area per acre and number of trees per acre with age, by s i t e and K groups. With them i t i s possible to predict the future r a t i o *3 C i o V I 0 it I I \ 5-\ 'C/OLSS, ~ * 1 • \ \\ A 1 . j \ / ' ZO AO bO do /OO /ZO 160 Stand A ga. (Ya araj /So ZOO Figure 17. Ratio (K) growth curves for an average s i t e . K of stands for which the present age, basal area and number of trees is known. Because of the shape of the curves in the younger age classes and the large variation in the number of trees i t is possible to encounter in young stands, these tables w i l l be subject to considerable error in these lower age groups. A large sample of young stands w i l l be required 44 to provide the information necessary to accurately con-struct this part of the growth curves. Application of the Growth Tables. By means of the three growth tables shown in Appendices III, IV and V and the table shown in Appendix II i t is possible to predict the future developments of any stand in terms of basal area, total cubic volume and number of trees per acre. To demonstrate the use of these tables let us assume a stand with the following specifications: a) Age - 53 years(1 foot stump height). b) Basal area per acre - 140 square feet. c) Total volume per acre - 2800 cubic feet. d) Average stand diameter - 4.0 inches. e) Number of trees per acre - 1610 effective stems. From Figure 2 i t is seen that, on the average, i t takes seven years for seedlings to attain a height of one foot. The total age of the stand w i l l thus be 60 years. The table in Appendix II indicates that on the average site a stand containing 1610 trees per acre would attain, at 60 years, an average stand diameter of 3.87 inches. The current stand diameter of 4.0 shows that the stand i s growing on a site with an index value of 103 percent. This f a l l s within the limits of site group II (Table III). At 60 years, site II, the basal area of 140 square feet places the stand in density class III. From the table ^5 in Appendix III, by interpolation, i t may be calculated that this basal area w i l l increase in the next 20 years by (156.2 - 134.8) or 21.4 square feet to a total of 161.4 square feet per acre at age 80 years. The present ratio of volume to basal area (R) f 2800] is \ 140/ or .20.0. At 60 years, site II, the stand w i l l be ratio class (R) IV. By interpolation in the table of Appendix IV i t w i l l be found that this ratio w i l l increase in 20 years by (29.6 - 21.4) or 8.2 units to 28.2 at age 80. The total volume of this stand 20 years hence w i l l therefore be (161.4x 28.2) or 4551.48 cubic feet. This indicates a net growth of 1750 cubic feet per acre in 20 years or an annual net increment of approximately 87 cubic feet per acre. At 60 years the ratio of number of trees to basal f 1610 \ area (K) i s \ 140 I or 11.5. On site II this places the stand in ratio class III. According to the table in Appendix V this ratio w i l l decrease in the next 20 years by (11.1 - 5-4) or 5.7 units to 5.8 at age 80. Basal area i s predicted to be 161.4 square feet per acre at the latter age. Thus the stand w i l l contain (5.8 x 161.4) or 936 trees at 80 years. Mortality during this period has averaged 34 trees per year. During the next 20 years the ratio (K) w i l l again decrease to 3.62 and the basal area w i l l increase to 173.0 square feet per acre. The stand at 100 years may therefore be expected to contain about 626 trees. Mortality has thus decreased over the latter twenty-year period to an average of 15 trees per year. As a second example: a) Present age - 105 years (total age), b) Basal area per acre - 128 square feet, c) Total volume per acre - 3142 cubic feet, d) Ratio (R) - 24.6. e) Average stand diameter - 6.7 inches. f) Number of trees per acre - 524. g) Ratio (K) - 4.1. Appendix I I shows that a stand of these specifications on an average s i t e w i l l have an average diameter of 7.9 inches. The above stand i s thus growing on a site with an index of 85$ which places i t i n site group I I I , The current basal area, according to the table in Appendix I I I places the stand i n density class I I I , In the next twenty-year period this basal area w i l l increase 8,9 square feet to a t o t a l of 136,9 square feet per acre. The present ratio (R) places the plot in ratio class I I (Appendix IV). Within the next 20 years this ratio may be expected to increase 3.1 units to a t o t a l ratio of 27.7. The future volume w i l l therefore be (27,7 x 136.9) or 3792.1 cubic feet. This represents a twenty-year increase of 65O cubic feet or an annual growth of 32,5 cubic feet per acre. The present ratio (K) of 4.1 places the stand i n ratio class I I I , In the next 20 years this may be expected to decrease to 2,97 and the stand at 125 years w i l l contain (2,97 x 136,9) or 406 trees. Mortality has averaged 6 trees *7 per acre annually during the twenty-year period. By similar routines i t i s possible to trace the development in basal area, total cubic volume and number of trees per acre for any stand within the range of the tables. Additional ratios to cover other expressions of volume for the whole or pa r t i a l stand may be developed in a manner similar to that already discussed and predictions made in these units. BIBLIOGRAPHY (1) Bruce, D. and Schumacher, F.X., Forest mensuration, Second Edition, New York, McGraw H i l l Book Co., 1952, pp. 220-228. (2) Chapman, H.H. and Meyer, W.H., Forest mensuration, First Edition, Toronto, McGraw H i l l Book Co., 1949. (3) Duerr, W.A. and Gevorkiantz, S.R., "Growth predictions and sit e determination in uneven-aged timber stands," Journal of Agricultural Research, Vol. 56, No. 2, pp. 81-98, 1938. (4) Dwight, T.W., "Refinements in plotting and harmonizing freehand curves," The Forestry Chronicle, Vol. 13, No. 2, pp. 357-370, 1937. (5) Fisher, R.A., S t a t i s t i c a l methods for research workers. 11th Edition, New York, Hafner Publishing Co., 1950. (6) Fisher, R.A. and Yates, F., S t a t i s t i c a l tables for biological, agricultural, and medical research. New York, Hafner Publishing Co., 1950. (7) Gaiser, R.S. and Merz, R.W., "Stand density as a factor in estimating white oak site index," Journal of  Forestry, Vol. 49, No. 8, p. 572, 1951. (8) Gevorkiantz, S.R., Growth and yield of .jack pine in the, lake states» Washington, Superintendent of Documents, 1947, United States, Department of Agriculture, Lake States Forest Experiment Station, Station Paper No. 7« (9) Gevorkiantz, S.R. and Duerr, W.A., Methods of predicting §rowth of forest stands, Washington, Superintendent of ocuments, 1938, United States, Department of Agri-culture, Lake States Forest Experiment Station, Economic Notes No. 9. (10) Gevorkiantz, S.R. and Duerr, W.A., Volume and yield of northern white cedar in the lake states, Washington, Superintendent of Documents, 1939, United States, Department of Agriculture, Lake States Forest Experiment Station, Progress Report. (11) Halliday, W.E.D., A forest classification for Canada, Ottawa, 1937, Canada,Department of Mines and Resources, Dominion Forest Service, Bulletin 89 • (12) McArdle, R.E., Meyer, W.H. and Bruce, D., Yield of Douglas f i r in the Pacific northwest, Washington, superintendent or Documents, xy 1^, united States, Department of Agriculture, Technical Bulletin 201, revised edition. (13) MacKinney, A.L., Schumacher, F.X., Chaiken, L.E., "Construction of Yield tables for non-normal loblolly pine," Journal of Agricultural Research, vol. 5»i.f pp. 531-b1^, 1937-(14-) Meyer, W.H., Yield of even-aged stands of ponderosa pine, Washington, superintendent or Documents, ±930*, United States, Department of Agriculture, Technical Bulletin 630. (15) Mulloy, C.A., Empirical stand density yield tableo, Ottawa, lyMi-, uanaaa, Department or Mines ana Resources, Dominion Forest Service, S i l v i c u l t u r a l Research Note 73* (16) Osborne, J.G., and Schumacher, F.X., "The construction of normal yield tables for even-aged timber stands," Journal of Agricultural Research, Vol. 511 PP' 5^7-bb% illustrated, 1935. (17) Parker, H.A., Dominant height and average diameter as a measure of site in untreated lodgepole pine stands, Ottawa, 19^2, Canada, Department of Mines and Resources, Dominion Forest Service, S i l v i c u l t u r a l Research Note 72. (18") Quenouille, M.A., Introductory statistics, London, Butterworth Springer Ltd., 1950. ' (19) Schnur, G-L-, Yield, stand and volume tables for even-aged upland oajc, Washington, superintendent or Documents, 1937, United States, Department of Agriculture, Technical Bulletin 5^0. (20) Wahlenburg, W.G., Methods of forecasting growth in irregular stands, wasnmgton, superintendent of Documents, 19^1, United States, Department of Agriculture, Technical Bulletin 796. (21) Barnes, G.H., The importance of average stand diameter as a factor in forecasting timoer yields, victoria, printed 07 (;.*•. tsanrield, 1931, British Columbia Department of Lands, Forest Service. Appendix I Standard Volume Table (Lodgepole Pine) (Volume in Total Cubic feet) . Dbh Total Height in Feet 10 20 30 40 50 60 70 80 90 100 110^  Volume (cu. ft.) 1 0.03 0.05 0.08 0.10 2 0.12 0.22 0.32 0.41 3 0.27 0.50 0.72 0.94 1.15 4 0.48 0.90 1.29 1.67 2.05 2.41 5 1.41 2.03 2.63 3.21 3.79 6 2.03 2.93 3.80 4.65 5.48 6.29 7 2.78 4.00 5.19 6.34 7.48 8.59 8 5.24 6.80 8.31 9.79 11.3 12.7 9 6.65 8.62 10.5 12.4 14.3 16.1 10 10.7 13.0 15.4 17.7 19.9 22.2 11 12.9 15.8 18.6 21.4 24.2 26.9 12 15.4 18.9 22.2 25.5 28.8 32.0 35.2 13 22.2 26.1 30.0 33.8 37.6 41.4 14 25.7 30.3 34.9 39.3 43.7 48.1 15 29.6 34.9 40.1 45.2 50.3 55.3 16 39.7 45.7 51.5 57.3 63.0 17 44.9 51.6 58.2 64.7 71.2 18 50.4 57.9 65.3 72.6 79.9 87.0 19 64.6 72.9 81.0 89.1 97.1 20 71.7 ao.s 89.9 98.8 107.1 (Prepared by A.W. Blyth, Dominion Forestry Branch, Calgary) Appendix II Average Stand Diameters attained by Stands of Lodgepole Pine on the Regional Average Site. Stand Number of Trees per Acre Age. 150 200 300 350 400 450 500 600 800 1000 1400 1800 2200 2600 3000 3400 3800 4200 4600 5000 5400 5800 6200 Average Stand Diameter (inch) 30 3.06 2.82 2.69 2.42 2.25 2.11 2.01 1.94 1.87 1.81 1.76 1.72 1.67 1.64 1.62 1.60 50 5.24 4.72 4.33 3.77 3.37 3.08 2.86 2.67 2.53 2.39 2.28 2.19 2.12 2.04 1.98 1.92 70 9.38 8.82 8.34 7.94 7.59 7.03 6.56 5.86 5.32 4.54 4.02 3.62 3.30 3.05 2.86 2.68 2.55 2.45 2.34 2.27 2.20 2.14 90 10.46 9.77 9.23 8.74 8.35 7.68 7.13 6.30 5.69 4.82 4.21 3.78 3.46 3.21 3.00 2.83 2.69 2.58 2.50 2.40 2.33 2.26 110 12.28 11.24 10.46 9.82 9.29 8.84 8.09 7.49 6.60 5.91 4.99 4.37 3.92 3.58 3.31 3.09 2.93 2.79 2.66 2.56 2.48 2.40 2.33 130 12.97 11.82 10.94 10.23 9.65 9.16 8.36 7.72 6.76 6.05 5.09 4.44 3.68 150 13.62 12.35 11.39 10.61 9.99 9.43 8.57 7.89 6.87 6.14 5.15 4.47 170 190 210, Data Incor nplete 230 H.89 13.42 12.30 11.40 L0.67 9.95 9.06 8.27 7.16 6.37 Data from 79 plots selected in pure stands of Lodgepole Pine: Aggregate Deviation - 0.8, Av. Deviation 6.26$ ro JO h-1 M M I—1 ro o 00 ON •P" JO O 00 ON •P-VJJ o o o O O o O o o o o r~> 1—1 1—1 M M M M M Vl Vl vi •P-•P-VjJ JO O OO •p-1—' vO vi »-• ON O JO M Vl o o • • • • • • • • - • • • O ON VI vi JO JO VJJ 00 JO o VI M 1—' I—1 H1 H M 1—1 M h-1 Co 03. 00 ON Vi VjJ vO VI •P-VJJ M vO ON JO Vl Vl ON VJJ • • • • • • • • • • • OJ. vi vi ON o 00 o JO o JO JO JO JO JO JO JO JO M (-< M f—' I-" r-> 1—' o o vO Vl vO o M JO JO JO 1—1 vO -p-VJJ JO • • • • • • . • • • • • VjJ o -0 -p-vi ON JO o o ro JO JO JO JO JO JO JO JO JO VJJ •p-•p-•P-Vl Vl VI •p-H VjJ vO JO 00 •P-VJJ •P-JO • • • • • • • • • • • •p-•p-VjJ •p-VI o ON O O O ro JO JO JO JO JO JO VJJ VJJ JO H* ON ON -o ->J 00 vO vO O O ON ->3 JO -0 JO 00 •p-O 00 JO JO ON OO • • • • • • • • • • • o O Vi JO •p--P-. Vl o VJJ M M l-» 1—1 M 1—' VjJ VjJ VJJ JO M o vO OO Vl JO 03 •p-O •p-00 vO 1—' Vl JO • • •' • • • • • • • o 03-JO ON o JO •P-M 00 1—1 H* H* M M M I-1 o \JI vi Vl •P-•P-VjJ M vO VI ON JO 00 r-> JO 00 Vl •-P-ON • • • • • • • • • • • -P" VJJ o Vl ON Vl JO O o M H M I—1 M H I—1 M 1—1 oo CO. 03-OO ->3 ON VI VJJ 00 •P-VJJ JO o 00 •P-ON •P-Vl • ' • • • • • • • • • • ' • vO vO VjJ Vl Q JO OO JO 00 o o JO JO JO JO JO JO JO H M o o O O o o o vO M JO ON 00 00 OJ-ON VJJ VJJ VJJ 00 •p-• • • • • • • • • • • o •p-JO o v» CO •P-o VI JO JO JO JO JO JO JO . JO JO JO VJJ w u> u> VJJ VJJ VjO M VJJ 03. •p-ON 00 vO 00 VjJ ON VI • • • • • • • • « .< • • o •P-JO •p-VjJ •P-vO O Vl o M M M M VJJ JO 1—1 H« O O vO ON VJJ JO 00 VjJ O O O OO 00 • • • • • • • • • • 1 ON JO JO o ON ->3 VI o H H* h-1 M |-» H» •p-VJJ W JO VJJ O 00 ON M vO ON Vl -*3 ON vO VjJ vO 1 • • • • • • • • • • 03--P-o ON Vl •P-ON VJJ VJJ M M M H H M M ON ON vi vi vi -P-VJJ 1—1 VO VJJ JO vO Vl M •P-•P-00 O 1 • • • • - • « • • • • • ro O O vi O h-1 Vl o ON M M t-> M H I—1 h-> M M 00 00 03. ^3 <J ON •P-M •P-JO JO M vO ON VJJ Vl 1 • • • • • • • • • • o o M vr JO JO o VJJ O O JO JO JO JO JO M 1—1 H M O o O o o vO vO •P-ON M VjJ •P" H 00 O Vl JO o 1 • • • • • • • • • • O o o -0 VO v» VJJ •>3 o CO a> a 5 CD 1 P> (W CD W 0> CO So > CD fl> »T3 CD > CD CO Density Class I Density Class II Density Class III Density Class IV Density Class V Density Class I Density Class II Density Class III Density Class IV Density Class V Density Class I Density Class II Density Class III Density Class IV Density Class V CO H- ct CD M CO H« ct CD CO H* Ct CD M J—I M > •-9 CD O <• Ct Jto CO 1—1 H- ct tn CD p CO B p. > o CD CD 3 {» CO H- Ct CD o M co CD CO CD cr CO •<< w CO p. CD to O O !T i-3 cr H CD VjJ VJJ VjJ VjJ VJJ M to M H" O VJI •p-vO vO -P" • • • • • • • • • • -P" CO. o VJJ 00. 00 ON ro ro ro o o o VJI O VJJ -p- oa oa o Ov O •P- o ro o o o oa o ov o -P- VJJ o o -p-•p-VJJ VjJ VjJ VjJ VjJ ro M o o vO 00 Ov •p-o •P-Ov oa VjJ • • • • • • • • • • • ON VjJ Ov -0 Ov VJI ro -P--p- •p-•P- VjJ •P- ro •P- M oa VjJ ro vO ro M i—1 t—1 Ov oa ^3 ro ro ro M VJI •p- vO -P" oa •P-ro VJJ •P- -P-VJ7 VjJ VJJ VjJ VJJ vO -P-• • vo ro ro • VJI t—1 -P- • • Ov VJV VJI •p-•P-•P-VjJ ro vO •P-VO O oa VO • • • • • • • • • • • ro Ov VJI oa o VJJ VJJ \J-1 VjJ VJJ VJJ VJJ ro ro ro M ro ro O vO Ov VJJ vO VJJ O • • • • • • • • • • oa VJI oa oa ro oa o O oa ro VJJ ON VJJ OV VJJ •P- oa VJJ VJJ VJJ O • • VJJ Oa ro ro ro h-» •P-VJJ VJt ov ro o •P-. •p-•P-VJJ VJJ VJJ VJJ ro 1—1 o o oa •P-1—1 VJl -0 • • • • • • • • • • • to oa o oa ro vO to VJl -P-VJJ VJJ •P-•p-•P-' •p-•P-VjJ VJJ ro ro •p-•P-ro oa VJl vo h-1 vO ro • • • • • • • • • • VJl oa o oa ro oa •P-Ov -P-o •p-•P-•p-•p-•P-•p-VJJ VjJ ro »-• vO vO oa Ov VjJ VO VJJ •P-ro ro • • • • • • • • • • • o o oa 1—1 O Ov oa Ov ro ro VjJ VjJ ro ro to ro ro M O vo oa Ov •P-ro VJl oa • • • • • • • • • • vji Ov vO o ro oa VjJ •p- ro VJJ VJJ VJJ ro VjJ o .ro oa VJl to -P- • oa oa vO VJJ VJJ • I ro VjJ VJJ VJJ VJJ VjJ VJJ to to M oa oa Ov •P-ro oa to •P--p-• • • • • • • • • • • . -Ov VjJ r-» •P-H-1 Ov Ov VJl -p-;  -p-to \-> • '• VJJ ~0 VJJ O VO • • vO VjJ oa VJJ VJl VJJ IO • VJJ ro Ov VJl ON oa • i ->0 •P- -P-Ov VJI • • ro ov •P- •P--P- VjJ VJJ VO VJJ ON VjJ o to o vO O • I VJJ 03 > <+ era p> CD 3 33 & O o h-1 to co CO p. O ts Ct Ratio Class I Ratio Class II Ratio Class III Ratio Class IV Ratio Class V Ratio Class I Ratio Class II Ratio Class III Ratio Class IV Ratio Class V Ratio Class I Ratio Class II Ratio Class III Ratio Class IV Ratio Class V co Ct CD co Ct CD M w to PJ CO co ct H» 0> H« Ct O CD io Ar Of CD a PJ To w t3 ct CO oi CD ct p) H» ct o C+ CD o Ac Cu M o -t cr J—1 las CD cr ic CO < CD o CO f Age, lume CD 3 a Appendix V Ratio of Number of Trees per Acre to Basal Area per Acre by Age, Site and Ratio Classes, Site I Site II Site III Stand Age 01 00 cd O cd tt M 03 03 cd O O •H 13 tt M 03 03 cd rH o O •H 13-tt CQ 03 cd rH O tt M M 03 03 cd H O 13 M M 03 03 cd H O 13 M 03 03 cd rH O 13 « M 03 03 cd iH O o •H 13 M M Hi 03 03 Cd rH O o •H 13 Ratio (K) Class Midpoint 30 40 60 80 100 120 140 160 180 200 220 81.0 48.6 14.6 4.1 2.1 1.36 1.03 0.86 0.78 0.74 0.73 42.0 3.45 30.0 12.20 11.3 3.56 1.90 1.27 0.98 0.84 0.76 0.72 0.71 7.3 3.19 1.72 1.18 0.93 0.80 0.74 0.70 0.69 107.0 64.8 20.5 7.2 3.35 2.62 1.98 1.60 1.36 1.18 1.07 53.9 39.5 15.7 6.4 3.53 :2.47 1.91 1.54 1.32 1.16 1.05 4.28 16.7 11.1 5.4 3.22 2.34 1.85 1.50 1.28 1.13 1.02 130.0 77.5 25.5 10.2 5.5 3.75 2*82 2.28 1.93 1.69 1.51 65.2 44.5 20.5 8.7 5.1 3.55 2,71 2.20 1.88 1,66 1.50 6.1 19.8 14.3 7.4 4.57 3.31 2.59 2>13 1.83 I.64 1.49 

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