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Incorporating Additional Tree and
Environmental Variables in a Lodgepole
Pine Stem Profile
Model
John C. Byrne, USDA ForestService,IntermountainResearch
Station, Moscow, ID 83843.
ABSTRACT. A new variable-formsegmentedstemprofile model is developedfor lodgepolepine (Pinus
contorta)treesfrom thenorthernRockyMountainsof the UnitedStates.I improvedestimates
of stemdiameter
bypredictingt•o of themodelcoefficients
withlinearequationsusinga measureof treeform, definedasa ratio
of dbhandtotalheight.Additionalimprovements
wereobtainedbyfitting thismodelto individualnationalforest
datasets.Othertreeandenvironmental
variablestestedbutfoundof littleusein improvingstemprofileestimates
werecrownratio, habitatseries,elevation,slopepercent,andaspect.West.J. Appl. For. 8(3): 86-90.
Inventory,
timber
sale
preparation,
and
growth
and
yield
esti-
tial)weresampled
untillodgepole
pinetreeswitha rangeof D,
mationrequireaccurate
andcompatible
methods
forestimating
individualtreedimensions
(volume,stemdiameter,andlength),
especially
for animportant
commercial
species
like lodgepole
pine (Pinuscontorta).A completesystemof equationsfor
H, and CR were found.
estimation of individual tree dimensions can be obtained from a
stemprofilefunction.Stemprofilemodelscommonlyrequire
onlyD (i.e.,diameteroutside
barkat4.5ft abovetheground)and
totalheight(H) measurements
asinputs.Othertreemeasurements,includingcrownratio (CR) and variousmeasures
of
form,havebeenusedaspredictor
variables
toimproveestimates
froma stemprofileequation
(ValentiandCao 1986,Newnham
1988).Theuseofcommonly
measured
environmental
variables
(i.e.,geographic
location,slopepercent,aspect,
elevation,and
habitattype),usedasindicators
of sitequalityby thePrognosis
Model(Wykoffetal. 1982),havenotbeenexplored
toanygreat
degree.
Larson(1963) andNewberryandBurkhart(1986)noted
thatsitequalityhassomerelationship
to treetaperandform.
Thispaperdescribes
thedevelopment
of astemprofilemodel
for lodgepolepine. Relationships
amongtree (crownratio,
form) and environmental
(geographiclocation,habitattype,
slopepercent,aspect,andelevation)variablesandlodgepole
pinestemprofileareexploredandareincorporated
intothestem
profileequationwhenfoundto be significant.
Sampletreeswerefelledandthencutatpredetermined
points
alongthe stem.At eachcut, four measurements
weretaken:
diameterinsidebark(dib),tothenearest
0.1 in.,of theminorax•s
of thestem;dibof themajoraxis(i.e.,90øfromtheminoraxis);
average
doublebarkthickness;
andtheheight(tothenearest
0.1
ft) fromtheground
tothecut.Cutsweremadeatthestump(from
0.1to 1.0ft aboveground),atD, andthen,starting
atthestump,
at approximately
8.25 ft intervalsup to a 5.6 in. dib.The next
buckingpointswereat5.0 in., 4.0 in., and2.5 in. dib.Cutswere
alsomadeat 1/2 and3/4 thetotalheight.D wasmeasured
(with
adiameter
tape)before,andtotalheightafter,thetreewasfelled.
Crown ratio (CR), wherebranchesin the lower crownwere
ocularlyrearranged
to achieveabalanced
crown,wasestimated
to the nearest10%.In addition,nationalforest,habitattype,
aspect
(by45øclasses;
e.g.,NE = 22.5ø - 67.5ø),andelevation
(infeetabovemeansealevel)atthestandlevelandslopepercent
(by 10%classes)
atthetreelevelwererecorded.
Treeswithdead
or forkedtopswerenotselected.
A totalof3,624treeswereavailable.
I systematically
splitthe
databy puttingeverythirdtreeinto a validationdatasetfor
testing,withtheremainingdatain a developmental
datasetfor
fittingthestemprofilemodels.Eachdatasubsethada similar
representation
of dataasthefull dataset.SeeTables1 and2 for
Methods
summaries of the data.
Data
Stem Profile Model
Data camefromfelledlodgepolepinetreesfrom ninenationalforests(NF) in Idaho,Montana,andWyoming(USDA
ForestService1973,1984b).Withineachnationalforest,stands
withinhabitattypegroupings
(groupedby productivity
poten-
The well-knownMax and Burkhart(1976) stemprofile
modelwasusedasthebasemodel(seeAppendix1). It is a
segmented
modelthatjoinsquadratic
equations
for threeparts
of the tree(i.e., thebutt,the middle,andthetop).The three
86
WJAF8(3) 1993
Table 1. Summary of full data set for pert,nent tree and environmental variables by national forest.
National forest
Variable/
Statistic
DBH (in.)
Average
Range
Total ht (ft)
Average
Range
Crownratio(%)
Range
Elev. (ft)
Range
Slope (%)
Range
Beaver-
head
Idaho
Panhandle
Nez
Flathead
Gallatin
Kootenai
Lolo
All
Peme
Targhee
Challis
data
7.4
0.9-19.8
8.9
0.5-22.3
8.7
1.3-17.6
8.3
1.7-21.7
8.1
0.8-21.0
8.2
1.0-18.3
10.0
7.0-18.9
9.5
5.0-24.1
8.5
5.0-14.6
8.3
0.5-24.1
48.1
8.3-98.8
53.3
7.2-95.3
57.1
11.6-98.7
55.4
11.7-104.5
59.1
10.4-110.4
60.7
8.3-112.1
78.3
47.5-107.1
61.8
34.5-96.4
51.3
32.8-84.8
56.7
7.2-112.1
10-90
10-90
10-90
10-90
10-90
10-90
20-80
10-90
20-80
10-90
6,4408,080
4,4006,160
3,2006,340
6,6008,100
2,7005,880
3,2006,560
3,8006,400
5,2008,000
6,6008,700
2,7008,700
0-40
0-60
0-50
0-30
0-40
0-70
0-50
0-30
0-30,
0-70
50,70
Aspect
1
Classes
All
represented
Habitattype
All except
All
All
All except
N, NE, NW
All
All except
level
All except
All except
level
NE
SE, S, SW
Series
2
PSME
ABLA
PSME
PSME
PSME
PSME
ABGR
PSME
ABLA
represented
ABLA
TSME
ABGR
PIEN
PIEN
ABGR
ABLA
ABLA
PICO
THPL
ABLA
ABLA
THPL
ABLA
All
All
PICO
ABLA
1All= allcodesrepresented
(i.e.,N, NE,E, SE,S, SW,W, NW,level).
2 Habitatseriesasdefinedin Pfisteret al. 1977,Steeleet al. 1981,Steeleet al. 1983,Cooperet al. 1987.
equations
aremadetobecontinuous
forthetreeboleattwojoin
points.Of the six coefficientsin the model,the join point
Incorporating Additional Variables in Stem Profile
elevation,slopepercent,aspect,andcrownmilo.I alsousedD
andH. Transformations
of thecontinuous
variables,
alongwith
interactions
betweenaspectandslope,areutilizedbecause
of
their importancein predictingdiameterand height growth
(Stage1976,Wykoffet al. 1982).SAS software(SAS Institute
Inc. 1987)is usedfor all analyses.
Importantadditionalvariablesweredetermined
usingthe
approach
of ValentiandCao(1986)forCRincorporation.
First,
Models
for eachof the continuousvariables(i.e., elevation,D, H, D/H
coefficients
(a1anda2)signify
changes
inthestem
profile.
The
al joinpointoccurs
nearwhere
buttswellends,
andthea2join
pointoccurswithinthecrown.Thismodelis conditioned
such
that d = 0 when h = H.
I usedthe techniquedescribed
by Burkhartand Walton
(1985) andValenti andCao (1986) to incorporate
additional
variables
intothestemprofile
model.Theydecided
onasuitable
equation
formforpredicting
a stemprofilecoefficient
froman
additional
variableandthendirectlysubstituted
thisequation
•ntotheprofilemodelandfit it to theentiredevelopmental
data
set. The followingadditionalvariableswere used:national
forest(as an indicatorof geographic
location),habitatseries,
[i.e.,(D/12)/H], andtwoaspect-slope
combinations),
thedevel-
opmental
dataweresorted
inascending
orderandthensplitinto
10groups
of approximately
equalnumbers
of observations.
For
theothervariables,I usedlogicalgroups(i.e., nationalforest,
habitatseries,10%crownratiogroups,10% slopepercentage
groups,
and45øaspect
groups).
Then,themodelwasfit toeach
of thegroupsdefinedfor a particularvariable.Thisprocedure
wasconducted
separately
for bothoutsidebark(ob) andinside
Table 2. Number of trees and height/diameter observations in the data sets by national forest.
Height/diameter
observations
1
Number of trees
National forest
Beaverhead
Idaho Panhandle
Flathead
Gallatin
Kootenai
Lolo
Nez Perce
Targhee
Full
Developmental
Validation
Full
data set
data
data
data set
data
594
166
238
382
198
575
141
81
296
83
118
191
99
287
70
40
8,024
2,535
3,603
5,727
2,905
8,429
2,787
1,041
5,343
1,687
2,395
3,912
1,913
5,632
1,862
696
890
249
356
573
297
862
211
121
Developmental
Validation
data
2,681
848
1,208
1,815
992
2,797
925
345
Challis
65
44
21
484
325
159
Totals
3,624
2,419
1,205
35,535
23,765
11,770
1Doesnotinclude
totalheight
observations.
WJAF8(3) 1993 87
bark0b) data.Substanual
reducuons
in Sumof Squared
Error
(SSE),astestedby anF-statistic
for a reduced
(withoutadditionalvariable)vs.complete(with additional
variable)model
(Ott 1977),indicatevariablesthatmaybe usefulin improving
the model.
Theimportant
continuous
variables
fromtheaboveanalysis
wereusedaspredictors
ofthestemprofilecoefficients.
Because
of thedifficultyof usingtheimportant
discrete
variables
from
theaboveanalysis
in themodel,I reserved
themforlaterin the
analysis.
Themodelwasfit to thefull developmental
dataset
withoneof thesixcoefficients
beingsubstituted
by a simple
equation
inoneoftheadditional
variables.
Threeequation
forms
were used(Valenfi andCao 1986):
Linear:
Pi=fl +f2(X)
Logarithmic:
Pi=fl +f2(lnX)
Hyperbolic:Pi=fl +f2(l/X)
(1)
(2)
(3)
where
P = a stemprofilemodelparameter,
fl,f2= regression
coefficients
tobepredicted,
X = an additional variable.
For eachcoefficient
in the model(includingthejoin point
coefficients),
eachof thesethreeequationforms
witheachofthe
important
additional
variables
wasdirectlysubstituted
intothe
modelandthenparameters
wereestimated
usingthedevelopmentaldata.New models,derivedby incorporating
the best
individualequation
form-variable
combinations
for different
coefficients,
werealsofit to the data.For importantdiscrete
variables,
themodeldeveloped
withimportant
continuous
variables above was fit to each class of the discrete variables. SSE
analysis,
aspreviously
described,
wasusedtodetermine
thebest
modelsthatincorporate
additionalvariables.
Comparisonof the Stem Profile Models
Thebestmodelsfit tothedevelopmental
datawereevaluated
furtherusingthevalidationdataandresidualanalysis.
These
modelswere usedto predictd for eachobservation
in the
validation
dataseteventhough
d2isthedependent
variable
in
Residuals
frommeasurements
takenatthreepointsonthetree
bole were graphed.Thesethreepointsweremidclearbole
(midwaybetween
thebaseof thestemandcrownbase),crown
base,andmidcrown(midwaybetweenthecrownbaseandthe
treetop).Foractualapplication,
I determined
finalcoefficients
byfittingthebestmodel
totheentiredataset(developmental
and
validationdatacombined).
Results
SSE analysisindicatedthatall of the additional
variables
tested
werestatistically
significant
atthe0.01probability
level.
The greatest
reduction
in SSEwasobtainedwiththeD/H raUo
(9%reduction),
forbothobandibdata.Variablesof lesser
value,
in orderof SSE reductionwereD, elevation,NF, habitatseries,
and crownratio for ob data;and elevation,NF, habitatseries,
crown
ratio,andDforibdata.Slopepercent,
aspect,
transformationsof slopeandaspect,
andH hadevenlowerSSEreduction
sowerenotusedasadditional
predictors.
The continuous
variables,D/H, D, elevation,andCR, were
thenusedto predict
thecoefficients
of thestemprofilemodel
Incorporating
D, elevation,
andCRforibdata,andelevation
and
CRforobdata,produced
onlyasmallreduction
in SSE.Greatest
reductions
in SSEwereobtained
whenbothb1 andb2 were
estimated,
withthelinearequationform,usingD/H orD asan
additional
predictor
(i.e.,D/HandDmodels,
respectively)
forob
dataandD/H asanadditional
predictor
forib data.
Todetermine
anyadditional
improvement
possible
usingthe
bestdiscretevariables,NF and habitat series,the D/H and D
modelswere fit to eachNF (i.e., D/H-NF and D-NF models,
respectively)
andhabitat
series
(i.e.,D/H-HSandD-HSmodels,
respectively).
All of thesenewmodelshadreduced
SSEand
werestatistically
significant,
butusingNF produced
a greater
reduction than habitat series. The D/H-NF andD-NF models for
obdata,andtheD/H-NFmodelforib dataappear
tobethebest
newmodelsdeveloped
fromthedevelopmental
datausingthe
available additional variables. The D/H and D models were
retained
forfurtheranalysis
toverifytheusefulness
of fittingto
individual NFs.
Using
thevalidation
data,
thethree
teststatistics,
(•), s,and
eachmodel.Thiswasdonebecause
dis a verycommonlyused
variableandtherefore
errorsin d weremoreeasilyinterpreted.
PVE (seeTable3), indicatedthattheD/H-NF modelis clearly
Threeteststatistics
werecalculatedfromtheresiduals(observed
are reachedfor both data sets.Also, to determinewhetherthe
developmental
dataarevery similarfor bothob andib. Not
surprisingly,
theibmodelsarelessvariablethanthecorrespondingobmodels,dueto theirregularity
of barkthickness.
Residuals
calculated
at10%heightintervals
forthetreestem
fromthedevelopmental
dataindicated
thattheD/H-NF model
wasthebestmodelalongtheentirestemfor theob data.The
othermodelshadaverage
residuals
fartherfrom0 andlarger
standard
deviations
thantheD/H-NF modelfor a majorityof
percentage
heightclasses.
Fortheib data,theD/H-NF model
hadaverageresiduals
closerto 0 for mostclasses
andlower
models
fit eachpartof theboleequallywell,teststatistics
were
calculated
withthevalidation
dataforpercentages
oftotalheight
results,aswell astheF-statistic
testfor SSE,verifythegainin
value- predicted
value),i.e., averageresidual(R), standard
deviationof theresiduals
(s),andpercentvariationexplained,
PVE (ByrneandReed1986).Averageresiduals
andstandard
deviation of residuals closest to zero and PVE closest to one
indicated
thebestprediction.
Thebestsystemwasselected
by
comparing
thesestatistics
forthebestmodels
fromthedevelopmentaldata.In addition,theseresidualstatistics
werecalculated
usingthedevelopmental
datatoverifythatsimilarconclusions
(10% classes).
To detectanyweaknesses
in themodel,residuals
fromthe
validation
dataweregraphed
against
theimportant
additional
variables(eventhosenot usedin the model),for the ib data.
88
WJAF8(3) 1993
the best for both ob and ib data. Test statistics based on the
standard deviations for all classes than the D/H model. These
predictive
abilitybyfittingtoindividual
NFs(D/H-NFmodel)
insteadof poolingthedata.As mightbe expected,
prediction
errortendstobehigheratthebaseof atreeduetothevariability
of the butt swell.
Table 3. ResidualI stat,st,cs for val,dation data set.
Outside bark data
Inside bark data
Standard
Standard
Average
deviation
Model
residual
residual
D/H
D
0.024
0.020
-0.032
0.570
0.535
0.586
0.968
D-NF
-0.036
0.568
0.970
D/H-NF
Average
deviation
PVE
residual
residual
0.970
-0.015
0.548
0.971
0.974
-0.017
0.516
0.974
PVE
I Residual
= observed
d- predicted
d, ininches.
Graphsof residualsfrom theD/H-NF modelagainstD/H
produced
a fan-shaped
pattern,withresiduals
increasing
asD/H
increasedat all threelocationson the tree.Thesegraphsalso
indicatedsomevariationin themagnitude
of theresiduals
from
the top to bottomof the tree.A functionwith D/H andh/H as
mdependent
variableswasdeveloped
asa weightingfactorto
eliminatethispatternof unequalresiduals.
Reparameterization
of theD/H-NF modelusingthisweightingfactorcorrected
the
patterned
residuals,
butresidualanalysisshowedno changein
thepredictions
fromthemodel.Therefore,theoriginalD/H-NF
theuseofanadditional
stemdiameter,
whichistime-consuming
and difficult to measure.
Usingseparatecoefficients
for eachNF seemsto help in
describing
thedifferingenvironmental
conditions
thatcontributeto varyingstemprofilesfor lodgepole
pinein theNorthern
Rockies. Different environmental and climatic stresses affect
treesontheNFsconsidered
in thisstudy.Forexample,
theIdaho
PanhandleNF (in northernIdaho) has a moister,maritimeinfluencedclimatewhereasthe Gallatin NF (in southwestern
Montana)hasa drier,continental
climate.EachNF is representedby a complexcombinationof environmental
factors,
includingvegetationpatterns,climate,landforms,elevation,
etc.Somecombination
of environmental
variables
mightdo a
betterjob in helpingto predictvaryingstemprofile,but the
availabledatareflectsthe usefulness
of usingNF aloneas a
surrogate
for sucha combination
of factors.
The patternsof overestimation
seenfor somehabitatseries
andhighcrownratiosmainlyreflectthelimitedoccurrence
of
thesecharacteristics
in theforest.The Engelmann
spruceand
mountain
hemlockhabitatseriesareonlyminorcomponents
of
thelandbasein thisgeographic
area(Byrneet al. 1988).Asfor
thebiasin heavilycrownedtrees,mosttreesnormallydo not
maintainsuchlargecrownswhengrowingin foreststands
and
thereforewerenot heavilysampledin thisstudy.Suchhigh
crownratiosindicatemoreopen-growntrees.
model coefficients were retained for use.
Residuals
werealsographed
fortheotherimportant
variables
(i.e.,nationalforest,habitatseries,elevation,andcrownratio).
No patterns
wereobserved
fornationalforestandelevation.
The
modelslightlyoverestimated
dfortheEngelmann
spruce
(?icea
engelmanniO
andmountain
hemlock(Tsuga
mertensiana)
habitatseries,especially
atthecrownbaseandmidcrownlocations.
For crown ratio, the midclear bole and crown baseresiduals
lookedgood,butforthemidcrownresiduals,
therewasa slight
overestimation
associated
with high crownratio, from 60%90% CR.
Discussion
The advantage
of incorporating
D/H intothe stemprofile
model has been shown. D/H can be considered an indicator of
tree form. Treeswith the sameD but differentheightswill
generallyhavesomewhatdifferentforms.The differingtree
formscausedby the rangeof D andH combinations
will be
Model Application
Datafromthedevelopmental
andvalidationdatasetswere
represented
byusing
theD/Hvariable
topredict
theb1andb2
combined,andtheD/H-NF modelfit to theentiredataset.Table
coefficients
inthestemprofilemodel.Thistechnique
allowsfor
•ncorporation
of somemeasure
of treeformwithoutrequiring
4 for ob data and Table 5 for ib data show the coefficients for each
NF, aswell ascoefficientsfor theentiredataset(all NF datasets
Table 4. Coefficients
for the D/H-NFmodel,outsidebarkdata.
Coefficient
National
forest
Beaverhead
IdAho
Panhandle
Flathead
Gallmin
Kootenai
Lolo
Nez Perce
al
a2
104.397948
b3
-2.183345
b4
0.070168
0.782064
-3.427675
fl
-54.912103
f2
1.168917
52.424091
134.441969
180.118890
114.054765
137.493492
114.917322
177.267918
-1.739870
-2.144045
-2.156758
-1.579915
-1.346061
-2.526703
0.060289
0.060662
0.067748
0.064608
0.069318
0.060030
0.786951
0,807914
0.781523
0.774465
0.760924
0.834903
-3.577941
-3,328044
-3.518426
-2.527736
-2.339391
-4.406124
-15.133016
-83.304301
-37.028555
-65.128102
-54.762665
-61.646518
1.362265
1,077652
1.182589
0.710065
0.622919
1.617829
25.289147
76.189012
42.128033
60.071600
52.394400
66.349579
-24.313446
f3
f4
Targhee
81.804828
-1,975942
0.065700
0.731487
-3.402352
Challis
Full
data set
43.244194
-1.666913
0.094450
0.772752
-1.588609
-140.653850
-0.260453
1.293340
128.277526
27.412999
156.425019
-1.766880
0.059515
0.780459
-3.228737
-33.504041
1.157767
35.376538
WJAF8(3) 1993 89
Table 5. Coefficients for the D/H-NF model, inside bark data.
Coefficient
National
forest
Beaverhead
b3
b4
al
a2
fl
90,568523
-1.962273
0,069688
0.764338
-2.988761
107.913802
156,963535
92.741639
127.445303
99.173326
139.184999
0,060584
0.056600
0.070212
0.061698
0.066329
0.061520
0.068208
0.097382
0.767100
0,783103
0.762345
0.759498
0.752011
0.813900
65.102088
35.329978
-1.578220
-1,988964
-1.989368
-1.432999
-1.352238
-2.323381
-1.918440
-1.568828
123.460573
-1.650902
0.060639
f2
f3
f4
-53.327745
1.002217
49.374054
-20.968552
-70,718097
-40.068917
-53.634131
-56.580619
-68.943455
-26.078065
-140.725558
1,129189
0,982754
1.017405
0.673612
0.605297
1.363008
1.248376
-0.285214
28,216212
67.609824
41.960071
51.480968
52.622730
69.380872
0.725536
0.749788
-3,092332
-3.044257
-3.095119
-2.341331
-2.230199
-3.817534
-3.223871
-1.289302
27.408256
122.967481
0.761115
-2.862620
-36.847296
1.003935
36.499678
Idaho
Panhandle
Flathead
Gallatin
Kootenai
Lolo
Nez Perce
Targhee
Challis
Full
data set
combined).
To usethemodel,fastestimate
theb1 andb2
coefficientsof the stemprofile model usingthe following
equations:
bl=fl +f2
(4)
PFISTER,
R.D., B.L. KOVALCHIK,
S.F. ARNO,and R.C. PRESBY.1977. Forest
habitattypesof Montana.USDA For.Serv.Gen.Tech.Rep.INT-34. 174
p.
SASINsTmrm,INC.1987.SAS/STATGuidefor PersonalComputers,
Version6 Edition.Cary, NC: SAS InstituteInc. 1028p.
STAGE,
A.R. 1976.An expression
of theeffectof aspect,slope,andhabitat
typeon treegrowth.For.Sci. 22(3):457-460.
S•ELE, R., R.D. PFIST•X,R.A. RY•R, andJ.A. K•rr•v•s. 1981. Foresthabitat
b2=f3+f4[(D/12)/H]
(5)
typesof centralIdaho.USDA For.Serv.Gen.Tech.Rep.INT-114. 138
p.
Sr•J•E, R., S.V. CooP[m,D.M. O•r•ov, D.W. ROEERTS,
and R.D. PFIS•R.
usingthe coefficients
for the nearestNF fromTables4 or 5.
Substitute
thecalculated
b1andb2coefficients
along
withb3,b4,
al, anda2intotheappropriate
equation
andsolve.
SeeByrne
and
Reed(1986)forcompatible
equations
to calculate
totalcubicfootvolume,merchantable
height,andmerchantable
cubic-foot
volumesto eithera heightor diameterlimit. Calculateboardfootor scaledcubic-footvolumesby applyingan appropriate
board-foot
orcubic-foot
ruletothedimensions
of logspredicted
usingthestemprofileequation.
Usediscretion
whenapplying
thecoefficients
outside
therangeofdataconsidered
inthisstudy.
1983.Foresthabitattypesof easternIdaho-western
Wyoming.USDA
For. Serv.Gen.Tech.Rep.INT-244. 122 p.
USDA FOREST
SInVICE.1973.Net volumefield procedures.
Div. of Timber
Manage.,Intermountain
Region,USDA For. Serv.37 p.
USDAFoe,•ST
SERVICe.
1984.TimberManagement
Staffinteroffice
memoon
lodgepolepinetreevolumeequations.
File 2440,May 4, 1980.Regionl,
Missoula, MT.
VALero,M.A., andQ. V. CAO.1986.Useof crownratioto improveloblolly
pinetaperequations.
Can.J. For.Res.16:1141-1145.
WwcoF•,W. R., N. L. CROOKSTON,
andA. R. STAGE.
1982.User'sguidetothe
standprognosismodel.USDA For. Serv. Gen. Tech. Rep. INT-133.
112p.
APPENDIX
Literature
Cited
TheMax andBurkhart(1976) stemprofilemodel:
BURKHART,
H.E., andS.B.Wta•TON.
1985.Incorporating
crownratiointotaper
equations
for loblollypinetrees.For.Sci.31:478-484.
BYRNE,J.C., andD.D. R•EV. 1986.Complexcompatibletaperandvolume
estimation
systems
for redandloblollypine.For.Sci. 32:423-443.
BYRNE,
J.C.,A.R. STAGE,
andD.L. RENNER.
1988.Distribution
of permanent
plotstoevaluate
silvicultural
treatments
intheInlandEmpire.USDAFor.
Serv. Res.Note INT-386.7 p.
COOPER,
S.V., K.E. NEIMAN,R. STEELE,
and D.W. ROBERTS.
1987. Forest
habitattypesof northernIdaho:A secondapproximation.
USDA For.
Serv.Gen.Tech.Rep.INT-236. 135p.
LARSON,
P. R. 1963.Stemformdevelopment
of foresttrees.For.Sci.Monogr.
5.42p.
M•x, T.A., andH.E. BURKHART.
1976. Segmented
polynomialregression
appliedto taperequations.
For.Sci.22:283-289.
NEWBERRY,
J.D., and H.E. BURKHART.
1986. Variable-formstem profile
modelsfor loblollypine.Can.J. For. Res. 16:109-114.
NEWNHAM,
R.M. 1988. A variable-formtaper function.PetawawaNat.
Forestry
Inst.Infor.Rep.PI-X-83.Forestry
Canada,ChalkRiver,Ontario.
33p.
OTT, L. 1977. An introductionto statisticalmethodsand data analysis.
DuxburyPress,Belmont,CA. 730 p.
90
WJAF8(3) 1993
1
h 2
(6)
+b
4(a2- -•h_)
212
]
H
where
al,a2,bl,b2,b3,b4=regression
coefficients
d-- stemdiameter(in inches)at heighth
h = heightabovetheground(in feet)to d
D = diameter(outsidebark)at4.5 ft aboveground,in inches
H = totaltreeheight,in feet
Ii--
1,/f
•_
h
O,/f -->ai;i=l,
H
f h<a
i
2