Stand Structure and Allometry of Trees During Self-Thinning of Pure Stands Author(S): C

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Stand Structure and Allometry of Trees During Self-Thinning of Pure Stands Author(s): C. L. Mohler, P. L. Marks, D. G. Sprugel Source: Journal of Ecology, Vol. 66, No. 2 (Jul., 1978), pp. 599-614 Published by: British Ecological Society Stable URL: http://www.jstor.org/stable/2259153 . Accessed: 04/03/2011 11:24

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http://www.jstor.org Journalof Ecology(1 978), 66, 599-614

STAND STRUCTURE AND ALLOMETRY OF TREES DURING SELF-THINNING OF PURE STANDS

C. L. MOHLER,* P. L. MARKS* AND D. G. SPRUGELt

Sectionof Ecology and Systematics,Cornell University, Ithaca, New York14853*, and Radiologicaland Environmental Research Division, Argonne National Laboratory, Argonne,Illinois 60439, U.S.A.t

SUMMARY

(1) Stand structureand mean weight-densityrelations of nearlypure, dense, even-aged, naturalstands of Prunuspensylvanica and Abiesbalsamea in thenorth-eastern U.S.A. were examinedand relatedto allometricgrowth. Values forthe exponents of allometric and mean weight-densityequations wereestimated by principalcomponents analysis of logarithm- icallytransformed data. (2) It is proposed thatsoon aftera stand of woody plantsbecomes established the size- frequencydistribution is a negativelyskewed, bell-shapedcurve; the distributionsub- sequentlybecomes positively skewed, and eventuallyapproaches normality after substantial thinning.Maximum positive skewness occurs at thetiine self-thinning begins. (3) The weight-frequencydistribution undergoes a parallelseries of stages: roughly normal at first,but quicklychanging to lognormal,with maximum skewness attained at the tiine thinningbegins. These curve-forinsare proposed only as approximationsto empirically observeddistributions. A consistenttendency toward bimodality is one commonly-observed departurefrom the idealized distributions. (4) The calculated exponentsof the inean weight-densityequations for Prunuspensyl- vanicaand Abies balsamea were - 1 46 and - 1 22, as comparedto theproposed value of - 15 (- 3/2power 'law' of self-thinning).In general,weight changes in plantparts during self-thinningdid not parallelthose for whole plants; in particular,the mean weight-density exponentfor foliage was approxiinately- 1 0. The exponentsof inean weight-density equationsfor total roots and totalshoots, however, approximately equalled theexponent for wholeplants. (5) It is concludedthat observed patterns of alloinetric growth are incompatiblewith inean weight-densityequations forwhole plants,unless a mutualadjustment between allometry and standstructure is assumed.

INTRODUCTION Intraspecificcompetition causes thinningin plant populations throughdeath of the smallestindividuals. In single-speciesstands small initial differences in seed size, timneof germination,growth rate and otherfactors lead to small differencesin above-ground statureand rootinghabit of individualplants soon afterthey become established,and these differencesbecome amplifiedas the stand develops (Black 1957, 1958; Black & Wilkinson 1963). Extremeexpression of these differenceseventually results in very dominantand verysuppressed individuals in a stand,although the abilityto withstand suppressionis knownto varywidely among species. 0022-0477/78/0700-0599$02.00(D 1978 BlackwellScientific Publications 599 600 Standstructure of treesduring thinning It is of interestto know thegenerality of theeffects of the thinningprocess in plants, especiallysince much of theexperimental work has been done on annual crop plantsand weeds.Do radishplants and pinetrees respond to thinningin thesame manner?It is clear fromthe literature(Yoda et al. 1963; White & Harper 1970) that both herb and tree species,when growing in single-speciesstands that are thinning,exhibit a linearrelation- shipbetween the logarithm of wholeplant weight and thelogarithm of density: log w=C'+klog p or w=Cpk (1) wherew is the inean weightof individualplants thathave survivedthinning, p is their density,C' is a species-specificconstant, and C is the antilogof C'. Yoda et al. (1963) proposedas a generalfinding that the exponent of (1) shouldbe - 3/2for plants that are thinningin purestands (the - 3/2power 'law'). White& Harper(1970) postulatedfurther thatthe exponent might also be - 3/2for plant parts(leaves, roots,etc.), whichwould implythat weights of individualparts show thesame relationshipto plantdensity as the weightsof entireplants. In thispaper we considerthe weightsand dimensionsof plant parts in relationto one another(allomnetry), and in relationto density,for naturally thinningpopulations of two treespecies, pin cherry(Prunus pensylvanica L.) and balsam fir(Abies balsamea (L.) Mill.). We also considerthe mechanisms of thinningin relationto developmentof stand structurefor thesespecies. For comparison,allomnetric relation- shipsare developedfor several additional species using data in theliterature. Unlikemany tree species, both pin cherryand balsam firare well suitedto examining the thinningprocess in naturallyoccurring stands. Pin cherryis a small,fast-growing, successionaltree that lives forabout 30 yearsin northernNew England (U.S.A.); it is commonafter cutting or othermajor disturbancethroughout the Northern Hardwood and Boreal forestsof easternNorth America. It typicallyforms truly single-age stands because of itsburied seed habitand germinationrequirements (Marks 1974).The buried seed habit means thatstand age reflectsboth timnesince disturbanceand age of all pin cherrystems in the stand. Wheredormant seeds are abundantin the soil, pure,dense standsof pincherry are foundsoon afterdisturbance, and thinningbegins in thefirst 2 or 3 years. Samplingplant weightin plots of varyingdensity in such pure, dense stands shouldyield results comparable with those from experiments. Balsam firis a stronglyshade-tolerant conifer with a lifespan of 60-80 yearsin the mountainsof thenorth-eastern U.S.A., whereit oftenforms nearly pure stands(> 99% fir)immediately below the timberlineat an altitudeof 1100-1300 in. Many of these high-altitudefir forests are characterizedby 'wave-regeneration,'a peculiar type of naturaldisturbance in whichtree mortality is concentratedin long bands (waves) which movethrough the forest in thedirection of theprevailing wind at 1-3 in peryear (Sprugel 1976). When a wave passes throughan area of firforest the overstorey trees die off,and are replacedby a dense,even-aged stand of fir seedlings. A largetract of wave-regenerated firforest thus contains numerous naturally regenerated stands of nearlypure fir varying widelyin densityand in thedegree to whichthinning has occurred.

METHODS Data collection Prunuspensylvanica Four-, six-,and fourteen-year-oldpin cherrystands, generally equivalent in aspect, C. L. MOHLER, P. L. MARKS AND D. G. SPRUGEL 601 slope and drainage,were selected for sampling in New Hampshire,U.S.A. (longitude71? 30' W, latitude440 00' N). The standswere small (0 25-05 ha in size), thusavoiding the patchydispersion of pin cherrycommon in largerstands (5-50 ha), and all standswere denselystocked withpin cherry.Mortality was already much in evidenceeven in the 4-year-oldstand. Density was measuredin randomlyplaced plots varyingin site and numberfrom fifteen1 x 1 in plots in the 4-year-oldstand to twenty2 x 3 in plots in the 14-year-old stand. Plant weightwas estimatedusing dimensionanalysis (Whittaker& Woodwell 1968).In each standa setof sample trees (about twenty-five)was chosento span therange of trunkdiameters observed in theplots, and thesesample trees were excavated carefully by hand to includeroots (roots smallerthan c. 0 5 mm in diameterwere not included). Trunkwood plus barkand rootsof all sampletrees were oven-dried and weighedwithout sub-sampling.Branches were sub-sampled by age-class, and leaves,current twigs (1 -year- old branchesplus 1-year-oldtwigs on older branches),and older partsof each sample branchwere weighed (dry), so thateach of thesetissues could be regressedon branch basal diaineterfor each branchage-class. After estimating dry weights of branchparts on all sampletree branches, another set of regressionswas calculatedto relatethe weight of each tissue(leaves, current twigs, older branch wood and bark,trunk wood and bark,and roots) to trunkdiameter, and theseequations wereused to calculatethe weight of each organon each pin cherryplant in each plot.

Abiesbalsamea Plantweight was estimatedby dimension analysis procedures generally similar to those describedfor pin cherry.Forty-seven trees in thewave-regenerated forest on Whiteface Mountain, New York (Sprugel 1976), were selected for samplingusing a stratified randomprocedure, in whichtrees were selected at randomfrom even-aged stands chosen to give a reasonablyeven distributionacross the range of ages (and thus densities) representedin the WhitefaceMountain firstands. Each sample tree was felledand sub-sampledto estimatethe weight (oven-dry) of each major above-groundcomponent (new foliage,old foliage,current twigs, older branches,trunk wood and trunkbark). Roots were not excavated. Regressionequations were thencalculated to predictthe weightof each componentfrom trunk diaineter (at 25 cm above ground)and age. Once the regressionequations had been developed, 10 x 10 in plots werelaid out in standson WhitefaceMountain ranging in densityfrom 0 15 to > 13 treesper in2. In the less dense plots ( < 1 treeper m2) the diaineterof everytree on the plot was measured, whileon thedenser plots five 2 x 2 m sub-plotswere laid out and diametersmeasured in these.The regressionequations were thenapplied to the diametersfrom these plots to estimatetotal weight of each componentfor each stand. The calculationsof inean weight-densityrelationships for pin cherryused densitiesof individualplots rather than inean standdensities; for balsam fir,densities of theplots in whichdiameters were tallied (i.e. eitherone of 10 x 10 in or fiveof 2 x 2 in) wereused. For bothspecies, calculations involving inean weight per plant and ineanweight of plant parts perplant in relationto densityused estimatedweight per plot from the dimension analysis regressionsdivided by plot density. Calculations of alloinetric relations among plant parts wereinade usingnot onlydata fromthe harvested sample trees of pincherry and balsam fir,but also publisheddata on otherspecies: Acer saccharum Marsh., Betula alleghaniensis Britton,Fagus grandifoliaEhrh. and Picea rubensSarg. fromWhittaker et al. (1974), and 602 Standstructure of treesduring thinning Populus deltoidesBartr. from Berlyn(1962); for Quercus nigra L. and Q. alba L., unpublisheddata of P. L. Marks and P. A. Harcombewere used.

Estimationof exponents To test theirproposed - 3/2 power 'law' Yoda et al. (1963) constructednumerous scatterdiagrams of log mneanweight v. log density.If it is assumedthat mnean weight is a simpleexponential function of density,then the slope of a fittedstraight line on such a log-log plot would be theexponent of the thinninglaw. PresumablyYoda et al. (1963) fittedlines to theirscatter diagrams by eye, but subsequentauthors (White & Harper 1970; Bazzaz & Harper 1976) have used linearleast-squares regression to estimatethe exponent.In general,if there is scatterin a setof data whichcannot be accountedfor by a linearregression equation, then the slope of the regression of y on x willnot be thesame as the slope for the regressionof x on y (Snedecor & Cochran 1967). In the case of the thinningrelationship, least squares regressionis not appropriatefor estimatingthe exponent,since neitherinean weightnor densitycan be consideredthe independent variable. In this paper principalcomponents analysis (PCA) on logarithmically-transformed data has been used to estimatethe exponents of themean weight-densityrelationship for wholeplants, plant parts, and severalallometric relationships. PCA is preferableto linear regressionin these cases, because the firstprincipal axis representsthe line which minimizesthe sum of thesquared perpendicular distances of points froin the line, and thus makesno assumptionsabout whichis thedependent and whichthe independent variable. The slope of the firstaxis is the ratio of the elementsof the firstaxis eigenvector.The percentageof thevariance accounted for by thefirst axis (% EV) is 100 timesthe ratio of thefirst eigenvalue to thesum of theeigenvalues. Although we foundit moreconvenient to determineslope using a standardPCA computerprogram (Gauch 1973), Sokal & Rohlf (1969, p. 526) provideformulae for computing slope and eigenvaluesfrom the varianceand covarianceof thevariables.

Problemsof bias in studiesof thinningin natural populations There are severalways in whichdata fromdense, pure, even-aged stands of a species mayfail to showa clear-cutthinning response. It is especiallyimportant to minimizethese effectswhen dealing with data fromnatural populations. First, under natural conditions plots receivedifferent numbers of seeds. In plots withfewer seeds, plants must attain a largersize beforethinning begins. If sampledprior to thinning,these plots will have a low densityfor their inean weight and willfall below the thinning line toward the middle of the usual log-logdiagrain. Second, in someyoung sample plots the population may not have attaineda sufficientsize forcrowding to have produceda changein density.On a scatter diagramnof log ineanplant weight v. log densitysuch plots will be below thethinning line at the rightside of the diagramn(Fig. 1(a) and (b)). Both of the above considerations indicatethat substantial changes in densityare requiredbefore the mneanweight of the populationwill be strictlya functionof density. Third, if density-independent mortality is actingon thepopulation, some plotswill have a densitybelow whatwould be predicted fromtheir mean plant weight. Although density-independent mortality can occurin plots of any age and stature,it is particularlycommon in olderplots wheremortality may be due to senescenceas well as to crowding.Plots of thissort will drop below theline, as lowerdensities are reachedwithout corresponding growth of thereinaining individuals. C. L. MOHLER, P. L. MARKS AND D. G. SPRUGEL 603

100000 (a) (b) 00

000000_ . | ?o o _ ? o

0 0~~~0 1000 _0* o _ o lao 00 D

0 oe~~~~ 100 _ * oO0_

10IIC I I 0 1 1 10 100 01 1 10 100 Density(number per in2)

FIG. 1. Scatterdiagrams of mnean whole plant dry weight versus density for (a) Prunuspen.sy/vanica and (b) Abiesbalsamea. Open circlesrepresent plots in whichmnean plant weight was judged to be mainlya functionof densityalone; filledcircles represent plots not used in the analysis,as explainedin thetext.

In each of thecases just described,departure from ideal circumstancesresults in data pointswhich lie belowthe thinning line on a log mean weightv. log densitydiagram (Fig. 1(a) and (b)); thatis, thethinning line represents the maximum mean weight attainable at a givendensity. The effectof thesedepartures on thethinning equation varies, however. Inclusion of plots not fullythinning due to lack of timefor growthwill resultin an estimatedslope whichis too steep;apparently Yoda etat. ( 1963) excludedsuch plots from theiranalysis. Inclusion of plotsnot thinningdue to low initialdensities may or maynot change the slope but will lower the intercept.Finally, inclusion of senescentplots will resultin an estimatedslope whichis too shallow;Schlesinger (1976) providesan example in whichdensity-independent mortality results in a log mean weightv. log densityline witha slope substantiallyshallower than -3/2. In analysingdata forbalsam firwe reliedon fieldnotes to eliminateplots whichhad clear density-independentmortality; such plots werealong the edges of waves and had > 50% of theoverstorey dead. Balsam firplots well below thelower right position of the thinningline (Fig. 1(b)) wereprobably not thinning, and wereeliminated also. For thepin cherrydata thefull complement of fifty-three plots was analysed;however, more reliance was placed on a second analysisin whichthe nineteen plots having relatively low mnean weightfor the observed density (Fig. 1(a)) wereeliminated.

RESULTS Weight-densityrelations Estimatedexponents of thethinning equations for whole plants and plantparts of pin cherryand balsam firare givenin Tables 1 and 2 respectively;unless otherwise noted, the slope and percentagevariance accounted for are estimatedby PCA usinglogarithmically- 604 Standstructure of treesduring thinning transformeddata. A numberof slopes estimatedby linearregression are also givenfor comparison.As mustbe the case, the regressionestimates are consistentlylower than thosefrom PCA, and themagnitude of thediscrepancy is relatedto thedegree of scatter in thedata. Thus thediscrepancy is negligiblein thecase ofbalsam fir, where the %EV and r2 values are quite large (Table 2). On the otherhand, regressionestimates may be quite misleadingwhere the r2 and %EV values are lower,as withthe complete pin cherrydata set (Table 1). Eliminationof outliersdoes not in generalchange the slopes appreciably. Whole plants,roots and trunksof pin cherry all have slopesabout equal to -1 5 (Table 1). Since roots have a slope about equal to thatfor whole plants,above-ground plant

TABLE 1. Slopesof thelog meanweight v. log densityrelationship for whole plant weightand weightof plantparts of Prunuspensylvanica; slopes were determined separatelyby principal components analysis and by regression for the complete set of fifty-threeplots; values are also shown for the thirty-four plots in which mean weight wasjudged to. be mainlya functionof density alone, as explainedin thetext

53 plots 34 plots Principal Regression Principal axis axis Slope %EV* Slope r2 x 100 Slope %EV* (a) Leaves -1 13 90 3 -0 88 64.5 -1 08 96.7 (b) Currenttwigst -0 81 80 5 -0 53 36 1 -0 70 85 2 (c) Older branchest -1 90 94 8 -1 50 73.3 -1 91 98.4 (d) Trunk -1 58 93 5 -1 25 71 6 -1 52 98 0 (e) Root -1 48 93 3 -1 19 72 2 -1 44 98 1 (f) (a+b) -102 896 - - -096 957 (g) (b +c) -1 42 93 6 - - -145 97 8 (h) Above-groundtotal -1 52 93 4 - - -1 47 98 0 (i) Total weight -1 52 93 4 -1 21 72 2 -1 46 98 1 * %EV is a measureof thevariation accounted for by thefirst principal axis (see text). t Currenttwigs include all twigsproduced during the year of harvest. t Older branchesinclude all branchesexcept current twigs.

TABLE 2. Slopes of the log mean weightv. log densityrelationship for whole plant weightand weightof plant parts of Abies balsamea (23 plots);slopes were determined separatelyby principalcomponents analysis and by regressionfor the twenty-three plotsin whichinean weightwas judged to be mainlya functionof densityalone, as explainedin thetext

Principalaxis Regression Slope %EV* Slope r2x 100 (a) New leaves -0 95 99 6 -0 95 98.4 (b) Currenttwigst -0 95 99 6 -0 95 98 6 (c) Older branchest -1-18 99 6 - - (d) Trunk -1 30 99 5 -1 28 98 0 (e) New+ old leaves -101 99 6 -100 98.6 (f) (a+ b) -0 95 996 - (g) (b + c) -1 15 99 6 - - (h) Above-groundtotal - 122 99 6 -1 21 98.4

* %EV is a measureof thevariation accounted for by the first principal axis (see text). t Currenttwigs include all twigsproduced during the year of harvest. t Older branchesinclude all branchesexcept current twigs. C. L. MOHLER, P. L. MARKS AND D. G. SPRUGEL 605 weightalso has a slope verynear the predicted - 1 5 value. The slope forleaves, however, is close to - 10 and the slope forcurrent twigs is even shallower;the slope for older branches,on theother hand, is considerablysteeper than -15. Total branchweight, the sumnof theweights of current twigs and olderbranches, naturally has a slope intermnediate betweenthe slopes of itscomponents. In thecase of balsam fir,slopes for whole plants (above-ground tissues only) and plant partsare quiteshallow (Table 2); in fact,the value of - 122 forwhole plants is one of the highest(i.e. closestto zero) of any value reportedin the literature.The veryhigh %EV values indicatethat these shallow slopes reflectintrinsic properties of balsamnfir rather thanany of thedistorting influences described in theMethods section. As withpin cherry, slopes forleaves and currenttwigs are farshallower than that for whole plants,and are quiteclose to - 10. The slope fortrunk weight is steeperthan the slope forwhole plants, but, conversely,the slope forbranch weight is slightlyshallower than the whole-plant value. Standstructure Figure2 showsfrequency histogramns of dbh and of radial growthrates for pin cherry

40 (a) 4-year-old c 30 _i I= 0 50 cm 220 20 t 1l__ p= 16.0 m- ~~~~~~~~n= 241

E (b) - E 100 3 5 -01 n=31

6-year-old

y 20 (c) I=0 44cm

140LUiI.IJJJLI.h.M l.n= 145 EfI o F Cd) 1i1|n= 326m 7- E 1[j yE 5 I- .IIIn=26

14-year-old I=0*80cm 20 Ce) p= 1-2 m-2 O 158 Vc n=

U- 0' E ( f)

0 2 4 6 8 10 Stem diameter(cm) FIG. 2. Frequencydistributions of stemdiameter ((a), (c) and (e)) and mean radial growthper annumas a functionof diameter((b), (d) and (f)) forthe 4-, 6- and 14-year-oldstands of Prunus pensylvanica.'I', the intervalof stemdiameter between successive bars in the histogram,was determinedby dividing the range of diametersobserved into twelve equal intervals,except in the 4-year-oldstand where this was not possible.'p' indicatesmean density. Radial growthrefers to thelast season's radial growthincrement. 606 Standstructure of treesduring thinning

6-year-old

c 10 II II=0-47m 10(a) 1 1| p 36m2 IlIllI m n=145

g 150 -(b) 100 n=26 r 50- 0

14-year-old

I=0*56m

sPF 1I im I .1iiI11I2 ii 0 10 m 8

FIG. 3. Frequencydistributions of treeheight ((a) and (c)) and meanheight growth per annum as a functionof height((b) and (d)) forthe 6- and 14-year-oldstands of Prunuspensylvanica. 'I', the intervalof treeheight between successive bars in thehistogram, was determinedby dividingthe rangeof heightsobserved into twelveequal intervals.'p' indicatesmean density.Height growth refersto thelast season's growthof theleader.

populationsat 4, 6, and 14 years;Fig. 3 showsfrequency histogramns of treeheight and of heightgrowth rates for the 6- and 14-year-oldpin cherrystands (height data are not available for the 4-year-oldstand). For an age sequence of balsamnfir populations, frequencydistributions of sterndiameters are givenin Fig. 4, and of wholeplant weights and log whole plant weightsin Fig. 5 (a) and (b) respectively.The data in Figs 2 and 4 corroboratethe frequentclaim that the diamneterdistribution is highlyskewed during thinning(Koyama & Kira 1956;White & Harper1970; Ford 1975),but show in addition a consistentdecrease in skewnessas thinningproceeds. The data in Figs 2-4 also stronglysupport Ford's (1975) claim that size-frequency distributionsof even-agedstands are bimodalwhen about twelveintervals are used. Use of fewer intervalsobscures the bimodality,as Ford predicts.There is consistent movementof thetrough toward larger sizes as one goes fromyounger to olderstands of fir(Fig. 4).

DISCUSSION Standstructure In any studyof thedynamics of theself-thinning process it is logicalto examinethe size structureof thepopulation. In a populationundergoing thinning it is generallyassumed thatit is chieflythe smallestplants thatdie duringany giventimne interval (Yoda et al. 1963; White& Harper 1970; Bazzaz & Harper 1976); Ford (1975) showedconclusively that thiswas the case in a spruce plantation.The structureof the population is also affectedby differencesin growthrates of plants of various sizes. The importanceof C. L. MOHLER, P. L. MARKS AND D. G. SPRUGEL 607

30- 3-yr I=0-20cm 20- p = 115 m-2 n = 280 I0- 0

20- 6-yr I = 0 28 cm p = 12-7m-2 10- n = 254

20 10-yr I = 0-46 cm III p = 12.9 m-2 io1IIi0 n = 259

S I0-il- 19-yr I = 0 79 cm liii p = 8 0 m2 l0_0 n = 159 &L 0 20I 35-yr I 1I73= cm

0 ~ ~~~~~~~~~~~~~~~~~~~ = 0 .9 m - 2

20- 54-yr I = 13 cm _ p = 0-6 m-2 10-j n = 59

30-19-yr I = 1H32 cm 20-I p =0-2m 2 10- , I , I-lal L n=22

0 5 10 15 20 25 30 Trunkdiameter (cm) FIG. 4. Frequencydistributions of trunkdiameter for Abies balsamea standsarranged in an age sequence. 'I', the intervalof trunkdiameter between successive bars in the histogram,was determinedby dividing the range of diametersobserved into twelveequal intervals.'p' indicates standdensity. relativegrowth rates for an understandingof standstructure is shownespecially well by thedata of Ford (1975), as wellas by thedata of Koyamna& Kira (1956) and thedata of thepresent study. The combinedeffect on standstructure of differencesin mnortalityand in relativegrowth rate is obscuredin theliterature by thediversity of size mneasuremnents, and by differencesin timneallowed forobservations relative to thelongevity of a species. Accordingly,the dynamic development of standstructure is now discussedin detail. Ignoringfor the moment the question of bimnodality,we proposethat the structure of a dense,pure stand of plantsfollows a developmnentthrough timne simnilar to thatshown in Fig. 6. Immediatelyafter germnination a bell-shaped distribution of plantweights is to be expectedsince seed weightsare oftenapproximnately normnally distributed (Koyamna & Kira 1956; Black 1957; Obeid, Machin & Harper 1967), and the size of veryyoung seedlingsis largelya functionof seed size (Grimne1966). It is significantthat weight distributionsat firstharvest of Raphanussativus (White & Harper 1970) and Tagetes 608 Standstructure of treesduring thinning

80 (a) (b) 80

60 60

40 3-yr p=ll.5m 2 40 n = 280 20 20 o 0 60 60 10-yr p =12.9 m-2 40 nn=259 40

20 - 20

0 60 -6 ~~~~~~~~~~0

40 19-yr p=8.0 m 2 40 n = 159 20 20

ol,, Iin.an.aranrr. j ..jLIIL .tI, 01

20- ~35-yr p=0_9M2 -2

2:0 I L54-yr p=0-6 M-2 -420

59-yr p =0*22mM1 2

0 24 6 810 12 2 46 810 12 Weight Log weight FIG.5. Frequencydistributions ofabove-ground plant weight (a) andlog of above-ground plant weight(b) in twelveequal classesfor Abies balsamea stands arranged in an age sequence.'p' indicatesstand density. Although weight distributions for young stands appear monotonic, the log weightdistributions demonstrate that there are fewervery small individuals than small individuals. C. L. MOHLER, P. L. MARKS AND D. G. SPRUGEL 609 patula (Ford 1975) wereroughly bell-shaped; in bothcases thestands were too youngto showsigns of thinning. Since stem size is proportionalto a rootof plant weight somewhat less than the cube root (Yoda et al. 1963; McMahon 1973), then,given a symmetric distributionof plantweights, the stem-size distribution will be negativelyskewed, i.e. will show moreplants of largesize. As the stand growsthere will be a trendtoward increasing positive skewness of the weightdistribution, while the size distributionwill inove toward norinalityand then becoinepositively skewed. This willoccur even in theabsence of coinpetition, inerely as a resultof the roughly exponential nature of early plant growth (Blackinan 1919; Shinozaki

(a) (b) (c) Establishment Transition Startof thinning Late thinning

, Diameter c)C a-

Weight FIG. 6. Idealized frequencydistributions of trunkdiameter and individualplant weightfor selectedstages in thedevelopment of a stand.

& Kira 1956;cf. Lawson & Rossiter1958). For exainple,as Koyaina & Kira (1956) point out, if theinitial distribution is exactlynormnal, then exponential growth will produce a lognormnaldistribution of plant weights;Koch (1966) presentsa forinalproof of this point.Comnpetition, provided it inerely increases the variance without changing the shape ofthe relative growth rate frequency distribution, speeds developinent of a skewedweight distributionbut does not affectits fundainental lognormnality (Koyaina & Kira 1956). The trendtoward increasingpositive skewnesswill continueuntil a inaxiinuinis reachedjust priorto thepoint when the suppressed plants begin to die (Fig. 2(a), (c) and (e)). The inaxiinuinskewness of the weightdistribution must occur at the point of mnaximnumnskewness of the size distribution.That the mnaxiinumnskewness of theweight distributioncorresponds to thetiine of first thinning.is supported by figures in thepapers by White& Harper(1970) and Ford (1975). When a dense standbegins to thinthe inost suppressed plants die, and thisresults in frequencydistributions that becoine less and less skewedas thinningproceeds (cf. Figs 2 and 4; Fig. 11 of White& Harper 1970; Fig. 6 of Ford 1975; Fig. 4 of Bazzaz & Harper 1976). Continuedmortality in the smallestcategories may eventuallyproduce a size- frequencydistribution that is essentiallysymmetrical (Fig. 6(c)); thisis seenin Fig. 4 and is also supportedin theforestry literature (cf. Hough 1932;Schnur 1934; Johnson & Bell 1975). As Koch (1969) pointsout, a normnalsize distributionimnplies an approximnately lognormnalweight distribution. Thus, the positivelyskewed distributionof weights observedin olderstands (Koyamna & Kira 1956;White & Harper 1970;Ford 1975;Bazzaz & Harper 1976) can be explicitlyrelated to size distributionswhich vary through timne 610 Standstructure of treesduring thinning fromnhighly skewed to symnmnetrical.In the idealized case we havea lognormnaldistribution of weightsthrough mnost of the lifeof thepopulation; before thinning begins this is the resultof exponentialgrowth acting on an initiallynormnal weight distribution, whereas afterthinning comnmnences itresults fromn plant weight being a powerfunction of plant size. Our own data, as wellas thoseof theseveral authors cited above, do not seemnto support themnonotonic weight distributions proposed by Hozurni(Hozurni, Shinozaki & Tadaki 1968;Hozumni & Shinozaki 1970;Hozurni 1971). There is considerableevidence to supportFord's (1975) claimnthat size and weight frequencydistributions for dense, even-agedstands are bimnodalwhen at least twelve categoriesare used (White& Harper 1970; Bazzaz & Harper 1976; our Figs 2-5). As a stand ages the largersize categoriesformn a distinct subpopulation: for examnple, in the 14-year-oldstand of Prunuspensylvanicaabout 50% of thepopulation is in theoverstorey (six largestheight classes in Fig. 3(c); ninelargest diameter classes in Fig. 2(e)), and the standis stratifiedinto a fast-growingoverstorey and a slow-growingunderstorey (Figs 2(f) and 3(d)). The youngerstands also followthis pattern (Figs 2(a)-(d); 3(a) and (b)), as firstdescribed by Ford (1975). Despite the biologicalsignificance and apparentlygreat consistencywith which bimodalityoccurs in data fromdense, even-agedstands, the degreeof bimodality is usuallyquite small. Consequently, bimodality can be regardedas a secondaryfeature superimposed on themodel described above.

Weight-densityrelationships White& Harper (1970) suggestedthat plant partsas well as whole plantsfollow the - 3/2thinning 'law' of Yoda et al. (1963). It is clearfroin Tables 1 and 2, however,that plant parts of fir and cherryshow considerablevariation in the slope of theirlog weight-logdensity relationships. A good deal of thevariation in Tables 1 and 2 seeinsexplicable. In particular,several studieshave shown that foliagebioinass and leaf area index of a pure stand tend to increasein tiineonly up to a certainpoint, after which both level off (Donald 1961; Satoo 1970; Sprugel1974; Marks 1974). In naturallyregenerating forest stands in theeastern U.S.A. thisplateau is reachedearly, often only a fewyears after seedling establishinent or release.Once such a foliageplateau is reachedthe relationship between foliage bioinass and sterndensity is givenby: Wf=Cfp (2) whereWf is theinean weight of foliageper plant, Cf is a species-specificconstant, and p is thedensity of survivingindividuals. Since totalfoliage weight for the standis constant, mneanfoliage weight per plant mnust be inverselyproportional to thenumnber of plants per unit area. The slope of -108 for leaves of pin cherryclosely approximnatesto the theoreticalvalue of - 1 0; thefoliage value forfir, - I01, inatchesalmnost exactly. Since a relativelyconstant proportion of balsam firneedles is new foliage(Sprugel 1974), the slope of thelog mneanweight-log density relationship for new foliage is also veryclose to - 10 (Table 2). Consideringthat current foliage is supportedon currenttwigs, the correspondence of theslopes for these two tissuesin balsamnfir is to be expected.The highvalue (i.e. close to zero) forcurrent twigs on pin cherryseemns anomnalous but can probablybe viewedas a consequenceof short-shootproduction. As a pin cherryages, an ever-increasingpropor- tionof thecurrent twigs are shortshoots with compressed internodes. Thus, theamount C. L. MOHLER, P. L. MARKS AND D. G. SPRUGEL 611 of twigrequired to supporta givenamount of leaves probablydecreases with time, and thiswould generatea shallowerslope of thelog mean weight-logdensity relationship. If therelation between whole plant weight and densityfollows eqn (1), thenthe weight of woody tissuesduring self-thinning can be foundby subtractingeqn (2) fromeqn (1): w. = p- ,(Cp,+ l - C) (3) whereww is the inean weightof woody tissuesper plant and the othersymbols are as above. This equation suggeststhat the logarithmof weightof woody tissuesbears a curvilinearrelationship to the logarithmnof density,with the curvilinearitymore pro- nouncedat higherdensity and forspecies with more negative values of theexponent k. Both linear regressionand PCA impose a straightline on the data regardlessof curvilinearity,and, since theslope forfoliage is shallowerthan that for whole plants, it followsthat the slope forat least one major woody tissuemust be substantiallysteeper thanthat for whole plants. The thinningexponents for both branches and trunksof pin cherryare greaterthan thewhole plantexponent; for balsam firit is only the slope for trunksthat is steeperthan the slope forwhole plants.

Allometry Sincethe weights of whole plants and foliageshow a regularrelationship to density,it is possible to show theirrelationship to one another,and thus to derivean allometric relationshipfor annual tissues.Kira et al. (1956) showed the relationshipbetween thinningfor plant parts and thelaw of allometry.If w = Cpr (4) and w= Cpk, then: W =W/kC -rk(5) wherew, and C, are themean weight per plant and thinningconstant respectively of part i, r is theexponent for part i, and theother symbols are as in eqn (1). Equation (5) can be reducedto Huxley's (1924) allometricequation: w,=H,w'1, whereH,=C,C-,/k and is a constantfor part i, and h = r/k(Kira et al. 1956). In contrastto whatwe and othershave shownfor tree foliage, White & Harper(1970) have shownthat roots and above-groundtissues of rape and radishboth follow the - 3/2 power'law' duringself-thinning. Slopes of theweight-density relationship for both roots (otherthan fine roots) and totalabove-ground tissues of pin cherry are also close to - 15 (Table 1). Ifthe exponent r ineqn (4) has thesame value fortwo plant parts, i andj, thenit followsthat w,/w, = C/C,, whichsays thatthe weight ratios of partsare constantduring thinning.For severalwoody speciesit has been shownthat root:shoot weight ratios are constantover a wide rangeof plantsize (Figs 4 and 5 of Whittaker& Marks 1975). It is possiblethat when an individualplant increases its shoot size a proportionateincrease in root structureis requiredfor support. There are manyinteresting alloinetric relationships apparently little explored in the literature.Table 3 shows two such relationshipspertinent to self-thinningin natural standsof treesin theU.S.A. The estimatedalloinetric exponents for the height-diameter relationshipin eightbroad-leaved tree species were found to be in therange of c. 1/2to 2/3 (Table 3(a)); thevalues of theexponent for two conifers, Abies balsamea and Picea rubens, weresomewhat larger, but stillsubstantially less than 1-0. The values forthe allometric exponentsrelating crown radius to trunkdiameter for five of thesespecies varied from 0 62 to 0 71 (Table 3(b)), withan apparentmode of c. 2/3. 612 Standstructure of treesduring thinning

TABLE 3. (a) Slopes of thelog heightv. log trunkdiameter allometric relationship for various species, and (b) slopes of the log crown spread v. log trunkdiameter allometricrelationship for various species; for both (a) and (b) %EV is a measureof thevariation accounted for by the first principal axis (see text)

Principal components Locality slope %EV n in U.S.A. (a) Tree height-trunkdiameter* Acersaccharum 0 521 98 1 21 New Hampshire Betulaalleghaniensis 0 480 97 1 21 New Hampshire Picea rubens 0 753 99.0 15 New Hampshire Fagus grandifolia 0 549 97 0 21 New Hampshire F. grandifolia 0 659 98 0 32 SoutheastTexas Prunuspensylvanica 0 487 97.6 29 New Hampshire Abiesbalsamea 0 842 98.3 47 New York Populusdeltoides 0 690 88 1 21 Iowa Quercusnigra 0 588 98 2 28 SoutheastTexas Q. alba 0 682 96 9 31 SoutheastTexas (b) Crownspread-trunk diametert Acersaccharum 0 714 99 4 19 New Hampshire Betulaalleghaniensis 0 636 99 3 21 New Hampshire Fagus grandifolia 0 620 99 3 21 New Hampshire Prunuspensylvanica 0 671 93 1 29 New Hampshire Abiesbalsamea 0 652 97 3 47 New York

* Trunkdiameter was measuredat breastheight, except in thecase of Abies balsamea wherediameter was measuredat a heightof 25 cm. t Mean lengthof the longesttwo brancheswas used as an estiinateof crown spread, exceptin thecase of A. balsameawhere crown spread was calculatedfroin the area of the crownprojection.

Relationof plant allometry to meanweight and density In derivingthe -3/2 power 'law' Yoda et al. (1963) assumed that(1) plant shape is constantduring self-thinning, and (2) thecrowns of adjacent plantsjust touchwithout overlapthroughout the thinning process. White & Harper(1970) presenta derivationin whichit is not necessaryto assume a single-layeredcanopy structure(second assumption); thefirst assumption of Yoda et al. is not relaxedin theirderivation, however, and requiresthat all alloinetricexponents relating one lineardimension to anotherbe 1 0. Two consequencesof thisassumption are necessaryfor the derivation of thepower law. The firstis thatthe volume (and thereforeweight) of planttissue will be proportionalto thecube of some linearplant dimension (e.g. trunkdiameter); the second is thatcanopy spread will be linearlyrelated to the same plant dimension.The two dimensionswhich togetherbest estimatetree weight are heightand trunkdiaimeter. Contrary to thefirst assumptionof Yoda et al. (1963), McMahon (1973) argueson mechanicalgrounds that theallometric exponent relating height to basal diainetershould be 2/3.Table 3 shows thaton thewhole theallometric exponent relating height to diaineteris farless than 1 0, and in fivecases itis substantiallyless than2/3. Perhaps more importantly with regard to self-thinning,the exponent for balsam firis muchcloser to 1 0 thanis thatfor pin cherry. This resultis at variancewith the fact that the slope of thelog inean weight-logdensity relationshipfor pin cherryclosely approximates to the theoretically-expectedvalue of - 3/2,whereas the slope forfir is muchshallower. Canopy spreadlikewise does notfollow theexpected relationship to stemdiameter; for the five species tested the exponent relating Standstructure of treesduring thinning 613 a measureof canopy radius to stemdiameter is in all cases substantiallyless than 1 0 (Table 3). The allometricrelations in Table 3 can be used as an alternativeto thefirst assumption of Yoda et al., to derivean equation relatingmean weightto density.If H a D9 is the allometricrelationship between plant height (H) and stemdiameter (D), whereq is the species-specificallometric exponent, and assumingw a HD2, then

waD2 +q (6)

Like White& Harper(1970), we assumethat R a p 1/2, whereR is canopyradius and p is density.If therelationship between canopy radius (R) and stemdiameter (D) is takenas R a Ds, wheres is theallometric exponent, then p aD-2s (7) Combiningeqns (6) and (7) givesa meanweight-density function of

W=Cp-(2 +q)/2s. (8) The exponentof eqn (8) willhave a largenegative value forlarge q and smalls, and a smallnegative value forsmall q and larges. Based on theextreme values in Table 3, the exponentranges from - 229 to - 175; values forpin cherryand balsam firare - 185 and -2 17. Not onlyare thesevalues a poor matchto theproposed - 15, but theyare at variancewith the empirically observed weight-density exponents (Tables 1 and 2). This apparentcontradiction may be resolvedby noting that stand structure changes during the courseof thinning.We suggestthat the exponent of theinean weight-density relationship is deterininedby mutual adjustmentof plant allometryand stand structureduring self-thinning.

ACKNOWLEDGMENTS We thankR. E. Furnasand R. K. Peetfor helpful comments on themanuscript and R. H. Whittakerfor providingoriginal data. The originalwork on Prunuspensylvanica and Abies balsamea was part of the Hubbard Brook EcosystemStudy supportedby the National Science Foundation. Preparationof the presentpaper was inade possible by McIntyre-StennisGrant No. 183-7551to P.L.M.

REFERENCES Bazzaz, F. A. & Harper,J. L. (1976). Relationshipbetween plant weight and numbersin mixedpopulations of Sinapisalba L. Rabenh. and Lepidiumsativum L. Journalof AppliedEcology, 13, 211-6. Berlyn,G. P. (1962). Some size and shape relationshipsbetween tree sterns and crowns.Ioua State Journalo] Science,37, 7-15. Black,J. N. (1957). The earlyvegetative growth of threestrains of subterraneanclover ( Trifoliumsubterraneum L.) in relationto size of seed. AustralianJournal of AgriculturalResearch, 8, 1-14. Black, J. N. (1958). Competitionbetween plants of differentinitial seed sizes in swardsof subterraneanclover (Trifoliumsubterraneum L.) withparticular reference to leafarea and lightmicroclimate. Australian Journal of AgriculturalResearch, 9, 299-318. Black, J. N. & Wilkinson,G. N. (1963). The role of tiineof emergencein deterininingthe growth of individual plantsin swardsof subterraneanclover. Australian Journal of Agricultural Research, 14, 628-38. Blackman,V. H. (1919). The compoundinterest law and plantgrowth. Annals of Botany,33, 353-60. Donald, C. M. (1961). Competitionfor lightin crops and pastures.Mechanisms in Biological Competition. Symposiaof the Societyfor ExperimentalBiology, Volume XV (Ed. by F. L. Milthorpe),pp. 282-313. AcademicPress, New York. Ford,E. D. (1975). Competitionand standstructure in someeven-aged plant inonocultures. Journalof Ecology, 63,311-33. 614 Standstructure of treesduring thinning

Gauch,H. G. (1973). The CornellEcology Programs Series. Bulletin of theEcological Society of America, 54(3), 10-11. Grime,J. P. (1966). Shade avoidance and shade tolerancein floweringplants. Light as an EcologicalFactor. Symposiaof the British Ecological Society, Volume 6 (Ed. byR. Bainbridge,G. C. Evans and 0. Rackham), pp. 187-207.Blackwell Scientific Publications, Oxford. Hough,A. F. (1932). Soine diaIneter distributionsin foreststands of northwesternPennsylvania. Journal of Forestry,30, 933-43. Hozumi,K. (1971). Studieson thefrequency distribution of theweight of individualtrees in a foreststand. III. A beta-typedistribution. Japanese Journal of Ecology,21, 152-67. Hozumi,K. & Shinozaki,K. (1970). Studieson thefrequency distribution of theweight of individualtrees in a foreststand. II. Exponentialdistribution. Japanese Journal of Ecology,20, 1-9. Hozumi,K., Shinozaki,K. & Tadaki,Y. (1968). Studieson thefrequency distribution of theweight of individual treesin a foreststand. I. A new approachtoward the analysis of thedistribution function and the -3/2th powerdistribution. Japanese Journal of Ecology, 18, 10-20. Huxley,J. S. (1924). Constantdifferential growth-ratios and theirsignificance. Nature, London, 114, 895-6. Johnson,F. L. & Bell,D. T. (1975). Size-classstructure of threestreamside forests. American Journal of Botany, 62, 8 1-5. Kira,T., Ogawa, H., Hozumi,K., Koyama,H. & Yoda, K. (1956). Intraspecificcompetition among higher plants. V. Supplementarynotes on theC-D effect.Journal of theInstitute of Polytechnics,Osaka CityUniversity, SeriesD, Biology,7, 1-14. Koch, A. L. (1966). The logarithinin biology.I. Mechanismsgenerating the log-normal distribution exactly. Journalof TheoreticalBiology, 12, 276-90. Koch,A. L. (1969). The logarithmin biology.II. Distributionssimulating the log-norinal. Journalof Theoretical Biology,23, 251-68. Koyama,H. & Kira, T. (1956). Intraspecificcompetition among higher plants. VIII. Frequencydistribution of individualplant weight as affectedby the interaction between plants. Journal of the Institute of Polytechnics, Osaka CityUniversity, Series D, Biology,7, 73-94. Lawson,E. H. & Rossiter,R. C. (1958). The influenceof seed size and seedingrate on thegrowth of two strains of subterraneanclover (Trifolium subterraneum L.). AustralianJournal of AgriculturalResearch, 9, 286-98. Marks,P. L. (1974). The roleof pincherry (Prunus pensylvanica L.) in themaintenance of stabilityin northern hardwoodecosystems. Ecological Monographs, 44, 73-88. McMahon,T. (1973). Size and shape in biology.Science, New York,179, 1201-4. Obeid,M., Machin,D. & Harper,J. L. (1967). Influenceof density on plantto plantvariation in fibre flax, Linum usitatissimumL. CropScience, 7, 471-3. Satoo, T. (1970). A synthesisof studiesby theharvest method: primary production relations in thetemperate deciduousforests of Japan. Analysisof TemperateForest Ecosystems (Ed. by D. E. Reichle),pp. 55-72. Springer,New York. Schlesinger,W. H. (1976). Biogeochemicallimits on twolevels of plant communityorganization in thecypress forestof OkefenokeeSwamp. Ph.D. thesis,Cornell University, Ithaca, New York. Schnur,G. L. (1934). Diameter distributionsfor old-fieldloblolly pine stands in Maryland. Journalof AgriculturalResearch, 49, 731-43. Shinozaki,K. & Kira,T. (1956). Intraspecificcompetition among higher plants. VII. Logistictheory of theC-D effect.Journal of theInstitute of Polytechnics,Osaka CityUniversity, Series D, Biology,7, 35-72. Snedecor,G. W. & Cochran,W. G. (1967). StatisticalMethods. Iowa State UniversityPress, Ames, Iowa. Sokal, R. R. & Rohlf,F. J. (1969). Biometry.Freeman, San Francisco. Sprugel,D. G. (1974). Vegetationdynamics of wave-regeneratedhigh-altitude fir forest. Ph.D. thesis,Yale University,New Haven, Connecticut. Sprugel,D. G. (1976). Dynamic structureof wave-regeneratedAbies balsamea forestsin the north-eastern Unitedstates. Journal of Ecology,64, 889-911. White,J. & Harper,J. L. (1970). Correlatedchanges in plantsize and nuinberin plantpopulations. Journal of Ecology,58, 467-85. Whittaker,R. H., Bormann,F. H., Likens,G. E. & Siccama,T. G. (1974). The Hubbard Brookecosystein study: forestbiomass and production.Ecological Monographs, 44, 233-54. Whittaker,R. H. & Marks,P. L. (1975). Methodsof assessingterrestrial productivity. Primary Productivity of theBiosphere (Ed. by H. Liethand R. H. Whittaker),pp. 55-118. Springer,New York. Whittaker,R. H. & Woodwell,G. M. (1968). Dimensionand productionrelations of treesand shrubsin the BrookhavenForest, New York. Journalof Ecology,56, 1-25. Yoda, K., Kira,T., Ogawa, H. & Hozumi,H. (1963). Self-thinningin overcrowdedpure stands under cultivated and naturalconditions. Journal of the Institute of Polytechnics, Osaka CityUniversity, Series D, Biology,14, 107-29. (Received15 November1977)