ForestSaence,
Vol 39, No 1, pp 7-27
Evaluationof the Size-Density
Relationships
for Pure Red Alder and
Douglas-FirStands
KLAUSJ. PUEVr•U•N
DAVID W. HANN
DAVID E. HIBBS
ABSTRACT.
Size-densitytrajectorieswere developedfor pure red alder (Alnusrubra Bong)and
Douglas-fir(Pseudotsuga
menziesii[Mirb.] Franco)standswith quadraticmeandiameter
of the standasthe tree-sizevariable.The resultingself-thinning
or maximumsize-density
line for red alderhada steeperslope(-0.64) thanthat for Douglas-fir(-0.52). The
assumption
of a common
slopeforallspecies
is thereforenotsupported
byourstudy.For
red alder,the shapeof the size-density
trajectoryandthe elevationof the maximumline
were notinfluenced
by initialdensityor standorigin.RedalderandDouglas-fir
mortality
startedat a relativedensityof 44% and58%, respectively.
FOR.SCL39(1):7-27.
AI)•)ITIO• KEYWOPd)S.Self-thinning,
size-density
trajectory,StandDensityIndex.
ODA
ETAL.
(1963)
found
that
maximum
average
plant
size
foragiven
plant
density
inevenagedmonospecific
plantpopulations
canbecharacterized
by self-thinning
line, expressedmathematically
as
log(w) = a1 + a2 * log(N)
(1)
where
w = ma3dmum
averageplant size,
N = plantdensity,
aI = the intercept,and
a2 = the slopeparameter.
Whenthe sizevariableis volumeor weight,andthe densityvariableis number
of plantsper unitarea, the self-thinning
linehasa slopeof - 1.5 for a varietyof
species(for review see White 1980), and is knownas the -3/2 power role.
Althoughthe universalityof the parametervaluesof the self-thinning
line (Zeide
1988, Barreto 1989) and the method used to establishthis line (Wetter et al.
1985, Wetter1987a,1987b,havebeenquestioned,
the self-thinning
linehasbeen
usedsuccessfully
in a numberof forestryapplications
(Reineke1933, Drew and
Fiewelling1979, Curtis 1982, Long 1985, Hyink et al. 1988, Hester et al. 1989,
Smith1989).Most applications
assumethatthe slopeof the self-thinning
lineis
basedon either the -3/2 powerrule (Yodaet al. 1963), wherew is definedas
F•BRUAR¾1993/7
biomass,or a Reineke(1933)slopeof -0.62, wherew is quadratic
meandiameter.
While the self-thinning
line expressesthe upperboundaryof all possiblesizedensitycombinations
(Yodaet al. 1963), the size-densityrelationship
focuseson
the time-trajectoryof individual
populations.
It coversthe fullspectrumof stand
development
fromphasesof growthwithoutmortalityover a curvedapproach
to
a linear phase(White 1980, 1981) (Figure 1), whichis commonlylabeledthe
"maximumsize-densityline."
Smith and Hann (1984) developeda size-densityequationusingdata from
standsthat had reached the maximumsize-densityline and standsthat were
self-thinning
butstillbelowthemaximum
line.Thisequation,therefore,describes
not only the locationof the maximumline but alsothe approachto maximum
size-densityandthusavoidsthe problemof subjectively
selectingstandsthat are
on the maximumline (Weller et al. 1985). By usinga regressionequation,the
analysisyieldsthe size-densitytrajectorythat averagestandswouldfollow.Individualstandswill fallaboveor belowthisline (seealsoFigures2 and3). Whilethis
"averagemaximum"needsto be distinguished
froman "absolutemaximum"(putHIGH
HIGH
%%
%
•
SIZE-DENSiTY
LINE
%%%%
o
HIGH
LOG (numberof trees)
FIGURE
1. Possible
size-density
trajectories
ofstandsundergoing
serf-thinning.
A. Standsmovealong
the maximumsize-densityline (ShapeA). B. Standsapproacha line connecting
the maximum
size-density
pointsandthenfallbelow(ShapeB).
8/FOgg.srsc•cE
ting a line aboveall datapoints)and an "upperskinmaximum"(puttinga line
throughthe upper5% of the datapoints),it was chosenbecauseit allowsinvestigationof our objectives.In addition,the size-densityrelationship
can be expandedinto a growthmodelby includinga growthor mortalityequation(e.g.,
Smith and Harm 1986).
In a theoreticalpaper,McfaddenandOliver(1988)presenttwo possiblepatterns of how standsmightapproachthe maximumsize-density
line (figure 1).
The figureshavebeenmodifiedto showa gradualapproach
rather thanan abrupt
transitionto the maximumsize-densityline. figure 1A showsstandsasymptoticallyapproaching
a commonline, whichis alsothe maximumsize-densityline.
This size-densityrelationship
with shapeA representsMcfadden and Oliver's
(1988) type I and II shapes.figure lB showsstandsfirst approaching
a line
representing
maximumpossiblesize-density
pointsandthen departingfrom the
boundaryto approach
parallellines,the interceptsof whichare negativelycorrelatedwithinitialdensity.This size-density
relationship
withshapeB representsa
Type III shapeas definedby Mcfadden andOliver (1988).
An importantissueconnected
with the development
of size-density
trajectories
andtheir application
to management
guidesis the onsetof competition-induced
mortality.In figure 1, this is the pointwhere the trajectoriesleave the vertical
andbeginto curveleft as they approach
the maximumsize-density
line.
We chosestandsof red alder(AlnusrubraBong.)andDouglas-fir(Pseudotsuga
rne•iesii [Mirb.] franco)astest systemsbecause
thosespeciesare sympatric
in
the PacificNorthwestandare of economic
importance.Red alderis a nitrogenfixingpioneer,andDouglas-fir
is a matureforestdominant
(franklinandDymess
1988). Both are commonassociates
in early successional
stages,especiallyin
forestplantations.
Our overallgoalwas to investigatethe size-density
relationships
of red alder
andDouglas-firby developing
analytical
modelsof their size-density
relationships
usingquadraticmeandiameter(QMD) of the standas the sizevariable.To meet
this goal, our specificobjectiveswere:
1. to determineif the elevationof the size-density
relationships
for purered alderstands
was inverselyrelatedto initialdensity6.e., to comparesize-density
rehtionships
in
Figures 1A and lB);
2. to examinewhetherthe size-density
relationships
of red alderstandswith different
initialdensitiesexhibitthe samecurvature;
3. to examinewhetherdensity-dependent
mortalityin red alder standsstartsat a constantrelativedensityregardlessof initialdensity;
4. to comparesize-density
relationships
for red alderin naturalstandsandin plantations;
and
5. to comparesize-densityrehtionshipsfor red alderandDouglas-firstands.
All data were from fixed-areaplotswhichhad no signsof past disturbanceand
exhibited
mortalityduringthemeasurement
periods.A standwasdefinedaspure
red alderor pureDouglas-fir
if at least80%of its basalareawasin that spedes
(Worthingtonet al. 1960, King 1966).
The red alderdataset consisted
of two subsetsof even-agedplots.Nineplots
(80 individual
measurements)
were froma spacing
studyin a plantation
locatedin
FEBRUARY1993/9
the CoastRangeof northwestOregonand15 plots(81 individual
measurements)
were from natural standsin western Oregonand Washingtonand southwest
BritishColumbia.
Datafromthe spacing
studywerekeptseparatebecause
these
plotshadinformation
abouteffectiveplanting
density(excluding
planting
mortality) thatwasneededto assessthe roleof initialsize-density
effects(Objectives
1
through3).
The datafor Douglas-fir
standswerefromthe controlplotsontheinstallations
of the RegionalFertilizerandNutritionalResearchProgram(Opalach
1989),the
Level-of-Growing-Stock
study(CurtisandMarshall1986), anda studyby J.E.
King (1973). To ensuregeographicsimilaritybetweendata sets for the two
speciesonlyplotslocatedin westernOregon,westernWashington
or southwest
BritishColumbia
were included.In addition,plotswith an annualmortalityrate
>6.16% (two standarddeviationsabovethe meanmortalityrate) for any measurementperiodwere assumedto haveexperienceddisturbance-related
mortality. FourteenDouglas-fir
plotsweredropped,
leaving58. All red alderplotshad
<6% averageannualmortalityrate. A detaileddescriptionof the data set is
presentedin Table 1.
The datawere fromplotswithdifferentsizes:<0.05 ha (N = 41); 0.05-0.1 ha
(30); 0.1-0.15 ha (4); 0.15-0.2 ha (1); >0.2 ha (6). In smallplots,estimatesof
standcharacteristics
havea highervariancebecause
growthandmortalitywithin
the plotare influenced
by adjacent,unknownstandconditions
(Smith1975, Curtis
1983). To adjustfor the patternof increasedvariance(Zumrawi1990) we used
plotsizeasweightin the regression
analysis.
StandQMD wasselectedto representmeantree sizebecauseit canbe measuredaccurately
andeasilyandis
closelyrelated to crown size (Brigleb 1952, Smith 1968) and tree biomass
(Hughes1971).Relativedensityvalueswere calculated
by dividingactualstand
densityby predictedmaximumstanddensityfromthe particularequation's
maximumsize-density
line.
The equationformsin our studywere nonlinear.We usedthe Marquardtalgorithm(Marquardt1963)andthe DUD algorithm(RalstonandJennrich1978)
of the nonlinearcomponent
of the SAS statisticalpackage(SAS Institute,Inc.
1987)to estimatethe parameters.
Conchsions
from previousanalyseswere neededfor succeeding
ones,so we
workedsequentially,
startingwith our first objective.The sequentialanalysis
leadsto concernaboutlossof degreesof freedomfor hypothesis
testing.A loss
of an unknownamountof degreesof freedomresultsin testingwith a higher
a-level than specified,whichcanlead to falselydeclaringmodelssignificantly
different.Whilethis hasto be considered
in the analysis,there are not enough
dataavailableto performmodelformdevelopment
andparameterestimation
with
independentdatasets.
A secondconcernin our analysisis the use of repeatedmeasures,whichcan
leadto a lackof independence
in the modelerrors.A highpositivecorrelationof
errors hasbeen shownto lead to underestimation
of the standarderrors (Seber
andWild1989).Whilethe numberof repeatedmeasurements
of individual
plots
wasinsufficient
to determinea correlation
structure(thelag-residual
plotsshowed
no trends),the highnumberof independent
plotsshouldbreakup correlation
betweenerror terms (SeberandWild 1989). In addition,a longertime period
between measurements has been shown to result in lower correlation between
the errors(Germer1985,SeberandWild1989).Thesepointsleadusto conchde
10/FORKs'rsc•cE
that the correlationbetweenmodelerrors anda potentialunderestimation
of the
standarderrors shouldbe consideredwhen evaluatingthe results,but that the
autocorrelation
is not highenoughto invalidatehypothesis
testing.
The size-densitytrajectorywas developedfrom log-logtransformeddata(Smith
andHaan 1984) andwas assumedto consistof two parts:a linearportion,which
is the maximumsize-densityor self-thinning
line [Equation(1)], anda nonlinear
portionin whichthe trajectoryapproaches
the maximumline asymptotically.
Wheninitialdensityis known,the size-density
trajectoryis modeledas the differenceof a linearandnegativeexponentialfunctionusingthe equationform of
Smith and Harm (1984):
(2)
Yi = al + a2 *Xi - al *a4* exp(a3* (Xo -X i))
where
logarithmof quadraticmeandiameter(cm)
logarithmof density(tpha)
logarithmof initialdensity(tpha)
//1 =
interceptof maximumsize-densityline
line
//2 ---- slopeof maximumsize-density
//3 =
shapeparameterfor size-density
trajectory
a4 =
adjustmentfor relativedensitywhenmortalitystarts.
i=
measurementidentifiers(1 = initial, 2 = second,etc.)
Equation(2) representsa size-densityrelationship
with shapeA (Figure 1A).
Initialvaluesfor the al anda2 parameters
werecalculated
by fittingEquation
(1)
to the size-densitycombinations
of the normalyieldtablefor red alder(Worthington et al. 1960)(a• = 7.3, a2 = -0.62).
TABLE
1.
Data description:
Mean valuesfor purered alderandpure Douglas-firstands
(rangesin parenthesis).
Red aideft
Numberof plots
Number of measurements
Tress/ha
Quadraticmean diameter(cm)
24
161
2550 (420-11030)
15 (3-32)
Douglas-tiff
58
282
1560 (2904660)
24 (9-57)
Age at first measurement(yr)
Measurementperiod(yr)
17 (1-34)
16 (4-30)
30 (6-43)
15 (6-35)
Siteindex(m)b
31 (19-35)
37 (23-45)
a Pure standsare definedas having80 to 100%basalarea in red alderor Douglas-fir.
bRedalderandDouglas-fir
siteindices
arebasedonWorthington
et al. (1960)andKing(1966),
respectively.
FEBRUARY1993/ 11
To comparethe fits of size-densityrelationships
with shapesA andB (Figure
1), an equationfor shapeB was formulatedby varyingthe interceptterm with
initialdensity:
Yi = al + bl * Xo + a2 * Xi - al * a4 * exp(a3* (Xo - Xi))
(3)
where the parametersare as in Equation(2), and
b1 = interceptadjustment
parameter.
The initialparametervaluesfor a2 and b• were set at -0.5 and -0.12,
respectively.These valueswere chosento reflecta standwhichfirst approaches
a common
maximum
size-density
linewitha Reineke(1933)slopeof - 0.62 (a2 +
bl) andthendivergesfromthatlineto asymptotically
approach
a linerepresenting
a basalarea that wouldremainunchanged.The particularvaluefor constantbasal
areawouldbe inverselyrelatedto the stand'sinitialdensity.
The results of fitting Equations(2) and (3) to the spacingstudy data are
presentedin lines A and B in Table 2. The mean squareerror (0.0004) for
Equation2, representing
shapeA, waslessthanthatfor Equation3 (0.0005).The
F-test comparing
the fit of thoseequations
wassignificant,
indicating
thatincluding
the relationship
betweeninitialdensityandthe intercept(bx)reducesmodelfit (P
< 0.05). Further analysiswas thereforebasedon the reducedmodelform which
assumesa size-densityrelationship
with shapeA.
Equation(2) assumesthat the approachto the maximumsize-densityline is
commonto all standsand does not vary with initialstandconditions.A unique
approach
for eachstandcanbe achieved
by allowing
a3anda4 to varyforindividual
plots:
Yii = al + a2* Xo'- al * % * exp(a3/*(X0•'- X0))
(4)
where parametersare as in Equation(2) and
j = plot identifier,j = 1, 2...
N.
By collapsing
the a3 parameterandfittingthe following
equation,we cantest
the differences
betweentheresidualsumof squares
for thefull(individual
a3) and
the reduced(commona3) equation:
Yo'= al + a2* Xo'- al * a4j* exp(a3
* (Xo•- XO.))
(5)
ComparingEquations(4) and (5) determinesif the shapeof the asymptotic
approach
to the maximumsize-density
lineis the samefor all plots.
The parameterestimates
forthefullequation
[Equation
(4)] are shownin Table
3, andthe estimatesfor the reducedequation[Equation(5)] are shownin Table
4. An F-test (Cunia1973) on the differencein residualsumof squaresbetween
the two equations
wasnotsignificantly
differentatP < 0.05 (linesC andD, Table
2) indicatingthat the size-densitytrajectoryof standswith a widerangeof initial
densitieshad a commonshape.
12/FOP,Ks7S.
C•
To test if the onsetof density-dependent
mortalityis associated
with a constant
relativedensity,we usedEquations(5) and(2) as the full andreducedequations,
respectively.Resultingparameterestimatesfor the full equationare given in
Table 4 andfor the reducedequationin line F, Table 2. An F-test on the differences between these equations(lines E and F, Table 2) was not significant,
indicatingthat the line at whichinitialmortalitystartsis paranelto the maximum
size-densityline.
The precedingequationsall containa variablefor initialdensity,whichis not
usuallyknownfor naturalstands.Equation(2) was thereforerestructuredto
eliminateinitialdensityas anindependent
variable.First, the relationship
between
initialdensityandthe densityat the timeof first measurement
was expressedas:
= kj * N¾
(6)
where
N¾ = trees/ha
at firstmeasurement
ofthejth plot
No•-= density
beforeonsetof density
induced
mortality
ofthejth plot,and
k• = adjustment
factorforthejth plot
RewritingEquation(2) andsubstituting
Equation(6) into it yielded
YO'= ai + a2 * XO.- ai * a4* (Nij * kj/No.)
a3i
(7)
whereN0.= number
of trees/ha
at measurement
i forthejth plot.Equation
(7)
can be rewritten
as:
Yo'= ai + a2* XO.- (ai * a4* k•-aai), (Nij/Nii)a3i
(8)
Setting
as•= a,•* ki-a3j
(9)
Yii = al + az* X0 - al * as•.
* exp(a3
i * (Xv - Xo'))
(TO)
resultsin followingequation
with parametersas in Equation(2), and
asi= adjustment
factor,and
Xli = logarithm
of density
at initialmeasurement
forthejth plot.
Theadjustment
factor(as)includes
theadjustment
fortheonsetof mortality
(a4i)andadjustment
forthedifference
between
initialdensity
anddensity
at the
timeoffirstmeasurement
(kj)foreachplot.Equation
(T0)cantherefore
beused
for analysisof standswith unknowninitialdensity.
FEBRUARY1993/ 13
14/FOe, Ks'rsc•'•CE
FEBRUARY1993/ 15
TABLE
3.
Parameterestimatesfor purered alderplantations,
usingindividual
a3 anda4
parameters.Standarderrorsin parentheses.
Parametersa• anda2 are
commonto all plots.
Parameter
Plot
a•
a2
aai
a41
1
2
3
4
5
6
6.79 (0.54)
-0.56 (0.07)
-6.55 (4.41)
- 10.60 (3.42)
-11.71 (15.22)
-2.23 (1.31)
-7.82 (1.82)
- 11.23 (0.08)
0.08 (0.012)
0.08 (0.008)
0.61 (0.471)
0.08 (0.038)
0.13 (0.022)
0.22 (0.033)
0.13 (0.028)
0.09 (0.022)
0.18 (0.034)
7
-8.89 (0.51)
8
9
-2.23 (0.12)
- 7.83 (1.82)
MSE = 0.0016.
To simplify
thisequation
areduced
formcanbeusedtotestwhether
as•andaai
in Equation(T0) are significantly
differentfor eachplot:
Yi = al + a2 * Xi - al * a5 * exp(a3* (X1 - Xi))
(TT)
with parametersas in Equation(2), and
as = adjustment
factorcommonto all plots.
A fit of Equations(T0) and(TT) to the datafor naturalred alderstands(Table
5 and line H, Table 2, respectively)indicatedthat residualsum of squaresfor
Equation(TT)wasnot significantly
differentfromthatfor Equation(T0), suggesting that shapeof the trajectorycurveandthe relativedensityat the initialmeasurementare similarfor all plotsin naturalstands.
TABLE
4.
Parameterestimatesfor purered alderplantations,
usingindividual
a4
parameters.
Standard
errorsin parentheses.
Parameters
a•, a2, anda3 are
commonto all plots.
Parameter
Plot
a•
a2
as
a4i
i
2
3
4
5
6
7
8
9
7.35 (0.35)
-0.61 (0.04)
-6.40 (0.001)
0.07 (0.011)
0.06 (o.oo9)
0.20 (0.031)
0.13 (0.056)
0.15 (0.019)
o.oo (0.0o03)
0.08 (o.o16)
0.08 (0.019)
0.13 (0.016)
MSE
= 0.0002.
16/FO•-F•rSca•qCE
TABLE
5.
Parameterestimatesfor naturallyregeneratedred alderstands,using
individual
aaanda4 parameters.Standarderrorsin parentheses.
Parametersat
anda2 are commonto all plots.
Parameter
Plot
a•
a2
aa•
a•
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
6.58 (0.35)
-0.50 (0.05)
-4.97 (9.24)
-7.12 (10.64)
-8.97 (8.53)
-5.23 (1.09)
-0.60 (21.36)
- 6.45 (10.76)
- 1.32 (7.18)
-8.32 (1.86)
-7.14 (2.71)
-2.11 (4.52)
-1.63 (8.73)
-4.33 (2.89)
-1.63 (3.69)
- 10.75 (11.04)
-4.47 (3.20)
0.034 (0.016)
0.019 (0.008)
0.013 (0.004)
0.018 (0.006)
0.036 (0.016)
0.024 (0.010)
0.025 (0.010)
0.020 (0.005)
0.010 (0.004)
0.016 (0.007)
0.018 (0.007)
0.017 (0.005)
0.015 (0.006)
0.008 (0.002)
0.016 (0.005)
MSE = 0.0005.
To examinewhetherplantations
andnaturalstandsfollowthe sametrajectory,
Equations(2) and (11) canbe combinedinto a singleequationandfitted to the
combineddataset. The followingfull equationusesplantedred alderstandsas a
baseequation(ai) andan adjustment
(bi) usingindicatorvariablesfor natural
stands:
Yi = a• + I• * b• + (a2 + I• * b2) * X i - (a•. + I• * bO* (a4 + I• * b•)
ß exp{[aa+ I1 * b3l* [(1 - 11)* Xo + I• * X 1 - Xil}
(12)
with parametersas in Equation(11),
bi = naturalstandadjustment
parameters
ontheredalderplantation
parameters
(al througha•), i = 1, 2, 3, 4 and
I1 = indicatorvariable(0 = plantation;
1 = naturalstand).
The followingreducedequation,whichis simplifiedby eliminatingthe stand
originadjustment
parametersandindicatorvariables(exceptb•), assumes
a commonsize densityrelationship
for plantations
andnaturalstands:
Yi = ax + a2 * Xi -al (a4 + Ix * b•)
ß exp{aa* [(1 - 11)* X o + 11* X 1 - Xi)}
(13)
with parametersas in Equation(12).
A comparison
betweenthe fits of Equations(12) and(13) testsif naturalstands
andplantations
havethe samesize-density
relationship,
makinganindicator
variablefor originunnecessary.
The parameterestimatesresultingfromfittingEquation (12) and the reducedform Equation(13) to the combinedplantationand
FEBRUARY1993/17
naturalstanddataset are presentedin linesI andJ, respectively,Table 2. The
residualsumof squaresof the reducedequationwas not significantly
different
fromthe residualsumof squaresof the fullequation.The development
of planted
andnaturalred alderstandscanbe representedby a commonsize-density
trajectory. For illustration
the datausedin analysisandthe maximumsize-density
line are plottedin Figure 2. Becauseof the statisticalconcerns,the confidence
intervalsof individual
parametershaveto be viewedwith caution.Thus, whilethe
slopeof - 0.64 technically
doesnot includeReineke's(1933) slope( - 0.62) in its
confidence
interval,thesevaluesare so closethat for practicaluse they canbe
assumedto be similar.The StandDensity Index (SDI) (Reineke 1933) of the
maximumsize-densityline for a standwith a QMD of 25.4 cm is 751 (304 in
Englishunits). The relativedensityat the onsetof mortalitywas 44% of the
maximumdensity.
As a representative
example,the correlation
matrixfor Equation(13) is presentedin Table 6. Whilea highcorrelation
betweenthe intercept(a0 andslope
(a2)parameters
of thelinearportioncanbe expected,thecorrelation
betweenthe
aa andas parameters
is dueto the mathematical
derivation
of as [seeEquation
(9)]. High multicollinearity
might lead to inflatedconfidenceintervals(Kmenta
1971). However, this effect wouldcounteractthe potentialunderestimation
of
confidence
intervalsdue to autocorrelation
andsequentialtesting.
SELF-THININING
LINE
NATURAL STAN DS
PLANTATIONS
30-
500
1000
TREES
Fmum• 2.
5000
10000
PER HECTARE
Plot measurements
andasymptoteof the size-densitytrajectoryfor red alder stands.
18/FORESTSCmNCE
TABLE
6.
Asymptoticcorrelationmatrixfor Equation(13) (see Table 2, G).
a2
a•
a2
aa
a4
0.97
a3
0.26
-0.1
a4
a5
0.12
-0.03
-0.25
0.25
-0.19
0.74
--0.66
The Douglas-firsize-density
trajectorycouldnotbe analyzedwiththe samescrutiny as thatfor red alderbecausedatafor initialdensitywasnot available.The
equationform for naturalred alderstands[Equation(11)] was appliedin the
analysis
of Douglas-fir
stands.The leastsquarefit resultsarepresentedin lineK,
Table2. The slopeof the maximumsize-density
linefor Douglas-fir
(- 0.52) does
not includethe slopedeterminedfor the red alder line (-0.64) or the slope
suggestedby Reineke(1933) (-0.62) in the 95% confidenceinterval. Even
accounting
for inaccuracies
in calculation
of thisconfidence
interval,the difference
betweenthe valuesis greatenoughto warrantthe conclusion
that the slopefor
Douglas-fir
is lower.At a QMD of 25.4 cm, the Douglas-fir
maximumsize-density
lineresultsin a SDI of 1196(485in Englishunits).For illustration,the datain this
analysis
andthe Douglas-fir
maximum
size-density
linearepresented
in Figure3.
To determinethe onsetof mortality,we chosethe slopeof -20 as the cutoff
pointwhere the size-density
trajectorystartsto deviatefrom vertical.This resuitedin a relativedensityof 58% of maximumfor the onsetof mortality.While
the cutoffpoint was chosensubjectively,slopesin this neighborhood
lead to
similarrelative densityvalues,becauseof the highcurvature.
The samegeneralapproachusedto analyzeobjective4 canbe usedto decide
if differentspeciesfollowthe samesize-density
trajectory.To do this, the red
alderequation
is usedasa baseequation,
andadjustments
for Douglas-fir
(ci)are
addedto every parameterin the baseequationusingindicatorvariables(12) to
formthe following
fullequation:
Yi = al + 12 * Cl + (a2 + 12* c2)* Xi - (al + 12 * Cl)
ß (a4 + I1 * as + 12 * c4)* exp{[a3+ 12 * c3]
ß [(1 -- I1) * Xo + I1 * X1 - Xi]}
(14)
with parametersas in Equation(13), and
ci = Douglas-fir
adjustment
parameters
onthe red alderparameters
(al through
a4), i = 1, 2, 3, 4, and
12 = indicatorvariable(0 = red alder,1 = Douglas-fir).
Equation(13)is alsothe appropriate
reducedequation
formfor testingwhether
Douglas-firhas a significantly
differentsize-density
trajectoryfrom red alder.
Therefore, a comparison
betweenthe fits of Equations(13) and (14) to the
combined
red alderandDouglas-fir
datasetwilldemonstrate
if alderandDouglas-
FEBRUARY1993/19
SELF-THININING
OBSERVED
LINE
STANDS
5O
E
n'
30
•
20
z
lO
500
1000
5000
10000
TREES PER HECTARE
FIGURE 3.
Plot meas•ements•d asympto• of Me s•e-density•ecto•
•r Dou•s-•
stands.
fir standshave the samesize-densityrelationship
makingindicatorvariablesfor
speciesunnecessary.
The resultingparameterestimatesfor the fullandreducedequations
are presentedin linesL andM, respectively,
Table2. A comparison
of the residualsum
of squaresof the fullandreducedequations
andthe significance
of allparameters
in the full equationshowedthat the size-densityrelationshipof both spedes
differedsignificantly.
The two-species
equationcouldnot be simplified.Spedes
indicators
andadjustments
on all red alderparameters
aspresentedin Equation
(14) were needed.The projectedsize-density
trajectoriesfor standswith different initialdensitiesare presentedin Figure4 andshowthe spedesdifferences.
The size-density
trajectoriesfor red alderandDouglas-fir
were established
using
nonlinearregression.Concernsthat the useof.datafromrepeatedmeasurements
andsequential
testingmightleadto falselydeclaring
modelssignificantly
different
seemunwarranted
because,withexception
of comparing
trajectoryshapesfor red
alderandthe red alderandDouglas-fir
model,all F-testsindicatedno significant
difference between models.
20/Fo•,EsTSC•'•C'E
•
RED ALDER
....
DOUGLAS-FIR
30-
20-
10I
I
I
500
1000
5000
I
10000
LOG (treesper hectare)
FIGURE
4. Projectedsize-density
trajectoriesfor red alderandDouglas-fir
standsinitiatedat 1000,
4000, 8000, and 12,000 trees/ha,projectedto 100 trees/ha.
Data for the red alder spacingstudy, where plantingdensitywas known,were
used to examinesome of the differentassumptions
that can be built into the
size-densitytrajectoryof SmithandHarm(1984).
First, we lookedat proposed
formsof the size-density
relationship
(Figure1A
andlB). The traditionalform (shapeA) showeda better fit to the datathanthe
formrepresenting
aninverserelationship
betweentheinterceptandinitialdensity
(shapeB) suggested
by McFaddenandOliver(1988). However,the red alder
spacingstudyis stillyoung(14 yr), sothe datadonotcoverlater stagesin stand
development
whenstandsmovealongthe maximumsize-density
line.
Second,the spacingstudydatawere usedto see if the shapeof the trajectory
curvesastheyapproach
the maximum
size-density
linewasindependent
of initial
density.Our resultsagreewith SmithandHarm(19tM), who founda common
trajectoryshapefor red alderseedlings
andfor red pine (PinusretinosaAir.)
stands.A commontrajectoryshapeseemsto be implicitin a speciesandis not
affectedby standdensity.
FEBRUARY1993/21
Finally,the analysisof the spacingstudydataindicatedthat the line connecting
the pointsat whichdensity-dependent
mortalitystartsis parallelto the maximum
line. Red alder standsstart density-dependent
mortalityat a relativedensityof
0.44. This fallsbetweenSmithandHann's(1984) valuefor red alderseedlingsof
0.31 and Hibbsand Carlton's(1989) valueof 0.5. In red pine, mortalitydid not
start at a constantrelative density,but at a lower relative densityif initial
density was low (Smith and Hann 1984). On the other hand, DeBell et al.
(1989) found that relative density at onset of mortality in standsof loblolly
pine (Pinus taedaL.) increasedin standswith low initialdensity.The mortality
thresholdline (definedas the densityat whichmore than 3% of the trees died)
was determinedby analyzingdata whichwere measuredat fairly long intervals (5 to 7 yr). Shorter intervalsmight have led to a differentconclusion.In
addition,DeBell et al. (1989) assumedthe slopeof the self-thinning
line to be
- 3/2. The slopethat we foundfor red aldermightgivedifferentresults.A more
thoroughinvestigation
wouldrequirespacingstudieswith replicateddensities
on a range of sites with frequentmeasurements
duringthe initial phasesof
growth.
An equationfor size-density
trajectorywithoutinitialdensitywas fitted to
naturalstands.However, the equationdidincludean adjustmentparameterthat
was a functionof boththe proportionof the initialdensitystillalivewhenthe first
measurement
was madeandthe relativedensityat whichmortalitystarts.These
two factorscould,therefore, not be separatedin the analysis.The adjustment
parametersare specificto eachdataset.
As in plantations,
the shapeof the trajectorycurveis independent
of densityin
naturalstands.In addition,the relativedensityat the initialmeasurement
was
similarfor all plots.The plotsusedin the analysiswere researchplotsthat were
establishedto observe stand growth and yield. The locationwas, therefore,
biasedtowardfully stockedstandconditions,
whichmay explainthe similarrelative densityfor the plotsat time of establishment.
The red alderdataset, whichwas a composite
of plantedandnaturalstands,
providedus with the opportunityto investigatethe influenceof regeneration
method.A commonsize-density
trajectorycharacterized
the development
of both
naturalandplantedstands.Standorigindoesnot seemto influencestanddevelopmentin termsof the size-density
trajectory,whichagreeswith resultsof our
analysis
in the firsttwo objectives.
However,researchplotsin naturalstandsare
oftenin homogeneous
areasof the standsthat mightbe similarto plantations
in
evenness.Plotlayoutcouldthereforehaveinfluenced
the similarityin size-density
relationship.
Also, all of the plantationdatawere from the samelocation.Additionalstudiescoveringa variety of sitesare needed.
The slopeof the maximumsize-densityline for red alder (-0.64) was only
slightlysteeperthan - 0.62, whichwassuggested
by Reineke(1933)for a variety
of speciesandhasbeengenerallyassumed
to applyto red alder(Hibbs1987,
HibbsandCarlton1989). The averagemaximumSDI of red alder(751) is within
the rangeobservedby otherresearchers.The maximumSDI of the normalyield
tablesdevelopedfor red alder by Worthingtonet al. (1960) is 571. Hibbs and
Carlton(1989) estimatedthe absolutemaximumSDI to be 1125 from temporary
red alder plots and reporteda maximumSDI of 675 for data from long termstudies.
22/FOmS'TSCm•CE
Sinceinitialdensitywas not availablefor Douglas-firplots,its effecton trajectory
shapeor the onsetof initialmortalitycouldnot be determinedin the sameway as
for red alder.
The approach
of the size-density
trajectoryto the maximumlineindicatesthat
Douglas-fir
mortalitystartsarounda relativedensityof 0.58, whichis higherthan
was foundfor red alder. Even thoughthis is an extrapolation
beyondthe data
range,it agreescloselywiththe commonly
assumed
valuefor Douglas-fir
of 0.6
(Drew andFiewelling1979, Long 1985).
The slopeof the maximumsize-densityline was shallowerfor Douglas-fir
(-0.52) thanthat proposedby Reineke(1933) andthanthat for red alder.The
resultingmaximumSDI for Douglas-fir(1195) wouldseemto be in the correct
range consideringthe absolutemaximumis 1470 (Reineke 1933). Drew and
Fiewelling(1979) alsofoundthat their maximumQMD was in agreementwith
Reineke'sat highdensitiesbut that it underestimated
Reineke'sQMD at lower
densities,indicatinga shallowerslopethanthat of Reineke(1933).
Possibleexplanations
for the discrepancy
betweenthe slopeof our maximum
size-density
line that generallyusedfor Douglas-fir
includedifferences
in data
analysismethodsandin underlying
assumptions
abouttrajectoryshapes,andthe
effectof dumpingwithinthe stand.They are discussed
here.
Different methodshave been used in determinationof the self-thinning
line,
complicating
the comparison
of results.The most commonmethodsare visual
location,prindpalcomponent
analysis,
extrapolation
fromotherspecies,andregressionanalysis.Reineke(1933) didnot locatean averagelinemathematically.
He relied on placinga line abovea numberof individualstandmeasurements
visually,andhe founda commonslopefor a numberof species.OsawaandSugita
(1989) fitted a line throughdata pointsconsideredto be on or near the selfthinningline. Using principal-component
analysis,they determineda slopeof
-0.64, whichis considerably
steeper than the one we found.Most other researchon self-thinning
in forestshas assumeda slopebasedon the 3/2 power
rule, often becauseof lack of a better data (Drew andFiewelling1979, Curtis et
al. 1981, Hyinket al. 1988, HibbsandCarlton1989).An exceptionto thisis the
work of Von Gadow(1987), who analyzedpine speciesgrownin plantations
in
SouthAfrica. He fit regressionlinesthroughmeasurements
consideredto be on
the maximumdensitylineandseparatedthe speciesintotwo groupsin termsof
self-thinning
patterns.GroupI consistsof Pinuspatula,P. taeda,P. elliottii,P.
radiata,andEucalyptus
grandiswitha slopeof -0.51. The secondspeciesgroup
consists
ofPinuspinasterandP. roxburgi•i
witha slopeof the self-thinning
lineof
-0.42. However,boththe analysis
by OsawaandSugita(1989)andthe work of
vonGadow(1987)usea subjectiveselectionof datapointsconsidered
to be onthe
self-thinning
line, whichmayhaveinfluenced
the resultsof their analysis(Weller
et al. 1985).
A size-densityrelationship
of shapeB (Figure 1) wouldresult in individual
standsfollowinga trajectorywith a slopeshallowerthan -0.62 whilethe overall
data set has a maximumsize-densityrelationshipthat agreeswith the results
suggested
by Reineke(1933):a collection
of Douglas-fir
standsare bounded
by a
linewiththe slopeof - 0.62. However,ouranalysis
of the differentsize-density
FEBRUARY1993/23
shapesfor red alderindicatedthat the size-densityrelationshipis better representedby shapeA.
The apparentdiscrepancy
betweenReineke's(1933)resultsfor Douglas-firand
ourscannotbe explainedby the theorythat clumping
leadsto lower stockability
and thus to a lower maximumsize-density
line. Studiesinvestigating
effectsof
dumpingon standdevelopmentindicatethat standsdeveloptoward a uniform
spacingover time (Stiell 1981, Hamilton1984). This would steepenthe sizedensitytrajectoryasthe standdevelops.Randommortalityevents,suchasinsect
attacksor windthrow,wouldleadto increasedclumpinganda shallowerslopeof
the size-densitytrajectory.However, earliereliminationof plotswith extremely
highmortalityratesshouldhaveexcluded
mostof the effectsof randommortality
events from this analysis.
A comparison
of size-densitytrajectoriesfor red alderandDouglas-firindicated
that eachspecieshada differenttrajectory.A singleasymptotecouldnot characterizedevelopment
for bothspecies.Thiscontradicts
the theorythatthe slope
of the self-thinning
line is independent
of spedes(Yodaet al. 1963, White 1980,
LongandSmith 1983).
A numberof aspects,singleor in combination,
mightbe responsiblefor the
differencein size-densitytrajectoriesof the two spedes. Even thougha final
answerrequiresmore detailedstudies,our resultsraisesomeinterestingpoints
for discussion.
The slopeof the maximumsize-density
linereflectsthe relationshipbetweenmortalityandgrowth.Standgrowth,if measuredas QMD, is a
composite
of increased
diameterof the surviving
treesandtheincreasein average
sizedueto mortalityof the smallertrees(Ford1975).Differences
in slopesmight
therefore be explainedby differentmortalitypatterns. However, the average
yearlymortalityratesof red alderwere lowerthanthe mortalityratesof Douglasfir, and,thus,the increaseof the QMD dueto mortalityof the smallertreesis not
likelya majorcontributorto the differencein slopes.
Suppressed
treesare morelikelyto succumb
to densitydependent
mortality
(Dahms 1983, Hamilton1986, Hann and Wang 1990). Those trees had been
becomingless competitivewith the survivingtrees as suppression
progressed
(Ford 1975, West and Burrough1983). Thus, a differencein slopeof the sizedensitylineis not simplyrelatedto a differencein competition
intensity.Instead,
the slopeof the red aldermaximumsize-density
relationship
indicatesthat diameter growthof red alderseemsto respondmore efficientlyto freed resources
thandiametergrowthof Douglas-fir.Thiscouldbe dueto differences
betweenthe
two speciesin how photosynthesis
efficiencyis affectedby availableresources
(Perry 1984)and/orpatternsof carbonallocation
(WaringandSchlesinger
1985).
Red alderhada lowermaximumsize-density
line anda lowerrelativedensity
at the onsetof mortalitythandidDouglas-fir(0.44 vs. 0.58). WeldenandSlauson
(1986) definedcompetition
as the inductionof strainas a directresultof resource
useby otherindividuals,
wherestrainis the sumof the physiochemical
changes
in responseto stress.Speciesdifferences
could,therefore,be due to different
strainlevel or tolerances,or to differentcompetition
intensitieswithinpopulations.
24/FOP,ESTSC•NCE
Shainsky(1988) foundthat red alderseedlingswere the superiorcompetitors
over Douglas-fir,i.e., red alderinfluenced
the growthof bothspeciesmorethan
did Douglas-fir.This wouldindicatethat competitionand strain intensityare
higherin red alderstands.Alternatively,the criticalstrainlevelat whichmortality
startsmightbe lower in red alder stands,not allowingcompetitionintensityand
standdensityto becomeas highas in Douglas-firstands.
As a finalpossibility,the competitionintensityandaveragestrainlevelscould
be similar,but the straindistribution
withina population
mightbe different.For a
similarcompetitionintensity,trees at the lowerendof the size-distribution
would
showmore strainin red alderthanin Douglas-firstands,leadingto a higherrisk
of mortality.
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Copyright1993 by the Societyo[ AmericanForesters
ManuscriptreceivedJuly2, 1991
KlausJ. Puettmannis AssistantProfessor,Departmentof Forest Resources,Universityof Minnesota,St. Paul,MN 55113,DavidW. HarmisAssociate
Professor,ForestResources
Department,and
DavidE. Hibbsis AssociateProfessor,Forest ScienceDepartment,OregonState University,Corvallis, OR 97331. Financialassistanceand the data were providedby the HardwoodSilviculture
Cooperative,OregonState University.The authorsthankDean DeBell for the use of unpublished
data. This is Paper2752 of the Forest ResearchLaboratory,OregonState University.
FEBRUbY 1993/27