UNIVERSITY OF CALIFORNIA COLLEGE OF AGRICULTURE AGRICULTURAL EXPERIMENT STATION BERKELEY, CALIFORNIA Yield, Stand and Volume Tables for Red Fir in California FRANCIS X. SCHUMACHER BULLETIN 456 August, 1928 UNIVERSITY OF CALIFORNIA PRINTING OFFICE BERKELEY, CALIFORNIA 1928 Digitized by the Internet Archive in 2012 with funding from University of California, Davis Libraries http://www.archive.org/details/yieldstandvolume456schu YIELD, STAND AND VOLUME TABLES FOR RED FIR IN CALIFORNIA FRANCIS X. SCHUMACHER* INTRODUCTION Investigations concerning the rate of growth and yield of California forests have, to date, been confined essentially to the species of outstanding commercial importance. Although several timber types which are now considered as of secondary value are being logged to a limited extent, demand for information basic to the management and utilization of these is sure to increase as logging progresses in the more popular and available types. Red fir (Abies magnified), 2 including the variety Shasta fir (Abies magnified shastensis) forms one of the so-called minor types. It occurs at elevations of 6000-9000 feet from the Cascade Mountains of southern Oregon southward along the western slope of the Sierra Nevada Mountains, and in the Coast Range from Lake to Siskiyou counties. 3 Lying above the main timber belt, it is relatively inacces- sible, hence utilization has been slight, although individual trees grow as big and stands as heavy as yellow pine and white fir on equivalent sites at lower elevations. Little information is available concerning the red-fir type. In California, it is found mostly within the national forests, where, according to the estimate of the United States Forest Service, the volume amounts to 12,935 million board feet 4 or 14 per cent of all timber within the national forests of the state. The following pages present the results of a study of the growth of well-stocked stands of red fir. lAssistant Professor of Forestry. 2Sudworth, G. B. Check list of the forest trees of the United States. U. S. Dept. Agr. Misc. Cir. 92:1-295, 1927. Scientific names used are taken from this publication. 3 Red fir is occasionally confused with Douglas fir (Pseudotsuga taxifolia) because the same common name has sometimes been applied to both. As Douglas fir occupies bottomlands and slopes of low to intermediate elevations, these species are never associated together. Botanically and commercially they are very different and should not be confused. *Ayres, R. W., and W. Hutchinson. The national forests of California. U. S. Dept. Agr. Misc. Cir. 94:1-34. 1927. UNIVERSITY OF CALIFORNIA EXPERIMENT STATION GROWTH OF RED-FIR STANDS The growth of a timber species is best shown by tables which state yields of even-aged stands over a period of years on lands of various degrees of productivity. But density of stand — or approximate number of trees to the acre — as well as age and timber-productive quality of site, must greatly affect timber yields. As there is no satisfactory way of expressing density in absolute terms, two types of tables are generally recognized (1) empirical yield tables based on density of stocking as actually found over a large area, and (2) normal yield tables based on the ideal density which produces maximum volume. It is at once evident that actual yields of fully-stocked stands, must be less than the normal yields stated therefor, as the latter are given in gross values including cull factors, such as decay in living trees, unused stumps and tops, sweep and crook in logs, and breakage in logging. Only under the best conditions of stand establishment, freedom from natural enemies and care, are normal stands possible over any considerable area. The value of normal-yield table lies mostly in this, that they furnish a basis for intelligent comparison between growth rates of species, as comparison may be made on equivalent stocking as well as on age and site. They also serve to measure the degree of stocking of any even-aged stand of the species in question. BASIC DATA The normal-yield tables for red fir which follow are based on measurements of 149 sample plots scattered through the geographical range of the species. Plot Selection. — Although little, if any, commercial logging has been conducted in the pure red-fir type, many small, even-aged stands of second growth, especially in the south-western part of Plumas County, date from the time of clearing or burning when mining began about 75 years ago. Older even-aged stands were located through a systematic search in the virgin timber. Within these stands, plots were established so as to enclose a comparatively complete crown canopy by excluding the larger open- ings which follow failure of reproduction or accident, and at the same time to include within boundaries the area equivalent to that BUL. 456] YIELD, STAND AND VOLUME TABLES FOR RED FIR 5 which seemed to be necessary for the growth of the enclosed timber. As practically all plots were well within large stands, this was in the main mechanically accomplished, attention having been given to securing a clear sight from one corner to the next rather than to balancing plot area with area used by the timber. Rectangular- shaped plots were not sought though acute angles were, in general, avoided. Plots were surveyed with staff compass and chain. Age Determination. — The age of each plot was determined by counting the annual rings on cores extracted (with Swedish incre- ment borers) from near the base of several trees. The number of rings on the core plus the number of years necessary for height growth to reach the point of boring (determined by an analysis of height growth of saplings) was taken as the age of the tree. Although there was seldom a difference of more than two or three years between the ages of the oldest and youngest tree examined, the age of the oldest tree was taken as the age of the plot, as it dates more nearly from the time of the removal of the earlier stand. Field Measurements. — Diameter breast high of every tree was measured with diameter tape and tallied by species and crown class (dominant, codominant, intermediate, or suppressed). The heights of fifteen to twenty-five trees were measured with the Forest Service hypsometer and plotted over diameter on cross- section paper in the field, the number of height measurements necessary being judged at the time by the range in diameters present and the dispersion of plotted heights around the free-hand curve. A short description of physiographic features completed the field work on each plot. Office Computations. — The computational work necessary for each plot is evident from following paragraphs. The construction of the yield tables is after Bruce, 5 while the stand tables are based on CharlierV method of calculating theoretical frequencies. NORMAL YIELD TABLES Table 1 gives the following data for the entire stand : Site index: the height that the average dominant red fir will attain, or has attained, at 50 years of age. Height curves used in determining site index of a plot are shown in figure 1. 5 Bruce, D. A method of preparing timber-yield table. Jour. Agr. Research 32:543-557, figs. 1-8. 1926. «Charlier, C. V. L. Die Grundziige der mathematischen Statistik. pp. 3-125. Lutke und Wulff, Hamburg, 1920. b UNIVERSITY OF CALIFORNIA EXPERIMENT STATION In well-established forestry practice it has been found that the timber-productive capacity of a forest area has a closer relationship to height of dominant stand for a given age than to any other readily measurable factor of timber growth. In this country, consequently, average height of the dominant trees is now generally accepted as the index to site quality. Age : the age of the oldest tree sampled. Because the establishment of a new natural stand is dependent on seed already on the area when the old stand was removed by logging or accident — such as fire or epidemic of insect or disease — or upon seed from neighboring trees, the new stand is seldom all of one age. Trees per acre: the number of trees that have reached a height of at least 4.5 feet (breast-height) above the average ground level. Basal area per acre: the sum of the cross-sectional areas at breast height in square feet. Mean diameter breast high: the mean of all tree diameters on an average acre. It is to be distinguished from average diameter, which is understood to be the diameter in inches of the tree of average basal area. 7 Height of average tree: the height from ground to tip of the tree of average basal area. Volume in cubic feet: the cubic volume of the entire stem of all trees from ground to tip but without limbs or bark. The volume table used (table 5) is given on p. 14. Average annual growth in cubic feet: the cubic volume of an acre of timber divided by the age. Maximum volume production is obtained by allowing the stand to grow to the age of greatest average annual growth, which is 140 to 150 years in red fir. Table 2 gives a number of the corresponding values for the trees in the stand which are 8 inches and over in diameter breast high, together with the following : Volume in board feet: this includes the board-foot contents of the trees by the International log rule of Vs-inch kerf between a stump height of one foot and top diameter, inside bark, of 5 inches, scaled in 16-foot logs with 0.3-foot trimming allowance to each. Gross volumes are presented, no account being taken of cull factors. The volume table used is given on p. 16. ?Mean diameter is used rather than average diameter, as the two terms are herein defined, because of association of the former with stem distribution as explained on pp. 21 ff. BUL. 456] YIELD, STAND AND VOLUME TABLES FOR RED FIR Height of average dominant in. feet 5«f?g>o>oto4s5) 30000000 CD & o o 3 I 2 W ° : >* B OQ. a> 0» « o g 5' Oj to p fi> OQ P « 3 § ► \ 1 \ 1 1W m^ \ \ \ \ ^v \ \ \ \ x *. G-VvS ■ \ l \ \ \ \ \ \ TT_Y_\ 1 \ <_ AiX vXX ^ \ \ E-M G V- ti l~WX T ti t4 . i a tj Site index in feet UNIVERSITY OF CALIFORNIA EXPERIMENT STATION TABLE 1 Normal Yield Table for Red Fir, Including All Trees Age Height of average tree Mean diameter breast high Total number of trees per acre Total basal area per acre Volume per acre Average annual growth Years Feet Inches Square feet Cubic feet Cubic feet Site index 60 feet at 50 years 30 20 2.5 1,560 150 1,600 53 40 29 3.7 1,210 235 3,200 80 50 38 5 3 957 295 5,000 100 60 48 7.0 760 345 6,950 116 70 59 9.0 605 386 9.000 129 80 72 11 2 478 420 11,400 142 90 87 13.8 367 451 13,850 154 100 103 16.3 279 478 16,700 167 110 120 18.8 214 502 20,100 183 120 135 21 2 170 523 23,900 199 130 148 23.5 141 543 27,200 209 140 160 25.5 119 559 30,000 214 150 169 27.4 102 573 31,900 213 160 177 29.3 89 585 33,150 207 Site index 50 feet at 50 years 30 17 2.1 2,030 135 1,250 42 40 24 3.2 1,580 211 2,600 65 50 32 4.4 1,250 266 4,050 81 60 40 5.9 990 310 5,600 93 70 49 7.6 790 347 7,200 103 80 60 9.5 620 378 9,050 113 90 72 11.6 480 406 11,100 123 100 86 13.7 362 431 13,400 134 110 99 15.8 279 453 16,100 146 120 112 17.8 220 473 19,150 160 130 123 19.8 181 490 21,700 167 140 132 21.5 155 504 23,950 171 150 140 23.0 133 517 25,500 170 160 146 24 3 116 528 26.500 166 Site index 40 feet at 50 years 30 13 1.7 3,240 125 1,000 34 40 18 2.5 2,510 195 2,050 51 50 24 3.6 1,990 248 3,200 64 60 31 4.7 1,580 289 4,350 73 70 38 6.1 1,250 323 5,700 81 80 47 7.7 990 352 7,200 90 90 56 9.4 762 378 8.800 98 100 67 11.1 580 400 10,550 106 110 77 12.8 442 422 12,700 115 120 87 14 4 350 440 15,100 126 130 96 16.0 289 455 17,150 132 140 103 17.4 246 469 18,950 135 150 109 18 6 211 481 20,100 134 160 114 19.6 185 490 21,000 131 BlJL. 456] YIELD, STAND AND VOLUME TABLES FOR RED FIR TABLE 1 — Continued Age Height of average tree Mean diameter breast high Total number of trees per acre Total basal area per acre Volume per acre Average annual growth Years Feet Inches Square feet Cubic feet Cubic feet Site index 30 feet at 50 years 30 10 1.3 5,900 119 800 27 40 13 2.0 4,560 185 1,600 40 50 18 2.8 3,600 234 2,450 49 60 22 3.7 2,860 274 3,400 57 70 27 4.7 2,290 306 4,400 63 80 33 5.9 1,810 334 5,550 69 90 40 7.3 1,400 358 6,800 76 100 47 8.6 1,050 379 8,150 82 110 54 9.9 800 399 9,800 89 120 61 11.2 640 416 11,650 97 130 67 12.4 525 432 13,200 102 140 73 13.5 445 445 14,550 104 150 77 14.4 386 456 15,600 104 160 81 15.2 337 465 16,100 101 Site index 20 feet at 50 years 30 6 0.9 10,400 114 600 20 40 8 1.4 8,150 178 1,200 30 50 10 2.0 6,400 225 1,800 36 60 12 2.6 5,100 262 2,450 41 70 15 3.3 4,050 294 3,200 46 80 18 4.1 3,220 320 4,050 51 90 22 5.0 2,490 345 4,950 55 100 26 6.0 1,880 366 5,950 60 110 30 6.9 1,430 385 7,200 65 120 34 7.8 1,130 400 8,450 70 130 38 8.6 935 415 9,650 74 140 41 9.3 790 427 10,600 76 150 43 10.0 690 438 11,350 76 160 45 10.5 600 447 11,800 74 Log run: the number of logs to the thousand feet board measure contained therein. (HECK OF BASIC DATA AGAINST YIELD TABLES Table 3 shows the check of the values of the 149 sample plots against the yield tables interpolated to nearest year of age and nearest foot of site index. That the standard deviation of basal area is less than that of any other variable is to be expected as it is the basis for judging normality of stocking. 8 Greater variation in number of trees and mean diameter s See page 18 ff. 10 UNIVERSITY OF CALIFORNIA EXPERIMENT STATION TABLE 2 Normal Yield Table for Red Fir, Including Trees 8 Inches and Over Age Years Number of trees per acre Volume per acre Board feet Average annual growth Board feet Logs per M. B. M. Site index 60 feet at 60 years 30 16 800 27 30 40 87 6,550 165 29 50 220 19,200 345 28 60 296 33,700 545 25 70 328 50,400 720 22 80 306 71,100 889 19 90 263 94,300 1,048 15 100 215 119,000 1,199 11 110 176 145,000 1,318 9 120 147 172,000 1,425 7 130 127 196,000 1,508 6 140 111 216,000 1,543 5 150 97 230,000 1,533 4 160 87 240,000 1,500 4 Site index 50 feel at 50 years 40 68 3,640 91 34 50 163 12,000 240 33 60 287 23,800 390 31 70 352 36,800 526 28 80 353 52,000 650 24 90 313 70,500 785 18 100 259 91,200 928 14 110 212 112,500 1,050 11 120 176 138,000 1,145 9 130 152 156,000 1,240 7 140 135 173,000 1,236 6 150 118 184,000 1,227 5 160 106 191,000 1,194 5 Site index Ifi feet at 50 years 40 25 1,030 26 42 50 135 6,050 121 42 60 253 14,400 240 41 70 388 25,000 357 40 80 447 37,100 473 35 90 430 50,200 588 26 100 368 65,700 695 21 110 303 84,000 793 16 120 256 104,500 871 13 130 226 122,000 938 10 140 196 136,000 971 9 150 172 145,000 967 8 160 154 151,000 944 7 BUL. 456] YIELD, STAND AND VOLUME TABLES FOR RED FIR 11 TABLE 2— Continued Age Years Number of trees per acre Volume per acre Board feet Average annual growth Board feet Logs per M. B. M. Site index 30 feet at 60 years 50 76 2,100 42 53 60 211 6,950 120 52 70 366 14,600 210 51 80 525 23,600 295 49 90 588 33,800 380 40 100 543 44,500 460 36 110 470 57,700 538 26 120 410 72,800 607 20 130 357 86,000 665 16 140 316 98,600 705 14 150 282 108,000 720 11 160 253 113,000 706 10 Site index 20 feet at 50 years 60 77 1,590 27 57 70 200 4,950 71 56 80 332 10,300 129 53 90 487 17,800 198 49 100 564 25,600 256 41 110 549 34,600 315 32 120 520 44,000 360 26 130 484 52,800 400 21 140 442 60,400 430 17 150 409 67,000 445 15 160 368 71,500 447 13 TABLE 3 Check Between Yield Tables and Basic Data All trees per acre Basal area Mean D. b. h. Volume in cubic feet Trees 8 inches and over Volume in board measure +0.74 33.7 ±1.86 +0.89 17.9 ±0.99 -0 67 21 5 ±1 14 +0.53 20.0 ±1.01 +0.33 30.2 ±1.67 — 1 41 24.1 Probable error of yield table value — ±1 34 * The aggregate difference is the sum of plot values expressed as a percentage difference from the sum of corresponding tabular values. t Standard deviation of plot distribution (tc«NlOOiMN COCOCO-^^T^lOlO O N. cc o o> t~ •o o N CO o to CO oa o oo OONOliHlOMOO OO-H^OOi-lTtlOOCM CM CO CO CO ■*■«*<■«# »C oo CM oo \ti "0 coo CO CM "5 "5 CO N ICO N •* »0 CO locoo CO 00-* 00 N N ■<*< CO oo OOCOOOOOOOCM lO00»HTtlt-~O5COCO (N IN M CO CO CO ^ t n co cs CO ncm CM CO CO co CO H-<*l00"0lONlO-*f< COlONO'^'fiOO'- 1 CMcMCMCOCOCOCO-^ iQ-OO c5 OS »C N- OO 00 OS r^^co-H CONN00 OkOOiO COCO t- OOCOU) I COO iOO"OlO osco coo ■** "0 »C CO CO CD MO CO O "5 O O O m "O co Ice i_ n|oo o -h oo n OO CM 00 "i CO lO NkO lOliO lO o o ■^ t>- O ■"* N l-i iflO) COOO colco ■>*< CO OS -1 i-H CM CM CM CO COCO ■«*< ■"* uolco N 00 OS o t- >o cm CO •>* CO oo INCONWON* l-«J< coo JNO-c* 00 CO COO 00 -* ^IN CO -^ ^ ^ i-l t-l CM CM CM CM CM CO CO CO CO « NCO N O Ml CO UO CO O ■* O Ifl N CO OO ■*! 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A stand that has been fully stocked over a period of several decades distributes its growth over many trees with a consequent dropping off in diameter growth of the average; whereas a stand of fewer trees that has but recently closed its crown canopy has also but recently approximated full stocking by basal area, and its growth is distributed over fewer trees to the greater advantage of each. It follows that a stand fully stocked by basal area may be made up of many slender trees or comparatively few stout ones, hence the greater variation in number of trees and mean diameter. STAND TABLES FOR RED FIR Although yield tables are basic to the solution of many forest management problems in a given timber type in that they express either total or average values attainable in properly established and protected stands, they are not complete without stand tables, which state the number of trees to be expected within each diameter class, as problems of valuation and utilization require knowledge of such stem distribution. Stand tables for red fir are given in table 4 9 . VOLUME TABLES FOR RED FIR Preliminary to the study of yields in cubic and board feet, volume tables in these units were prepared. The basic tree data of the tables presented are from measurements taken by the Division of Forestry in Plumas and Sierra counties from several previously measured, even-aged sample plots. The ages of the trees measured were from 30 to 110 years on the stump. Table 5 is the volume table in cubic feet, and states the entire volume of the stem, including stump and top, but without bark. It was prepared by the form-factor method. Table 6 is the volume table in board measure. It includes the board-foot contents of the trees between a one-foot stump and top diameter inside bark of five inches. It was prepared from an analysis of the board-foot — cubic-foot ratio of the trees. 9The analysis of stem distribution and construction of stand tables is explained in pp. 21 ff. 16 UNIVERSITY OF CALIFORNIA EXPERIMENT STATION ffl.8 S NiooofainoiMN Hinon CM CM CM MCOW*' i 1*1 m m (o— I if t^- 0> N 1(5 05 N «5 05 (N COO^OO t-h CM CM* CM C- O CM >C NOCOIOOIINIOX N!D»CC • O CO CO Ol (M 1(5 ,_( _i,h,_! rtiMN CM CM cO~CO CO~CO if - »0 "5 CO CO OOOOO O OOO OOO O OO _ " g 32 S »" Q 9° ?? ¥5 OS OM* COC CM K5NOCN10 00. 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OOOOOOOO OOO _ CO -H - ncON 00 050 *H NCO^CONOOOrt CO 1*1 CO t^ IHU5O0NN I CM CM CM CM CM CM mom >omo m oooooooo ooo< CM Oi CO MHO OO 00 00N00 0)05ON CO *0 CT> < ■**U) CO t^ 0O 00 OlOHMM^CON OOOlOl f^ Oi «— I CO OO CO < I CO if 1*1 iC CO CO • 00 OS O O —l CM CO !»■* «) CO N OVO CM 1*1 CO C O CM 1*1 CO l CM CM CM CM CM CO CO CO CO CO §9 ifcOOOOCM TftOoOO 1*1 if if io io ifliomco BUL. 456] YIELD, STAND AND VOLUME TABLES FOR RED FIR 17 Table 7 shows the check of the basic tree data with the volume tables. TABLE 7 Check Between Volume Tables and Basic Data Cubic foot volume table Board foot volume table +0.17 8.9 ±0.35 - 0.43 16.2 Probable error of volume table value — ± 0.69 APPENDIX BASIC DATA The sample plots on which the yield and stand tables are based were measured by the Division of Forestry in 1926. Out of 156 plots originally measured, 7 were discarded for reasons to be discussed later. The 149 actually used are from the watersheds listed in table 8. TABLE 8 Distribution op Plots by Principal Watersheds Watershed Number of plots 2 Upper Sacramento River Battle Creek.. .. 20 1 51 31 9 6 1 1 Upper San Joaquin River 27 Total 149 The composition of the plots by basal areas of the various species included is shown in table 9. 18 UNIVERSITY OF CALIFORNIA EXPERIMENT STATION TABLE 9 Composition by Basal Area of the Plots Used Species Basal area in percentage of total Red fir 97.41 White fir 2.24 0.23 0.08 0.03 Incense cedar and sugar pine... 01 Total 100.00 The distribution of the plots by site and age classes is given in table 10. TABLE 10 Distribution of Plots by Site and Age Classes Site index- -height in feet of the average dominant tree at 50 years 16-25 26-35 36-45 46-55 56-65 Total 26- 35 1 1 4 5 2 7 3 6 4 2 I 4 36- 45 2 46- 55 11 7 5 56- 65 4 1 5 6 5 2 20 66- 75 1 3 14 76- 85 12 86- 95 3 3 10 2 6 6 5 9 2 3 3 18 96-105 23 106-115 8 116-125 6 126-135 3 1 2 10 136-145 7 146-155 i 1 11 156 165 3 166-175 2 1 6 Total 5 25 39 67 13 149 REJECTION OF ABNORMAL PLOTS The analysis of the data for abnormalities was based on (1) basal area to the acre, (2) number of trees to the acre, and (3) mean diameter breast high. Basal Area.— Preliminary curves of basal area over age were fitted to the data of the original 156 plots and the percentage deviation of each plot from the curves, interpolated to nearest year of age and nearest foot of site index, arranged as in table 11. BUL. 456] YIELD, STAND AND VOLUME TABLES FOR RED FIR 19 TABLE 11 Deviation of Actual Basal Area of Plots From Theoretical Basal Area of Preliminary Curves Taken to Nearest Foot of Site Index and Nearest Year of Age Percentage deviation Number of plots -35 to -44 2 -25 to -34 11 -15 to -24 26 - 5 to -14 40 5 to - 4 26 6 to 15 23 16 to 25 15 26 to 35 7 36 to 45 6 Total 156 The standard deviation of distribution is 18.2 per cent. Four plots which exceeded two standard deviations (36.4 per cent) from the mean for age and site index were scrutinized in order to discover, if possible, a reason for their high deviation. One boundary of two of them was partly in the open, and because of the possibility that this boundary of each was drawn in too close to the timber, giving exaggerated figures on the area basis, these two were discarded. Number of Trees. — Percentage deviations in number of trees of the remaining 154 plots from the curved number of trees for age and site index were next computed and are shown in table 12. TABLE 12 Deviation of Actual Number of Trees on Plots From Theoretical Number of Trees of Preliminary Curves Taken to Nearest Year of Age and Nearest Foot of Site Index Percentage deviation Number of plots -46 to - -65 13 -26 to - -45 28 - 6 to - -25 32 14to- - 5 29 15 to 34 23 35 to 54 13 55 to 74 7 75 to 94 4 95 to 114 3 115 to 134 1 135 to 154 1 Total. 154 20 UNIVERSITY OF CALIFORNIA EXPERIMENT STATION The standard deviation of distribution is 40.0 per cent, and the deviation of seven plots exceeded twice the standard deviation. On examination of these it was found that five contained a high per- centage of trees in the low-diameter classes, indicating a possibility of the presence of two age classes. These five plots were accordingly discarded. Since no abnormal plots were discovered in the analysis of devia- tions of mean diameter breast high, the remaining 149 plots were used in the construction of the yield and stand tables. THE EFFECT OF PLOT AREA At the time each plot was located, effort was made to enclose, as nearly as could be judged, the equivalent of the area occupied by the growing timber within plot boundaries. The personal factor may have played a small part in the location of boundaries with respect to boundary trees, but where practically all plots were well within larger stands, as is the case with the red-fir plots, the resulting error is probably negligible. Table 13 shows the distribution of plots by area classes. TABLE 13 Distribution of Plots by Area Classes Area in acres Number of plots Less than 0.100 acre 25 0.100-0.199 53 0.200-0.299 41 0.300-0.399 24 0.400-0.499 8 0.500-0.599 1 600-0.699 3 0.700-0.799 1 Total 149 Average acrea in acres 0.222 The correlation coefficient (r) between plot area and plot basal area as a per cent deviation of yield table basal area for age and site was found to be r = — 0.22 ±0.053 indicating that with increase of plot area there is a tendency toward decrease in basal area to the acre, the 59 plots having an area greater than 0.25 acre, average 3 per cent less basal area to the acre than the yield-table figures. The reason for the correlation, though of BUL. 456] YIELD, STAND AND VOLUME TABLES FOR RED FIR 21 little significance in this case, is important in sample-plot work. Should all plots have been of the same size? It is not probable that the boundaries of small plots were drawn in too close to the trees, because special care, combined with previous experience in work of this nature, was taken with them. Suitability of plots as samples was judged by number of trees thereon, after the stand was deemed of about normal stocking and found to be even-aged, and not by area. It was assumed that 100 to 200 trees would be representative of diameter distribution for site, age, and density, and the enclosing of about these numbers was sought regardless of size of timber; hence plots small in area are, as a rule, samples of small timber. Where rather abrupt changes in density occurred in such stands the policy of locating two or more plots therein, each consistent in its own density, was adopted. Thus average density of small plots — hence plots of small timber — is made up of a wide range in the density of individual plots. In large timber, small plots are scarcely acceptable, for, on account of the size of individual trees and distance between them, there is less assurance that plot area is equivalent to the area occupied by the enclosed timber. As plots are enlarged, however, especially when boundaries are entirely within the stand, their areas approach true stand area more certainly. An ocular estimate was made in the field of the stocking of each plot by basal area and this was later compared with actual stocking. The correlation coefficient between estimated and actual stocking is r = 0.68 ± 0.038 which is satisfactory, since the basis of ocular comparison (100 per cent stocking) had not yet been worked out. The correlation shows that dense plots were usually known to be dense at the time of meas- urement and yet were considered, in the field, acceptable as basic data for the yield study. Furthermore most of the denser plots were considered among the surest measured, in that boundaries might have been changed at will without affecting the average spacing of the enclosed timber. CONSTRUCTION OF THE STAND TABLES If the distribution of number of trees by diameter classes follows the normal curve of error, the construction of stand tables becomes relatively simple, as it is based on but two constants, (1) mean diameter, and (2) standard deviation of distribution. If, on the 22 UNIVERSITY OF CALIFORNIA EXPERIMENT STATION other hand, frequency distribution is not normal, the computation of at least two other constants becomes necessary, namely (3) a measure of skewness or the departure from symmetry of distribution, and (4) a measure of kurtosis or excess, that is, the form of the frequency curve in the general region about its central tendencies, which may be pointed or flat-topped, rather than normal. It is obvious that the numerical values of these constants must progress smoothly with increase of age for a given site index, and with site index for a given age. At once the question comes up as to whether the last three constants of distribution may not show as high correlation with the first, the mean diameter, as with age and site, since these variables, in stands of normal density, may be considered as causal factors of mean diameter itself. If that is true the compu- tational work may be considerably lessened, as fewer curves would be required. The constants may be considered separately: Mean Diameter. — This must necessarily be correlated with age and site as given in table 1. Standard Deviation. — Standard deviation of stem distribution was correlated (1) with age and site, and (2) with mean diameter. Percentage deviation of the standard deviation of each plot from the two curved standard deviations was next computed and analysed with the following results: ( 1 ) Standard deviation of plot standard deviations, measured from the curves of standard deviation for age and site index=20.9 per cent. (2) Standard deviation of plot standard deviations, measured from the curve of standard deviation for mean diameter = 21.1 per cent. It becomes apparent that the correlation of dispersion with mean diameter is as high as it is with age and site. Skewness. — In the trial correlation of skewness with age and site as against mean diameter, skewness of each plot was computed by the formula. 3(M-Md) skewness = — a in which M is mean diameter, Md is median diameter and o- is standard deviation. This formula combines sufficient accuracy with ease of calculation, and as the calculated quantity is in terms of standard deviation, skewness of large timber is comparable with that of small. Correlated with (1) age and site, and (2) mean diameter, following is the comparison: BUL. 456] YIELD, STAND AND VOLUME TABLES FOR RED FIR 23 (1) Standard deviation of skewness of the stem distribution of each plot measured from the curve of skewness for age and site = 0.446. (2) Standard deviation of skewness measured from the curve of skewness for mean diameter — 0.398. In this case there is apparently closer relationship between skew- ness and mean diameter than there is between skewness and the age-site variables. Kurtosis. — Since kurtosis or excess of each plot is based on moments higher than that from which standard deviation is derived with consequent greater probable error of its calculated value, no trial measure of kurtosis was correlated with site and age, as against mean diameter. Stand Tables Based on Mean Diameter. — As the correlation of standard deviation is as high with mean diameter as with age and site, distribution of stems may be considered as independent of the latter variables, except in so far as age and site are causal factors of mean diameter. Given two stands, then, of common mean diameter, one of good site quality but young, the other of poor site quality and perhaps 60 years older, the form of their stem distribution curves should be identical. The plots were therefore sorted into 2-inch mean-diameter classes. But in order to prevent greater than actual range in diameters on account of such rather broad grouping, frequency by diameter class of each plot was taken, not by frequency in each one-inch diameter class, but by frequency, in percentage of total, in each unit of one- half standard deviation measured from mean diameter. This was easily accomplished by constructing a cumulative frequency curve for each plot and marking it at the upper limit of each one-half standard-deviation unit (fig. 2). By subtraction between upper limits of adjacent units, cumulative frequency was broken down into ordinary frequency. After trial by Pearson's 10 and Charlier's 11 methods of fitting frequency curves to distributions, the "Type A" of the latter was adopted because less volume and exactness of computational work is involved. With Pearson's "Type I" system into which all the dis- tributions fall, the interdependence of half a dozen constants calls for the use of calculating machines for computations to several decimals — a volume of work not warranted by the accuracy needed. loElclerton, W. P. Frequency curves and correlation, pp. 1-167. Layton, London. 1906. nCharlier, C. V. L. Die Grundziige der raathematischen Statistik. pp. 3-125. Lutke und Wulff, Hamburg. 1920. 24 UNIVERSITY OF CALIFORNIA EXPERIMENT STATION Charlier's "Type A" frequency fits distributions which are unimodal and not extremely asymetrical. 12 The following constants, with mean diameter, are required: 100 90 t 1.70 x 60 t C ft) £ 40 ■?? jo | 20 « .0 Site index 45ft. Age ^oyrs Mear\d.b.h5".9*m- = Upper limit of each ^standard deviation class Interval. Ur 6 6 10 ia 14- 16 Id Diameter breast high in inches 20 2.2 Fig. 2. — Graph of cumulative frequency of number of trees of a sample plot, showing a step in the conversion of frequency by diameter class in inches to frequency in units of one-half the standard deviation. 12 Charlier's "Type A" frequency curve has the form N \ ] F(x)=— U (z)+/Wz)-f/M>4(z) a I > in which F(x) = frequency of x (in this case frequency per unit of one-half standard deviation measured from mean diameter). N = total frequency. a = standard deviation. 1 -x 2 *,(*) =V^e- T >*(x) d 3 o : dx* .rfVo dx i These are tabulated for unit frequency with x in terms of standard deviation in Charlier, loc. cit. pp. 123-125. Coefficient of asymetry, /3 3 = — -x\ ( 6tr J Coefficient of excess, /3 4 _ 1 (v, , _ 2lV7 4 ' ) (F4- the 3rd moment measured from mean). the the 4th moment measured from the mean). BUL. 456] YIELD, STAND AND VOLUME TABLES FOR RED FIR 25 (1) Standard deviation (fig. 3). The plotted points are weighted means of individual plot values with standard deviation in inches. A 9 ce • ~ 7 i 9 \ I />a II / ^ 5^ ^ 1 -°5 \ \ > / . 1/ \ I 10 1 s i/ 13 // /* 1U & 'As I9> l5 <^ 7 I Z 4 6 © »o 12 14 16 18 20 22 24 26 26 30 32 Mear\ diameter breast high in inches Fig. 3. — "Relation between standard deviation and mean diameter of stand. (2) Coefficient of asymetry (/3 3 ) (fig. 4). These were computed after plots were grouped by 2-inch mean-diameter classes and fre- quencies in units of one-half standard deviation. IS 14 16 16 20 22 ZU- 26 26 30 32 Mean diameter breast high m inches Fig. 4. — Relation between coefficient of asymmetry and mean diameter of stand. 26 UNIVERSITY OF CALIFORNIA EXPERIMENT STATION (3) Coefficient of excess (/? 4 ) (fig. 5). These were also computed after the latter grouping. 02 ^ 02 N, 19 /V 15 <\ V 15 ) ^Co {/ \ a*\ 15 <^ •v IO "■2 10 5^ "H 2 "^P- 5 U 6 & 10 12 14 16 18 20 22 Mear\ diameter breast K\gk irv inckes 24 26 28 30 32 Fig. 5. — delation between coefficient of excess and mean diameter of stand. From the calculated frequencies for each 2-inch mean diameter the values of table 4 were calculated by transcribing percentage frequencies to number of trees according to the mean diameter and total number of trees for a site-age class as given in table 1. RELATIONSHIPS WITHIN THE STAND If it is possible to express the stocking of an even-aged stand in terms of probable maximum stocking for site and age of the species in question, a quantitative measure of the mean stocking of the basic plots used in the construction of the yield tables of the species is the first requirement. Now it is evident that stocking by basal area or volume depends not only upon number of trees to the acre for age and site but upon average size of the trees as well. But since average size of tree can be controlled only indirectly by limiting the number of trees to the acre, the variation of stocking should show correlation with the number of trees. This was accordingly tried. From the relation between mean diameter and total number of trees to the acre (figure 6) it is evident that, within the limits of the data, the mean diameter varies inversely with the number of trees. This may be checked by comparison of the correlation coefficient, r — a measure of linear relationship, and the correlation ratio -q — a measure of true relationship whether linear or curvilinear, in which r = — 0.80 ± 0.020 and >7 = 0.80± 0.020 showing that the relationship is linear. BUL. 456] YIELD, STAND AND VOLUME TABLES FOR RED FIR 27 20 I . c «4- C % ~ IS- in 1 o \ \e ^ ^* 17^ N^ 13 te IS ^ ,9 II ^4. v*^_ -^ * *^o\ 70 60 30 40 30 20 10 - + 10 20 30 40 50 60 70 SO 90 Deviation of total number of trees on plots from yield table.m per cent Fig. 6. — Kelation between mean diameter and number of trees to the acre. The relation between basal area to the acre and total number of trees to the acre is of a different type. In figure 7 it is apparent that as the number of trees is increased, the basal area increases up to a certain point beyond which it tends to fall off. That the relation- ship is curvilinear is brought out by the difference between the correlation coefficient and correlation ratio, in which r = 0.439 ±0.044 v = 0.540 ± 0.039 The peak of a curve fitted to these data should represent probable maximum stocking by basal area that the species can attain, but as the course of the plotted data is not rigidly defined, a free-hand curve is likely to be too subjective; hence parabolic regression is fitted by the method of moments. 13 The relation of cubic volume to total number of trees (figure 8) is similar to that of basal area but is not as strong, as indicated by the following correlations : r = — 0.006 ±0.056 ^ = 0.373 ±0.047 13 Pearson, Karl. On the general theory of skew correlation and non-linear regression. Mathematical contributions to the theory of evolution XIV. Drapers' Company Research Memoirs, London University, pp. 3-54. figs. 1-5. 1905. 28 UNIVERSITY OF CALIFORNIA EXPERIMENT STATION -a 5 1 - 3* £ S O * .0 o (0 ? 80 *0-23 30 ii A /t t /I v i \ "A y ■-T -^ \ \ I6> >I3 4 > 7 i\ y lo l 13 r 14 ' \ ^ // / / / ii > 35 70 60 50 40 30 20 10 - + 10 20 30 HO SO 60 70 60 QO Deviation of total number of trees on plots from yield table tr\ per cent Fig. 7. — Eelation between basal area and number of trees to the acre. Greater variation in cubic volume for a given number of trees is due, in part at least, to greater variation in height of trees in crown classes below the dominant class. o c "i 1 A \ 4 A 16 •& \ l?N Nn, 6 v^/ A \ '°\ 9 ' 3^ \ \ \ ' .7 I x y / ,3 \ \ I V \ \ \ °l 70 60 30 40 30 20 10 - + 10 20 30 40 50 60 70 60 90 Deviation of total number of trees on. plots from yield table, in. per cent Fig. 8. — Eelation between cubic volume and number of trees to the acre. BUL. 456] YIELD, STAND AND VOLUME TABLES FOR RED FIR 29 The effect of number of trees on volume board measure shows the same general tendency (figure 9). But since r = — 0.006 ± 0.056 and v = 0.278 ±0.052 the relationship is weak, being partly upset, like that of cubic volume, by greater variation in height of trees below the dominant-crown class, and partly by the number of trees below the 8-inch diameter limit which do not at all contribute to board-foot contents. 1? 0— II / / / \ i 16 \ Y 14- 4 A V , u - -*s \-r ^S- V / t 13 ii ^9\ ^ A / \ / \ 13 1? V ? \ \ \ \ \ /' ! \ b 1 w V 3 Fig. 70 60 50 40 30 20 10 - O + 10 £0 30 4-0 50 60 70 SO 90 Deviation of total number of trees or\ plots from yield table in per cevt 9. — Relation between volume board measure and number of trees to the acre. If the regressions as graphically shown, indicate the effect of increasing the number of trees on basal area and volume, then the average stocking of the basic plots as measured from the maximum for the species may be judged by the difference between the height at culmination of the curves and the zero base. On these premises yield-table figures give roughly 9 per cent less than maximum basal area, 6 per cent less than maximum cubic volume, and 3 per cent less than maximum board foot volume. The tables, then, state gross- volume close to the productive possibility for red fir, and the figures given are attainable only when the entire area is producing timber. STATION PUBLICATIONS AVAILABLE FOR FREE DISTRIBUTION BULLETINS No. No. 253. Irrigation and Soil Conditions in the 389. Sierra Nevada Foothills, California. 390. 262. Citrus Diseases of Florida and Cuba Compared with those of California. 391. 263. Size Grades for Ripe Olives. 268. Growing and Grafting Olive Seedlings. 392. 277. Sudan Grass. 393. 278. Grain Sorghums. 394. 279. Irrigation of Rice in California. 283. The Olive Insects of California. 304. A Study of the Effects of Freezes on 395. Citrus in California. 310. Plum Pollination. 396. 313. Pruning Young Deciduous Fruit Trees. 397. 324. Storage of Perishable Fruits at Freez- ing Temperatures. 398. 328. Prune Growing in California. 400. 331. Phylloxera-resistant Stocks. 402. 335. Cocoanut Meal as a Feed for Dairy 404. Cows and Other Livestock. 405. 340. Control • of the Pocket Gopher in 406. California. 407. 343. Cheese Pests and Their Control. 344. Cold Storage as an Aid to the Mar- keting of Plums, a Progress Report. 408. 347. The Control of Red Spiders in Decid- 409. uous Orchards. 348. Pruning Young Olive Trees. 349. A Study of Sidedraft and Tractor Hitches. 410. 350. Agriculture in Gut-Over Redwood Lands. 353. Bovine Infectious Abortion, and As- 411. sociated Diseases of Cattle and New- born Calves. 412. 354. Results of Rice Experiments in 1922. 357. A Self-Mixing Dusting Machine for Applying Dry Insecticides and Fun- 414. gicides. 358. Black Measles, Water Berries, and 415. Related Vine Troubles. 416. 361. Preliminary Yield Tables for Second- Growth Redwood. 417. 362. Dust and the Tractor Engine. 363. The Pruning of Citrus Trees in Cali- 418. fornia. 364. Fungicidal Dusts for the Control of 419. Bunt. 366. Turkish Tobacco Culture, Curing, 420. and Marketing. 367. Methods of Harvesting and Irrigation 421. in Relation to Moldy Walnuts. 422. 368. Bacterial Decomposition of Olives During Pickling. 423. 369. Comparison of Woods for Butter Boxes. 424. 370. Factors Influencing the Development of Internal Browning of the Yellow 425. Newton Apple. 426. 371. The Relative Cost of Yarding Small and Large Timber. 427. 373. Pear Pollination. 374. A Survey of Orchard Practices in 428. the Citrus Industry of Southern California. 375. Results of Rice Experiments at Cor- 429. tena, 1923, and Progress in Experi- 430. ments in Water Grass Control at the 431. Biggs Rice Field Station, 1922-23. 377. The Cold Storage of Pears. 432. 380. Growth of Eucalyptus in California Plantations. 433. 382. Pumping for Draininge in the San Joaquin Valley, California. 434. 385. Pollination of the Sweet Cherry. 386. Pruning Bearing Deciduous Fruit 435. Trees. 387. Fig Smut. 388. The Principles and Practice of Sun- Drying Fruit. Berseem or Egyptian Clover. Harvesting and Packing Grapes in California. Machines for Coating Seed Wheat with Copper Carbonate Dust. Fruit Juice Concentrates. Crop Sequences at Davis. I. Cereal Hay Production in Cali- fornia. II. Feeding Trials with Cereal Hays. Bark Diseases of Citrus Trees in Cali- fornia. The Mat Bean, Phaseolus Aconitifo- lius. Manufacture of Roquefort Type Cheese from Goat's Milk. Orchard Heating in California. The Utilization of Surplus Plums. The Codling Moth in Walnuts. The Dehydration of Prunes. Citrus Culture in Central California. Stationary Spray Plants in California. Yield, Stand, and Volume Tables for White Fir in the California Pine Region. Alternaria Rot of Lemons. The Digestibility of Certain Fruit By- products as Determined for Rumi- nants. Part I. Dried Orange Pulp and Raisin Pulp. Factors Influencing the Quality of Fresh Asparagus after it is Har- vested. Paradichlorobenzene as a Soil Fumi- gant. A Study of the Relative Value of Cer- tain Root Crops and Salmon Oil as Sources of Vitamin A for Poultry. Planting and Thinning Distances for Deciduous Fruit Trees. The Tractor on California Farms. Culture of the Oriental Persimmon in California. Poultry Feeding: Principles and Prac- tice. A Study of Various Rations for Fin- ishing Range Calves as Baby Beeves. Economic Aspects of the Cantaloupe Industry. Rice and Rice By-Products as Feeds for Fattening Swine. Beef Cattle Feeding Trials, 1921-24. Cost of Producing Almonds in Cali- fornia: a Progress Report. Apricots (Series on California Crops and Prices). The Relation of Rate of Maturity to Egg Production. Apple Growing in California. Apple Pollination Studies in fornia. The Value of Orange Pulp for Production. The Relation of Maturity of fornia Plums to Shipping Dessert Quality. Economic Status of the Grape Industry. Range Grasses of California. Raisin By-Products and Bean Screen- ings as Feeds for Fattening Lambs. Some Economic Problems Involved in the Pooling of Fruit. Power Requirements of Electrically Driven Manufacturing Equipment. Investigations on the Use of Fruits in Ice Cream and Ices. The Problem of Securing Closer Relationship Between Agricultural Development and Irrigation Con- struction. Cali- Milk Cali- and No. 436. I. The Kadota Fig. II. Kadota Fig Products. 437. Economic Aspects of the Dairy In- dustry. 438. Grafting Affinities with Special Refer- ence to Plums. 439. The Digestibility of Certain Fruit By- products as Determined for Rumi- nants. Part II. Dried Pineapple Pulp, Dried Lemon Pulp, and Dried Olive Pulp. 440. The Feeding Value of Raisins and Dairy By-Products for Growing and Fattening Swine. 441. The Electric Brooder. 442. Laboratory Tests of Orchard Heaters. 443. Standardization and Improvement of California Butter. 444. Series on California Crops and Prices: B ULLETI NS— ( Continued) No. 445. Economic Aspects of the Apple In- dustry. 446. The Asparagus Industry in California. 447. The Method of Determining the Clean Weights of Individual Fleeces of Wool. 448. Farmers' Purchase Agreement for Deep Well Pumps. 449. Economic Aspects of the Watermelon Industry. 450. Irrigation Investigations with Field Crops at Davis, and at Delhi, Cali- fornia. 451. Studies Preliminary to the Establish- ment of a Series of Fertilizer Trials in a Bearing Citrus Grove. 452. Economic Aspects of the Pear In- dustry. CIRCULARS No. No> 87. Alfalfa. 265, 117. The selection and Cost of a Small 266. Pumping Plant. 127. House Fumigation. 267 129. The control of Citrus Insects. 136. Melilotus Indica as a Green-Manure 269. Crop for California. 270. 144. Oidium or Powdery Mildew of the 273. Vine. 276 157. Control of Pear Scab. 277 164. Small Fruit Culture in California. 166. The County Farm Bureau. 278 178. The Packing of Apples in California. 202. County Organization for Rural Fire 279. Control. 203. Peat as a Manure Substitute. 281 209. The Function of the Farm Bureau. 212. Salvaging Rain-Damaged Prunes. 215. Feeding Dairy Cows in California. 282 230. Testing Milk, Cream, and Skim Milk for Butterfat. 284. 231. The Home Vineyard. 286. 232. Harvesting and Handling California 287! Cherries for Eastern Shipment. 288. 234. Winter Injury to Young Walnut 289 Trees During 1921-1922. 290. 238. The Apricot in California. 292! 239. Harvesting and Handling Apricots 293. and Plums for Eastern Shipment. 294! 240. Harvesting and Handling California 296! Pears for Eastern Shipment. 241. Harvesting and Handling California 298. Peaches for Eastern Shipment. 243. Marmalade Juice and Jelly Juice 300. from Citrus Fruits. 301. 244. Central Wire Bracing for Fruit Trees. 302. 245. Vine Pruning Systems. 304! 248. Some Common Errors in Vine Prun- 305! ing and Their Remedies. 307. 249. Replacing Missing Vines. 308. 250. Measurement of Irrigation Water on 309! the Farm. 310. 252. Support for Vines. 253. Vineyard Plans. 311. 255. Leguminous Plants as Organic Fer- 312. tilizers in California Agriculture. 257. The Small-Seeded Horse Bean (Vicia faba var. minor). 258. Thinning Deciduous Fruits. 259. Pear By-Products. 261. Sewing Grain Sacks. Plant Disease and Pest Control. Analyzing the Citrus Orchard by Means of Simple Tree Records. The Tendency of Tractors to Rise in Front; Causes and Remedies. An Orchard Brush Burner. A Farm Septic Tank. Saving the Gophered Citrus Tree. Home Canning. Head, Cane and Cordon Pruning of Vines. Olive Pickling in Mediterranean Countries. The Preparation and Refining of Olive Oil in Southern Europe. The Results of a Survey to Deter- mine the Cost of Producing Beef in California. Prevention of Insect Attack on Stored Grain. The Almond in California. Milk Houses for California Dairies. Potato Production in California. Phylloxera Resistant Vineyards. Oak Fungus in Orchard Trees. The Tangier Pea. Alkali Soils. The Basis of Grape Standardization. Propagation of Deciduous Fruits. Control of the California Ground Squirrel. Possibilities and Limitations of Coop- erative Marketing. Coccidiosis of Chickens. Buckeye Poisoning of the Honey Bee. The Sugar Beet in California. Drainage on the Farm. Liming the Soil. American Foulbrood and Its Control. Cantaloupe Production in California. Fruit Tree and Orchard Judging. The Operation of the Bacteriological Laboratory for Dairy Plants. The Improvement of Quality in Figs. Principles Governing the Choice, Op- eration and Care of Small Irrigation Pumping Plants. The publications listed above may be had by addressing College of Agriculture, University of California, Berkeley, California. 8m-9 ,'28