Crown Competition--A Measure of Density

Com?etitio Measureof Density

BY JOHN E. KRAJICEK KENNETH A. BRINKMAN SAMUEL F. GINGRICH

THE MEASUREMENTof standdensity has be usedby a tree of small diameteris, of been one of the most trying problemsin course,less than that for a tree of larger forestry. Many measureshave been pro- diameter. The problem of determining posedbut a completelysatisfactory one has the growing-space/tree-sizerelations is not yet beenfound. • In an effort to help complicatedby the effectsof presentand solvethe problem,a new measureof stand past density on both stem diameter and density,developed at the Central States crown size. So for a givenspecies of speci- Forest Experiment Station, is presented fied d.b.h., the size of the crown may vary here. greatly.(Although trees compete for mois- Density,as foresterscommonly use the ture and nutrients as well as for light, it term, is the relation between the number would be impracticalto measureand re- of trees or some volumetric or areal unit late size of root systemsto degreeof com- to a specificarea, usually1 acre. In effect, petitionbetween trees. In fact, it is prob- then, a densitymeasure is intendedto help able that crown area may be an indicator determinethe relationof the averagetree of root area and thus a measure of such •n the standto the maximumgrowing space subterraneancompetition. 2) The onlyplace •t could utilize on the one hand, and the where the crown/d.b.h. relationsare not m•nimumgrowing spacenecessary to live confoundedby competitionfrom othertrees on the other. It is within this range that is in the o•Sen-growntree. So the new the forester wishesto maintain densityto method described here is based on the rela- fully utilizethe sitefor maximumproduc- tion of crown areato d.b.h. of open-grown non of desired usable volume. trees? The maximum growing spacethat can 2Bi2sgen,M. andMi2nch, E. The structure •The authorsrecognize the vastamount of and llfe of foresttrees, page 273. New York, literature on the subject of stockingand 1931. density, but believe a review of this work 3A briefpresentation of this subject is given would add very little to the presentationof in StationNote 108 "Crown development:an the new measuregiven in this report. Good index of stand density" by Krajicek and &scusslonsof stockingand density measures are Brlnkman. Central StatesForest Experiment g•ven by: Station, U. S. Departmentof Agriculture, Spurr, StephenH. Forestinventory. The 1957. Ronald PressCo. New York. 476 pp. 1952 and The authorsare respectivelyResearch For- Bickford,C. Allen, et al. Stocking,nor- ester, Silviculturist, and Research Forester, mallty, and measurementsof stand density Central StatesFo.rest Experiment Station, For- (Report of S.A.F. committeeon stocking). est Service,U.S. Departmentof Agriculture. J For. 55:99-104. 1957. Manuscriptreceived Nov. 24, 1958.

volume7, number1, 1961 / 15 Development of Crown-V•tidth/ D.B.H. 5. No evidenceof pruning,shearing, brows- Relations ing, decadence,storm damage, or sen- To establish the relation between width of ousinsect damage. crown and d.b.h. for open-growntrees, 6. Tree apparentlynot of sproutorigin. 340 open-growntrees were measuredin The smaller hardwood sample trees easternIowa: 88 whiteoaks (Quercus alba were foundin old fields,on roadrights-of- L.), 60 blackand red oaks(Q. velutina way, and on recently cutover, ungrazed Lam. and Q. rubra L.), 4 35 hickories areas where the sample trees originated (Caryaovata (Mill.) r. Koch),and 157 after cutting.The larger hardwoodsample Norwayspruce (Picea abies (L.) Karst.). trees were found in timbered pastures, As nearlyas could be determined,each tree cemeteries,and, to a limited extent, in haddeveloped without competition through- residentialareas. Norway sprucesample out its entirellfe. To maintainuniformity trees were found principallyin cemeteries in selectingsample trees, only thosethat and parkslocated in or near small towns, met the following specificationswere in- in recently establishedwindbreaks, and cludedin the sample: occasionallyin residentialareas. Crown width of each sampletree was 1. Crownfree of competitio.non all sides. measuredtwice, the secondmeasurement 2. Limbsextending to the groundon small treesand nearlyso on largertrees. at right anglesto the first, and the results 3. Lowestbranches the longest,or at least averaged. Regressionanalyses of the data aslong as those above. (This eliminated showeddefinitely that the crown width of trees that had been released from com- an open-growntree is closelyrelated to •ts petitionin the pastand on which the d.b.h. lowerportion of thepresent crown orig- Site qualitywas not determineddirectly inatedfrom epicormicbranches.) for the locationswhere the sampletrees 4. For smalltrees (less than 16 feet tall), were found. For the speciesexamined, no forking in the entire length; for open-growntrees were somewhatshorter larger trees,no forkingof the bolebe- than their forest-growncounterparts of low 16 feet. (This limitedthe sample the same diameter on similar soils and as- to trees that were of the type that pects. Most of the hardwoodsample trees would be favored in the foreststand, were found on forest soils where site •n- exceptfor lirahiness and excessive taper.) dicesranged from 45 to 75. The restwere found on prairie or transitionalsoils •n which the productivityof the similar forest soilswas known to range from poorto good. Differences in the relation between crownand stemdiameter among the hard- woodspecies sampled were small (Fig. 1) The differences between the red oaks and white oak and between white oak and hickory were not statisticallysignificant at the 5-percentlevel. The differencebetween hickoryand red oakswas significant,but

WHITE OAK just barely so. For simplicityin applica- tion all hardwood data were combined (Fig. 2). Crown width and d.b.h. for Norway F]CT.JP•: 1. Relationof crownwidth to d.b.h. for ope,n-grownhickory, white oak, and 4Hereafter"red oaks"refers to both Q. black oaks. velutinaand Q. rubra.

36 / ForestScience To determine MCA values for each tree in a standwould be time consuming.How- ever, from the least squaressolution of crown width and d.b.h., a formula can be derivedthat simplifiesthe procedure: The MCA value of a tree with a crown diameter of CW is (cw) MC /1 '-- 4 435.6 '-- 0.0018 From the rcgresslonanalysis of the com- binedoak-hickory data (Fig. 2), FretroE 2. Open:-grow•crow• width-&&& CW': (1.829D q- 3.12) relation for combinedhickory, white oak, where D '= d.b.h. and black oaks. and (CW) 2 = 3.345D"• q- 11.413Dq- 9.734 Therefore, sprucealso were highly correlated(Fig. MC/] '----0.0060D2q- 0.0205D q- 0.0175 3.). The differencesbetween the hardwoods and Norway sprucewere highly Similarly,the formula of MCA values.for significantat the 1-percentlevel. Norway spruceis MC/1 '= 0.0031D 2q- 0.0239D q- 0.0460 Sincesuch a closerelationship exists between the crown diameter and d.b.h. of With the MCA values.determined, it is open-growntrees, and this relationshipis not difficultto computethe minimumnum- nearly constantwithin a species,it may be ber of open-growntrees per acre that are •nferred that the crown of a tree of given theoreticallyrequired to producea com- d.b.h. cannotoccupy more than a certain plete canopy(assuming that crownsare areaeven with unlimitedgrowing space. distortedsufficiently to fill all voidsin the canopy,but •vithout reducingthe crown Development of an Index of areaof the individualtrees). If a tree has Crown Competition a d.b.h. of 16 inches and an MCA of 1.22 (asin theprevious example), then 82 open- Knowing the averagecrown width and assumingthat crownsof open-growntrees tend to be circular, the approximatearea coveredby the crown can be readilycom- puted. The percentageof an acre occupied by a vertical projectionof the crown is obtainedby dividingthe areain squarefeet by 435.6. For example,an open-grown Norway sprucewith d.b.h.of 16 incheshas a crown width of approximately26 feet (Fig. 3), or crownarea of 531 squarefeet (1.22 percentof an acre). This valuehas been termed "maximum crown area" (MCA), becauseit expresses,in terms of percentof an acre, the maximum area that Diome•er of B•eost Height (Inche•) could be occupiedby the crown of a tree Fmv•E 3. Relationof crownwidth to d.b.h. of specifiedd.h.h. for open-grownNorway spruce.

volume7, humget1, 1961 / 37 TABLE 1. Determinationof the CCF for an even-ageduI)land oak stand,site index 60, age50 (Schnur1937).

D N DN DeN Checkusing MCA values (Numberof trees (d.b.h.) peracre) ßMCA per tree Total M.CA

Percent Percent I I0 10 I0 0.044 0.440 2 29 58 116 .082 2.378 3 50 150 450 .133 6.650 4 63 252 1,008 .196 12.348 5 77 385 1,925 .270 20.790 6 78 468 2,808 .356 27.768 7 66 462 3,234 .455 30.030 8 49 392 3,136 .566 27.734 9 31 279 2,511 .688 21.328 I0 17 170 1,700 .822 13.974 I1 7 77 847 .969 6.783 12 3 36 432 1.128 3.384 13 1 13 169 1.298 1.298 •otal 481 2,752 18,346 174.905 (CCF)

CCF = 0.0060(18,346) -4- 0.0205(2752) -4- 0.0175(481) • 174.91

grown trees of that size could cover an 1 acre completely(82 X 1.22 '-- 100 per- CCF .-- cent,or sumof MCA valuesequals 100). d UsingMCA valuesfor otherd.b.h. classes, [0.0060(52D•2N•)q- 0.0205(52D.oV0 any combinationof open-growntrees of + 0.0175(N0] various diameters whose MCA values.total and for Norway spruce, 100 for an acre theoreticallycould have a 1 closedcanopy. CCF ,-- To avoidconfusion in usingone term-- d MCA--to expressthe arearequirement of [0.0031(52D,2N0q- 0.0239(52D•N0 both a singletree and a sum of trees,an- -Jr-0.0460 (52N0 ] other expression,"Crown Competition where Factor" (CCF), hasbeen coined to rep- D• •-- individual d.b.h. or d.b.h. class resent the sums of the MCA values for an N•'-- number of trees in d.b.h. class acre. Crown CompetitionFactor is defined .d '= area in acres as the sum of MCA values for all trees in An exampleof the applicationof the the stand,divided by the area in acres. It CCF to a specificstand is givonin Table 1. is usedas an expressionof stand density. All the trees in the stand are used to de- These summationscan be expressedby us- termine the CCF. In the stand of trees ing the previousformulas in the following shownin Table 1, nearly60 percentof the form: Crown CompetitionFactor is containedin For oak-hickorystands, trees less than 8 inches d.b.h. To consider

ForestScience only thosetrees larger than a specified At leastthree things account for differ- lower diameter limit will result in a low encesamong species in the maximumCCF CCF and will not representthe actual they can obtain. One is their crown area competitionin the stand. It is known, of developmentwhen thereis no competition. course,that the larger treeshave an ad- Another is the basicshape of the crown, verse effect on the growth rate of the suchas conical,parabolic, etc. Also, toler- smaller. It is more difficultto assignan ancemay be a factor. For example,meas- effect of the small trees on the growth of urementsover an 8-yearperiod in heavily larger ones. Where moistureis a limiting stockedoak-hickory plots show a slightly factorat any time duringthe growingsea- higherCCF than thosefound in pure oak son, the smaller trees probablyhave an stands,probably because of the greatertol- adverseeffect on the growthof the larger eranceof hickory. Someof the Ohio plots trees. cited above also had CCF values in ex- Oneother precaution isnecessary. When cessof 200, but thesetoo containedsome computingthe CCF from standtables, the of the moretolerant species. . actual number of trees in each diameter To further confirm the consistencyof class should be used rather than the diame- the valuesfound in field tests,the formula ter of the tree of averagebasal area and for oak-hickorywas appliedto standtables total numberof trees. If the latter proce- for even-agedupland oaks. Except for dure is followed,excessive errors may re- highvalues in youngerstands on the poorer sult when the diameter distribution deviates sites,the CCF valuesshow great uniform- greatlyfrom the normalcurve. ity (Table 2). Results and Discussion No Norway sprucestands were avail- ablefor field testingof the formula. The It is obviousthat maintainingstand density below CCF 100 from time of establish- similarityof crown widths among the oak speciessuggested that perhapsa given ment to maturitywould normalIy produce crownwidth/d.b.h. relation may be rather low-qualitytrees, especially if they were not commonwithin a genus.With thisthought, prunedartificially. Greater density (with a the Norway spruceformula was applied to h•gherCCF value)would be accompanied standtable data for Sitkaspruce and west- by a reductionof the actual or effective hve crownof the averagetree in the stand. Densitywill not increaseindefinitely, of course.A numberof even-agedpure oak TABLE 2. Applicationof CCF to standsin Iowa consistentlyshowed a ceil- standtable data• for even-agedup- mg densityof aboutCCF 200, when all land oaks. treesin the standwere included. Similarly, (CCF valuesby ageclass and siteindex) the averageCCF for 52 even-agedoak Age Site •ndex plots,each 0.2 acrein size,located through- out southeastern Ohio and selected for full (years) 40 50 60 70 80 stockingwas exactly200fi A consistent 10 301 249 219 190 166 20 218 210 188 180 181 ceilingprobably exists for all other species 30 184 183 178 176 167 also,but it is probablydifferent from that 40 171 170 168 165 160 found for oaks. 50 169 169 175 173 173 60 171 171 176 175 169 70 165 168 172 170 165 5Theseunpublished data on whichthe com- 80 168 176 .173 172 164 putationsare basedwere collectedand fur- 90 175 179 177 174 164 nishedby Willard Carmeanand StephenG. 100 178 182 177 175 166 Boyce, Soil Scientistand ResearchForester, 1Schnur•G. Luther. Yield• stand,and volumetables respectively,Central States Forest Experiment for even-agedupland oak forests. U.S. Dept. Agric. Station. Tech. Bull. No. 560, Table 37, 1937.

volume7, number1, 1961 / 39 ern hemlock(Table 3). These data are with stand-densityindex. Althougha CCF primarilyfor standsin Washingtonand of 100 (which might indicateminimum Oregon,yet the CCF valuesfor the age- densityfor full siteutilization) is approxi- site classesenclosed by the brokenline are matelyassociated with a stand-densityin- quiteconsistent, except for age 140, sitein- dexof 110 percentfor standswith average dex 160. The latter maybe an error in the diameters as shown, the relationshipof originaltable. numberof treesto averagestand diameter A comparisonof oak-hickoryand Nor- is curvilinearon log-logpaper for any given way sprucestands of the samedensity as CCF value, whereasstand-density index is measuredby crown competitionreveals a linear. However, this doesnot representa striking differencein their structure. For deterrentto the applicationof CCF. Ap- example,a Norway sprucestand composed proximatefull stockingin termsof stand- of 16-inch trees would have more than densityindex (240) resultsin CCF values half again the numberof treesand basal from 197 to 262 as shown in column 6 of areaper acre asan oak-hickorystand with Table 4. Again, one could concludein trees of like size and the same CCF. generalthat a CCF of 200 (whichmight representmaximum densityfor full site Comparison •Vith Other Measures utilization) is approximatelyequivalent to a stand densityindex of 240 for stands A thoroughcomparison of CCF •vith other with an averagediameter of from 5 to 15 measuresof stand densitywould require inches. much space. A brief comparisonserves to illustratethe relation between CCF, basal In compar/ng"D plus"and "D times" area per acre, and Reineke'sstand-density spacingø with CCF spacing,it is apparent index (Table 4). It is of interestto note 6Rulesof thumbused as thinningguides' that when CCF is held constant over a Using the "D plus" rule, spacingof treesin range of average stand diameters,basal feet is equalto the diameterof the treesin area per acre changesat a rate that is inchesplus a constant;using the "D times" typical of most even-agedstands. CCF rule, spacingin feet is equalto tree diame- does not show a consistentrelationship ter in inches times a constant.

T.dBLE 3..dpplicationof CCF to standtable data • for even-agedSitka spruce and westernhemlock. (CCF valuesby ageclass and site index.) Age Site index (years) 60 80 100 120 140 160 180 200

4O 1184 635 412 345 320 I 292 275 269 6O 427 351 332 299 288 276 270 267 8O 356 327 306 293 284 283 280 276 100 326 299 296 288 284 285 285 285 120 304 296 291 286 285 285 287 289 140 291 289 286 286 286 [ 241 .I 290 287 160 286 285 287 287 287 291 296 ...... 180 283 282 281 283 288 288 292 2OO 279 280 280 284 289 293 294

1Meyer, W. H. Yield of even-agedstands of Sitka spruce and western hemlocks. U.S. Dept. Agrlc. Tech. Bull. No. 544, Table 36, 1937. Dash lines indicate limits of basic data.

40 / ForestScience TABLE 4. Comparisonsof CCF with basalarea and stand-densityindex • for theoreticaloak-hickory stands of variousaverage diameters. When stand-density Basalarea per acre When CCF is 100: index is 240: when: Average Stand- d.b.h.2 Treesper Basalarea density Treesper Equivalent CCF -- SDI'-- (inches) acre peracre index acre CCF 200 240 Number S]uarefeet Percent Number Percent Squ•efeet Squ•efeet 2 1,220 26.8 92 3,177 262 53.7 69 5 370 50.3 122 730 197 101 100 10 122 66.5 122 240 197 132 131 15 60 73.6 115 125 209 146 153 20 35.3 77.0 107 78.9 223 154 172 25 23.4 79.8 102 55.1 236 159 188 1Reineke,L. H. Perfectinga stand-densityindex for even-agedforests. J. Agric. Res. 46: 627-638. 1933. gCalculatedfrom averagebasal area per tree. (Used in the exampleto facilitate comparisonswith stand- densityindex. As statedin the text, the actualnumber of trees in each diameterclass should be usedto compute CCF.)

TABLE 5. Spacingof treesfor a CCF of 100 and equivalent"D plus" and "D times"constants for theoreticalstands of oak-hickoryand Norway spruceof variousaverage diameters. Oak-hickory Norway spruce Averaged.b.h. x CCF "D plus" "D times" CCF "D plus" "D times" (inches) spacing constant constant spacing constant constant Feet Feet Unit Feet Feet Unit 2 6.0 4.0 3.00 6.8 4.8 3.40 5 10.8 5.8 2.16 10.3 5.3 2.06 10 18.9 8.9 1.89 16.1 6.1 1.61 15 26.9 11.9 1.79 21.9 6.9 1.46 20 35.1 15.1 1.76 27.7 7.7 1.38 25 43.1 18.1 1.72 33.6 8.6 1.34 1Calculatedfrom averagebasal area per tree.

that spacingbased on CCF will not result It shouldbe emphasizedthat CCF is not •n a constant factor for stands of different essentiallya measureof crown closure. averagediameters (Table 5). An examina- Theoretically,complete crown closurecan tion of these "constants" substantiates the occur from CCF 100 to the max/mum for findingsof Stahelin7 and others that diameter plusa constantresults in too muchbasal the species(in oaks,approximately CCF area with large trees and too litde with 200). Insteadof estimatingcrown closure, small trees. CCF estimates the area available to the averagetree in the standin relationto the 7Stahelin,R. Thinning even-agedloblolly and slashpine standsto specifieddensities. maximum area it could use, if it were J. For. 47:538-540, illus. 1949. open grown.

volume7, numberl, 1961 / 41 Conclusions sentation. There is goodevidence to indicatethat the The CCF is assimple to applyas basal- CCF methodof measuringdensity can be area tables,but, unlike basal-areacontrol, appliedto even-agedstands of hardwoods interpretationof valuesdoes not depend and conifers. Moreover, althoughfield upon diameter distributionof a specific tests of the method have been confined to type or range. The Crown Competition even-agedstands, in theory it can be used Factor, basedon a verifiedrelationship be- for uneven-agedstands equally well. The tween crown width and d.b.h., should methodadjusts accurately to variationsin measurecompetition more acuratelythan diameterdistribution and to the potential the methodspresently used. crown spacerequirements for the individual trees. Summary Traditionally, researchershave insisted Measurements of several hundred oak, that the idealmeasure of densityshould be hickory,and Norway sprucedisclosed a correlatedwith growthand yield and inde- high degreeof correlationbetween crown pendentof standage. On thispremise CCF width and d.b.h. for open-growntrees of hasshown up well in preliminarytests. In thesame species. Apparently this relation is standdensity studies in even-agedoak net independentof age and sitequality. From basalarea growth (accretionless mortal- the open-grown-treedata, formulaewere ity) is at a maximumwhen CCF is 135. derivedwhich expressthe maximumper- This holdstrue for stands25, 50, and 90 centageof an acre the crown of an open- years old, with correspondingaverage growntree of a givendiameter and species basalarea per acre of 71, 86, and96 square couldoccupy. feet respectively.The growth data cover Theoretically,competition for crown an 8-yearperiod for the two youngerage spacebetween trees in a standbegins when classesand a 4-yearperiod for the other. all crownspace is occupiedand eachtree Severalimportant points should be em- crown is equalin area to that of an open- phasized: growntree of the samed.b.h. This con&- 1. Relatively few measurementsof tionis taken as a baseof 100 for expressing truly open-growntrees are requiredto pro- competitionamong crowns for growing vide a basicCCF equationfor a species. space.From this base,stand density may 2. The ceilingof aboutCCF 200 for be expressedas a percentage.For con- oakdensity should not beassumed for other venience,this percentageis termed the species,even when crown widths are found Crown CompetitionFactor (CCF), be- to be similar. Relativetolerance and shape causeit is an estimateof the competition of crown (conical,parabolic, etc.) deter- among crownsfor growing space. Pre- mine the maximum densitiesattainable by liminary trials indicatethat the methodis a species. applicableto even-agedstands, and theo- 3. In standshaving mixtures of species retically can be used to advantagein with different crown characteristics and uneven-agedstands. These trials show tolerance,the determinationof CCF may strongevidence that siteand standage do needto be handledby proportionalrepre- not influence CCF.

42 / ForestScience