Influence of Vertical Foliage Structure on the Distribution, of Stem Cross-Sectional Area Increment in Western Hemlock and Balsam Fir John A. Kershaw, Jr., and Douglas A. Maguire
ABSTRACT.A set of westernhemlock(TsugaheterophyllaRaf. [Sarg.])and balsamfir (Abiesbalsamea [L.] Mill.) from western Washington, USA and western Newfoundland, Canada, respectively, were destructively sampled to examine relationships between vertical foliage structure and distribution of
stem cross-sectionalarea increment. A series of stem growth distributionmodels describingthe relationship between amount of foliage area above a given height and stem cross-sectional area
increment at that height was derived from the pipe model theory. These models were derivedwith increasinggeneralityso that two commonlyheld assumptionscould be explicitlytested: (1) linear increase in cross-sectional area increment with increasing foliage area; and (2) constant crosssectional area increment below base of the live crown. Overall, the models performed very well, accountingfor over 80% of the observed variation in cross-sectionalarea incrementfor both species. The results show that, while cross-sectionalarea increment increases with increasingcurrent foliage, the increase is not proportional(i.e., linear). Furthermore,the rate of cross-sectionalarea increment below base of the live crown was found to increase in the smaller balsam fir trees and decrease
in the
larger western hemlock trees. These results suggest that cross-sectional area increment per unit foliage below the live crown may change as trees grow. FoR.Sc•.46(1):86-94.
Additional Key Words: Pipe model theory, stem form development,physiologicalmodeling.
to stemsize,shape,andgrowth.In termsof growthdistribution,Larson(1963) summarized thegeneralpatternasonein
T HE RESPONSE OF STEM GROWTH tO silvicultural treatmenthasbeena primaryfocusof silvicultureresearchfor well over a century(e.g., BQsgenand
MQnch 1929, Assmann1970). The main motivationfor such
long standinginterestis the economicimportanceof the stem.Despitea large body of research,thereare very few generalitiesor hypotheses relatingcrown structureto stem form development. Larson(1963), buildingon work initiatedby Duff and Nolan (1953) and other forestryresearchers,provideda comprehensive reviewof theliteraturerelatingcrownfactors
whichgrowthgenerally increases fromapextothevicinityof
maximumcrowndevelopment(heightof maximumfoliage area)andthengenerallydecreases belowthislevel.Theexact positioningof this patternis dependenton whetherring width,ringarea,or ring volumeis beinginvestigated andon otherfactorssuchassocialpositionandpastdamage.Larson (1963) considered thatmuchof theconflictingobservation in differentpatternspossiblyarisesout of differentmeasurementtechniques appliedto bothcrownandstemvariables.
JohnA. Kershaw,Jr., AppliedStand Dynamicsand ManagementLab, Facultyof Forestryand Environmental Management,Universityof New Brunswick,P.O. Box44555, Fredericton,NB E3B6C2, Canada--Phone:(506) 4534933; Fax:(506) 453-3538;
[email protected]. Douglas A. Maguire,Collegeof Forestry,OregonState University,Corvallis,OR 97331.
AcknOwledgments: Fundingforthisworkwas provided bythe Mclntyre-Stennis funds,University of Washington andthe CanadianForestryService, IFPMGreenPlanProject.TheauthorsthankMikeLavigneandthe RegionalForestNutritionResearchProjectfor use of their data andstudysites. LibbySoden,DaveLarsen,JimMcCarter,GlenGalloway,andAnnCamphelpedcollectdata. DeniseHart,ChadOliver,Dougsprugel,andMikeLavigne providedcriticalreviewsand/or usefuldiscussionduringthe developmentof this paper. ManuscriptreceivedAugust8, 1996. AcceptedJuly21, 1999.
86
ForestScience 46(1)2000
Copyright ¸ 2000 by the Societyof AmericanForesters
In this article, the pipe model theory is utilized as a foundationto statisticallyforrealizerelationships between crown structure(the verticaldistributionof foliage) and the distribution of cross-sectional area increment over the
bolesof two conifer species.The specificobjectivesof this paperare (1) to derive a seriesof modelsdescribing the distribution
of cross-sectional
area increment based on
thepipemodeltheory(Shinozakiet al. 1964a, 1964b)and (2) comparethe resultsand assumptionsof the models developedhereto thoseutilized in otherderivationsof the pipe model(e.g., Valentine 1985, 1988, 1990, Chiba et al. 1988, Osawa et al. 1991). Model
Derivation
The pipe model theorywas originally formulatedas a staticdescriptionof the relationshipbetweenstemcrosssectional areaat a specificheightandtotalfoliageabovethat height(Shinozaki.et al. 1964a,1964b).Withinthelive crown, stemcross-sectional areawashypothesized to increaselinearlywithincreasing totalfoliage.Thisproportional increase wasfurtherhypothesized to betheresultof a constant amount of conducting tissuenecessary to provideanadequate water supplyto eachproportionalunit of foliage(Shinozaki_et al. 1964a). Below the live crown, the amountof conducting tissuewasconsideredfixed, andtheincreasein stemdiameter
wastheresultofanaccumulation of"disused" pipes(Shinozaki et al. 1964a).Thepipemodeltheoryhasbeenwidelyapplied as a meansof estimatingtotal foliage (Waring et al. 1982, Espinosa-Bancalari et al. 1987, Dean et al. 1988,Long and
live crown.In all applications of thepipemodeltheoryasa growthmodel,theseassumptions generallyhavebeenexplicitlystated,butnotformallytested.The modelsderived herearenotdependent on assumption (1) andareaimedat explicitlytestingassumptions (2) and(3). As originallyformulated,the pipe modeltheorypostulateda constant relationship betweencross-sectional areaof conducting tissue(sapwood)at a given heightand total foliageabovethatheight(Shinozakiet al. 1964a): SCAHr,• TOTFOL•r ;
(1)
where SCAHT issapwood cross-sectional area(cm 2) at height HT andTOTFOLHT is totalfoliage area(cm 2) aboveHT. The originalrationaleforthepipemodeltheory was a proposedfunctional relationshipbetweenfoliage and sapwood;to supporta unit of foliage with the neces-
sarywater and nutrients,theremustbe a corresponding proportionalunit of conductingtissue(Shinozakiet al. 1964a).Underthisrationale,thepipemodeltheorywould imply that a changein foliage amountwould require a correspondingproportionalchangein sapwoodarea for functionalsupport;therefore,sapwoodcross-sectional area growth would be the first derivativeof (1) with respectto time,andthebasichypothesis of a growthmodel derivedfrom the pipemodeltheorywouldbe:
ASCAHr= [•o' ATOTFOLtrr ;
(2)
whereASCAHT ischange in sapwood cross-sectional area
Smith 1988, 1989, Bidlake and Black 1989, Woods et al. 1991).
atHT(cm 2yr-l),ATOTFOLHT ischange intotalfoliage area(cm 2)above HTand[30 isaconstant ofproportional-
The pipe modeltheory,with variousmodifications,also hasbeenputforthasaphysiological modeldescribing carbon allocationandstemgrowth(Mitchell 1975,Hari et al. 1985,
ity. ASCA includesboth cross-sectionalarea increment (CSAI) and sapwoodmortality and ATOTFOL includes bothcurrentfoliage growth(CURFOL) andfoliage mortality. Sincethe pipemodeltheory,in its strictestinterpretation,wouldassumesapwoodmortalityis proportionalto foliagemortality,then CSAI can be substitutedfor ASCA andCURFOL canbe substitutedfor ATOTFOL to produce
Valentine 1985, 1990, M•ikel•i 1986, Chiba et al. 1988,
Osawaet al. 1991).Valentine(1985, 1988,1990)developed a seriesof standandtreelevel growthmodelsbasedon the pipemodeltheoryinterpreted in itsstrictest sense: a constant linear relationshipbetweenstem sapwoodcross-sectional areaandfoliagearea.Thesemodelspredicta poolof carbon basedonamountof foliageandthenallocatethecarbontothe majorgrowthsinks:foliage,roots,andstems.The allocation is formulated to maintaintheconstant relationship between sapwood cross-sectional areaandfoliageareabothtemporallyandverticallywithintreesandstands(Valentine1990). Chibaetal. (1988)reformulated thepipemodeltheory,as depictedin profile diagrams(Monsi and Saeki 1953), to incorporate timeintotherelationship anddevelopeda theory of totaltreegrowthrelatedto crowndevelopment. Osawaet al. (1991)reformulated Chibaet al.'s (1988)theoryto incorporatedistance fromthetip of thetreeintotherelationship andcalledthisthe "profiletheoryof treegrowth." Theseextrapolations of the pipe modeltheoryrely on three basicassumptions: (1) there is no changein relative verticalfoliagedistributionover time; (2) sapwoodcrosssectionalarea or cross-sectional area incrementper unit foliageis constant overbothtimeandtheentirestem;and(3) cross-sectional area increment is constant below base of the
a cross-sectional
area increment
model:
CSAIur = [•o' CURFOLur ;
(3)
whereCSAIHTis cross-sectional areaincrement at HT, CURFOLHT iscurrent foliageareaaboveHT and[•oisa constantof proportionality.Underthishypothesis[Equation (3)], CSAI increasesat a rate proportional to the increase in CURFOL from tree tip to the base of the actively growingcrown (the heightbelow which no new foliageis produced).At the baseof the actively growing crown, CSAI is maximumand remainsconstantalong the bole below this point. Theassumption of constantCSAIbelowcrownbasehas beenutilized in severalgrowthmodelsderivedfrom the pipe model (e.g., Mitchell 1975, Valentine 1985, 1990, Osawaetal. 1991);however,with theexceptionof Mitchell (1975), all of thesemodels have utilized TOTFOL rather than CURFOL as a meansof predictingCSAI. Empirical evidence supportingthe assumptionof constantCSAI below crown basehas producedvarying and often conForestSctence 46(1)2000 87
flicting results (Larson 1963, 1969, Oliver and Larson 1990, Kozlowski et al. 1991). Dependingon theperspective taken,therelationship be-
A
B Quad. Mean
--- FoliageHt. ---.
tween CURFOL and CSAlcan be viewed asfunctional(new
sapwoodis neededto supportnew foliage)or a cause-andeffect(currentfoliageproduces thecarbonnecessary toform newsapwood). Regardless of theperspective taken,Equation (3) impliesthatCURFOLhasthesamefunctionalneedsand/ or productioncapabilitiesthroughoutthe crown,or, conversely,that sapwoodhasthe sameconductance capability and/orcarbondemandbothalongtheentirestemandacross differentagedtings;however,bothphotosynthesis (Woodman 1971) andrespiration(Brookset al. 1991) havebeenshown to vary within thecrown.Sapwoodconductance alsovaries throughout thestem(Pothieret al. 1989)andbetweendifferentagedrings(M'fikel•iet al. 1995).To allow for differential ratesof CSAI per unit CURFOL within the live crown,but constantCSAIbelowthelive crown,model(3) wasmodified
-- Crown Base --.
C
D Quad. Mean
--- Foliage Ht. ---Crown Base --,
as follows:
F
CSAIwr = [30 ßCURFOL•t3r ;
(4)
Quad. Mean
---Foliage Ht.----
whereCSAIHr, CURFOLHr, andHTareasabove, [50isa constant ofproportionality and[5l accommodates thedif-
~- Crown Base--.
ferential rate of increasein CSAI per unit CURFOL. Equation(4) ensuresthat CSAI is greatestat baseof the activelygrowingcrownandremainsconstantbelow(Fig-
)
',,
ure 1A); however,if 0 < [51< 1, thenCSAIper unit CURFOL is maximum at the tree tip and decreasesto the baseof the actively growingcrown and remainsconstant below (Figure lB). Further, Equation (4) allows an explicit test of the assumptionof constantgrowthper unit
foliage.If [51is significantly (P < 0.05)smaller than1.0, the assumption of constantgrowthper unit foliagewould not be substantiatedin this study. Total volumeproduction,thusCSAI alongthe stem,is influencednotonly by foliagearea,butalsoby thevertical distributionof foliage (Jack and Long 1992). In the general patternof growth over the stemidentified by Larson (1963), CSAI was generally found to be greatestat the point of maximum crown development(i.e., height of maximum foliage area). To evaluate the influence of vertical structureon growthdistributionand the assumption of constant CSAI below base of the live crown,
Equation(4) wasmodifiedto incorporatedistancebelow maximum crown development(assumedto be the quadratic meanfoliage height,c.f., Long et al. 1981):
CSAI(cm2yr-1)
CSAI per Unit Foliage
(cm2yr-l/cm2) Figure 1. Predictedgrowth (CSAI)and growth per unit foliage area (CSAI/CURFOL) from Equations (4) and (5) for various
valuesof parameters(•1and (•z:(A) CSAIfor(•1= 1 ( ) and •1 ½= 1 (----) predictedfromEquation(4);(B) growthper unit foliagefrom(A);(C)CSAIfor•1=1and•z 0 (-- -) from Equation(5);(D) growthper unitfoliaga from(C);(E)CSAIfor{•lcland{•z 0 (---) fromEquation(5);and(F)growthperunitfoliage from (E).
increasingfoliage (compareFigure 1A to Figure 1C); however, below the quadraticmean foliage height, the patternbecomesdependenton the value associatedwith
[52(Figure1C).For[52< 0, therateofincrease inCSAlper unit foliage beginsto decrease(Figure 1D). Maximum CSAI is at crown baseand then declinesbelow (Figure
1C).For[52> 0, CSAIandCSAIperunitfoliageincreases from treetip to groundlevel (Figures1C and 1D). For the
caseof0 < •] < 1,then,if[•2islessthan0, theCSAlpattern
½I•+I¾1e'OlSrQFOLur) '' (5) follows Larson's (1963) generalpatternwith the rate of CSAI•tT = •o ' CURFOL•tT indicatorvariablefor positionrelativeto quadraticmean
CSAI increasingfrom the tree tip to the quadraticmean foliageheightthendecreasingbelow(Figure 1E). Further, CSAI per unit CURFOL is maximumat the tree tip and
foliage height (IQ= 0 if above andIQ= 1 if below); and
decreases toground level(Figure1F).If •2 isnotsignifi-
where CSAI•t • CURFOL•t • andHTareasabove, IQisan DISTQFOL•t Tisthedistance thatHTis fromthequadratic meanfoliageheight.Depending onthemagnitude of [52,
cantlydifferentfrom0, thenthepredictionsof model4 are
Equation(5) produces anallocationpatternverydifferent from the basicpatternscommonlyheld underthe original
extrapolations ofthepipemodeltheory. If [5• = 1,thenin
quadraticmeanfoliageheightto thegroundlevel (Figure 1C). Therefore, an explicit test of the assumptionof constantgrowthbelowcrownbaseis providedby testing
the uppercrown,thereis a linear increasein CSAI with
[52forsignificant (P < 0.05)differences fromzero.
88
ForestScience 46(1)2000
sufficient. If [52isgreater than1thenCSAlincreases from
Methods
Data from two differentforestproductivitystudieswere analyzedhere.The firstdatasetwasfromwesternWashing-
ton (47015' N, 121o45 ' W) and was compiledfrom two westernhemlock(Tsugaheterophylla Raf. [Sarg.])fertilization trials (Universityof Washington1979). The primary purpose of thisstudywasto examinelongtermchanges in crownstructureand stemform developmentfollowingfertilization(Kershaw1993)aspartof a largerregionalforest productivitystudy(Maguireet al. 1991).The secondstudy wasin northwestern Newfoundland, Canada(49o22' N, 54ø19' W) and is part of a regionalstudyof differenteffectsof precommercial thinningin balsamfir (Abiesbalsamea(L.) Mill.).
Althoughthetwodatasetsrepresent verydifferentunderlying treatmentsand used different field methods in the collection of data, the associatedcrown and stem data from
bothstudieswereamenable to testingvariations onthepipe model theory. The two datasetswere usedbecausethey represent commerciallyimportanttreespeciesin easternand westernNorthAmerica.Comparisons betweenspeciesare restricted todifferences associated withpipemodelformulationsandtheinfluenceof treesizeonpipemodelpredictions ratherthandifferences associated with speciesand/orclimatic conditions. Western Hemlock
The two westernhemlockfertilizationtrialswerepartof the RegionalForestNutritionResearchProjectPhaseIV young westernhemlockfertilization trials (University of Washington1979). The two siteswere locatedwithin the White River drainagein the CascadeMountainsof western Washington.The first sitewaslocatedat an elevationof 550 m andhada northerlyaspectwith slopevaryingfrom 2540%. The second site was located at an elevation of 1100 m
andhada southwest aspect withslopevaryingfrom10-20%. BothsiteswereintheTsugaheterophylla Zone(Franklinand Dyrness 1988) and regeneratednaturally following clearcuttingand slashburning.Westernhemlockwas the dominantspecies, comprising 90% of thestemsat thefirst siteand98% of thestemsat thesecondsite.The remaining stemswerea mix of Douglas-firandgrandfir. Siteonewas8 yr old(breastheight,1.37m) at thetimeof thinningandfertilization(1980),andinitialdensityof trees > 7.5 cm dbhwasestimatedto be 1200stemsha-1. Site two was23 yr old(breastheight,1.37m) at thetimeof thinning andfertilization (1980),andpriortothinning, initialdensity of trees> 7.5cmdiameter atbreast height(dbh)wasapproxi-
mately 1600stems ha-1.Priortofertilization, initialdensity of trees_>7.5 cmdbhwasreducedto anaverageof 750 stems
ha-• onallplots. Theprincipalexperimental designwasa generalized randomizedblockwith two treatments: a control(no fertiliza-
tion)and225kgha-• nitrogen in theformof urea.Each treatment wasreplicatedthreetimes(0.04 ha plots)at each site.All treesonall plotsweresampled for crowndevelopment and growth measuresat the time of destructivesam-
pling;however, tomaintain continuity of theregionalfertili-
zationexperiment,destructive samplingof individualtrees was restrictedto one control(nonfertilized)plot and one fertilizedplot at eachsite. Basedonpertreeperiodicannualdbhincrement,sitetwo hadan initial positiveresponse to fertilizationwhile siteone hadnosignificant(P > 0.05) response. Thefertilizationeffect wastheresultof aninitialdoublingof foliageareaonsitetwo (Kershaw1993);however,theinitial growthincreaseceased within5 yr followingfertilization.By thetimethetreeswere destructively sampledfor thisstudy,thedifferences in foliageareabetweensitesandplotswithinsiteswerenotsignificant (P > 0.05). The treeswere sampled9 yr following fertilization, and the long-termeffectsof fertilizationwere testedon the crowndevelopment-growth datacollectedon all treesandontherelationships presented in thisarticle,but nosignificantdifferences werefound(Kershaw1993);therefore,analyses of fertilizationeffectsarenotpresented in this paper.
Thirty-five westernhemlocktreeswere selectedfor detailedcrownand stemgrowthmeasurements acrossthe four site-treatment combinations. All trees were selected from the
centralportionof each20 m by 20 m plot.Table 1 showsthe meanplot and sampletree characteristics. Startingwith thelowestlive branchandpruningupward, all branches below9 m in heightweremeasuredfor heightat the pointof attachment(nearest0.03 m) while still attached to the standingtree. Following pruning,each branchwas stackedsequentially forfurthermeasurement. Branches above 9 m in heightweresimilarlymeasured by fellingthetreeand standingthetop sectionupright. Branchbasaldiameter,total branchlength,green(foliated)length,maximumgreenwidth,andlengthof deadtip(if present)were measuredon all live branchesfollowingremoval from the stem. Branch diameters were measured to the
nearest0.1 cm, andlengthsandwidthsweremeasuredto the nearest 15 cm. A total of 7,603 western hemlock branches were measured. A subset of branches was selected for further
dissection andbroughtbackto the laboratoryfor analysis. One branchout of everygroupof 20 consecutive branches was randomlyselectedto ensurean adequatesampleof branchesfrom all positionswithin thecrown.Brancheswere placedin cold storageandseparatedinto needleandbranch materialwithin 72 hoursof pruning.A setof equationswas Table 1, Mean plot and sample tree characteristics (standard errors in parentheses)for western hemlock and balsam fir data.
Parameter
Meanplotcharacteristics Numberof plots
Density (trees ha-I) Basalarea(m2ha-I) Dbh(cm) Height(m) Sampletreecharacteristics Samplesize Dbh (cm) Totalheight(m) Heightto lowestlive branch(m)
Crownlength(m)
Western
Balsam
hemlock
fir
12
25
702 (45) 3,450 (732) 22.4 (4.67) 19.9 (4.37) 19.8 (1.67) 15.8 (0.54)
8.4 6.4
(1.65) (1.22)
35 20.2 (0.55) 15.9 (0.23) 3.3 (0.15)
25 8.7 6.3 1.3
(0.62) (0.37) (0.18)
12.7 (0.19)
5.0
(0.29)
ForestScience 46(1)2000 89
developed fromthefoliagedatacollectedonthesebranches to predicttotalandcurrentleaf areaandleaf mass(Kershaw andMaguire1995).Theseequations were usedto predict total and current leaf area for all live branches.
Disks were removed from the stem at 0.5 m intervals
beginning at0.5m belowbreastheight(1.37m) upwardtothe 2-yr-oldportionof thestem.Thediskswerebrought backto thelab andsandedwhilegreen.The longestradiusfreeof branch distortions was located on each disk. The shortest radius free of branch distortions between 50 ø and 140ø of
eithersideof the longestradiuswas alsolocated.For each radius, the distancefrom the center of the disk to the outside
of thebarkandeachannualringwasmeasured to thenearest 1.0mm. Ringwidthsfor thecurrentleaderandthe2-yr-old leaderweremeasured directlyfrom theleaders.Cross-sectionalareaincrement(CSAI)wascalculated usingthegeometricmeansof thelongandshortradii.Currentfoliagearea aboveeachdisk(CURFOL)andtotalfoliageareaaboveeach disk (TOTFOL) werecalculatedby summingthe predicted branch-level foliageamountsabovetheheightof eachdisk. Balsam Fir
A totalof 25 treeswasrandomlyselected for destructive sampling fromtheprecommercially thinnedplotswithinthe Cotmack,Newfoundland thinningstudy(Donnellyet al. 1986) locatedin SectionB28b of the CanadianBorealForest
Region(Rowe 1972). The area was clearcutin 1962 and naturallyregeneratedto balsam fir. In 1982, plots were precommercially thinnedto spacings of 1.2m (6950tree/ha), 1.8m (3100trees/ha),2.4 m (1750 trees/ha),and3.0m (1100 trees/ha).Plotswere40 m by 40 m andthinningtreatments wererandomlyassigned andreplicated threetimes.Fivetrees fromeachtreatmentwererandomlyselectedfrom all of the treesacrossall plots.In an initial analysisof thesedata,the thinningtreatmentswere found to influencedensityand individualtreesize,buthadnosignificant effectsonanyof the relationshipsbetweenstem cross-sectional area increment and foliage area.Table 1 showsa summaryof the averagecharacteristics of plotsandselectedtrees. Selected trees were felled and divided into annual stem
segments. Branchesfor eachannualsegmentwereremoved. Foliagewasseparated intotwoageclasses: currentandolder.
level,datawereaggregated byspecies andtheequations were fitted to obtainestimatesfor the speciesaverage.Transformedlinearandbothnonweighted andweightednonlinear regression analyseswereusedto estimateequationparameters. The best fit was selected on the basis of Furnival's
Index, a modifiedlikelihoodcriterion(Furnival 1961), and residualpatterns.
At boththeindividualtreeandspecies levels,Equations (4) and(5) werefittedto theCSAIdatausingbothCURFOL andTOTFOL.TOTFOLwasusedtocompare thederivations herewith thoseappliedprior to this work (e.g., Valentine 1985, 1990,Chibaet al. 1988,Osawaet al. 1991). The two crownmeasures werecomparedbasedon rankof Furnival's Index and meanFurnival's Index. Inclusionof DISTQFOL wasevaluatedatboththeindividualtreeandspecies levelby testingwhetherparameterestimatesweresignificantlydifferentfrom 0 (Neter et al.1988). Results
Cross-sectionalarea increment (CSAir) increasedwith increasing CURFOL(Figure2) onindividualtrees;however, thisincrease,in general,wasnotproportionalto theincrease in cumulativefoliageareaabove;that is, the trendwas not linearthroughthecrown.In general,CSAItendedto reacha maximumrateandthenleveloff with increasing depthinto crown,despiteacontinued increase in foliagearea(Figure2). At boththe individualtree andspecieslevels,weighted nonlinearregressions produced thebestfits for bothspecies regardless of modelform[Equation (4) or (5)]. Differences betweenlogarithmicallytransformed linearregressions and weightednonlinearregressions wereslight,butthenonlinear regressions consistently producedsmallerFurnival'sIndices indicatinga closerfit to thedata. Both CURFOL and TOTFOL explained significant amounts of variation in CSAI
at both the individual
and
specieslevels(Tables2 and3). For bothspecies, thepseudo
nonlinear adjusted R2's(Neteretal.1988)weregreater than 0.80 (Tables2 and3). At thespecieslevel, CURFOL consistentlyproducedbetterfits, asindicatedby smallervaluesof Furnival's Index (Tables 2 and 3). At the individual level, A. Western Hemlock
B, Balsam Fir
A subsample of bothfoliageageclasses wastakento deter-
minespecific leafarea(cm2 g-t) foreachageclass ateach branchwhorl.Theremainingfoliagewasdriedandweighed. Currentandolderfoliageareawithineachbranchwhorlwas calculated bymultiplying specificleafareabythedryweights for each age class.
A disk wasremovedfrom the midpointof eachannual stemsegment(internode).The width of eachring wasmea-
suredto thenearest0.1 mm alongoneradiusfor eachdisk, and CSAI was calculated. CURFOL
and TOTFOL
were
calculatedby summingtheassociated foliageareafrom the whorls above each disk. Model Evaluation
Equations (4) and(5) werecompared attheindividualtree levelandatthespecies level.Equations werefittedseparately to the datafrom eachindividualtree,while at the species 90
Forest Science 46(1)2000
Current Fofiage Area A•ove (1000'cm 2)
Figure 2. Relationship between cross-sectionalarea increment
(CSAI,cm2yr-1)andcumulative currentfoliageareafromtreetip to crownbase(CURFOL, cm2)by species: (A)westernhemlock; and (B) balsam fir. Linesrepresent trends within individual trees.
Table2. Parameter estimates, standard errors(inparentheses), R2s,andFurnival's (1961)Indexformodelsdescribing the relationshipbetween foliage area above (CURFOLor TOTFOL)and cross-sectionalarea increment (CSAI)fitted
totheaggregated westernhemlock data.TheR2spresented arethepseudo nonlinear adjusted R2s(Neteret al.1988). Equationno.--
Parameterestimates
foliage type
R2
Fumival's Index
(4)-42urrent
0.84
4.782
(4)--Total
0.82
5.143
(5)-42urrent
0.85
4.740
(5)--Total
0.83
4.803
CURFOL wasconsistently thebestpredictorof CSAI, especially ifDISTQFOL wasincludedin themodel(Table4). The
parameter estimates for½1weresignificantly lessthan1.0for bothspeciesat the specieslevel (Tables2 and3) andat the individuallevel (Figure3) for bothCURFOL andTOTFOL. Basedon Furnival's Index, Equation(5) predictedactual valuesmoreaccurately thanEquation(4) (seeFigure4 for a selectedwesternhemlockandbalsamfir tree).
Based on thesignificance (t-test,P < 0.05)of the½2 parameterestimates,inclusionof DISTQFOL, significantly improvedregressions at the specieslevel for CSAI versus both CURFOL and TOTFOL (Tables2 and 3). At the indi-
vidualtreelevel,however, theparameter estimates for½2 werenotsignificantlydifferentfrom zerofor 54% of western
hemlockand24% of thebalsamfir treessampled(Table5).
Forwestern hemlock, 14of35trees hadestimates of•2 that weresignificantlylessthanzero,while 19 of 25 estimatesof
½2forbalsam fir weresignificantly greater than0. Discussion
and Conclusions
The basicpremiseof theoriginalpipemodeltheorywas that a unit of foliage required a similar unit stemwood
"pipe"to supportthe foliage.Treatinggrowthasthefirst derivativeof the pipe model,additionof a unit of foliage would require additionof a unit of stemwood.As predicted from the pipe model theory, cross-sectionalarea increment(CSAI) at a givenheightwas highly correlated with bothcurrentand total foliage areaabovethat height (Tables 2 and 3); however, the relationshipswere not linear, ashascommonlybeensuggested for CSAI at base of the live crown (Chiba et al. 1988, Osawaet al. 1991).
Theestimates of thepowercoefficients (½2)weresignificantly less than 1 for both speciesat both the species
130
13•
0.06443
0.4729
(0.005940) 0.1318 (0.01109) 0.05405 (0.005510) 0•07491
(0.007699) 0.3791 (0.006459) 0.4906 (0.008790) 0.4321
(0.004771)
(0.008024)
13,
-0.001720
(0.0003950) -0.004601
(0.0003900)
average and individual tree levels (Tables 2 and 3 and Figure 3). When total foliage area above(TOTFOL) was includedin the model (Tables 2 and 3 andFigure 3), these coefficients
were even smaller.
While theseresultssuggestthata strictinterpretationof the pipe model theory for modeling the distribution of growth may be inappropriate, alternative explanations exist for the departuresobserved.The lack of a direct linear relationshipbetweencross-sectionalareaincrement andfoliage areaabove(either CURFOL or TOTFOL) may be explained by sapwooddynamics.The basic premise here was that new foliage must have new sapwood.Following this premise forward would imply that when the foliage formed this year died, its associatedsapwood would alsodie. Thus,the numberof sapwoodringswould equal the numberof foliage age classes.In most conifer trees,the numberof sapwoodrings varies greatly depending on age, site or crown position (Waring et al. 1977, Whitehead 1978, Maguire and Hann 1987, Pothier et al. 1989, OjansuuandMaltamo 1995). As originally formulated (Shinozaki et al. 19648), the pipe model theory predicteda constantrelationshipbetweensapwoodcrosssectionalarea and foliage area above. A dynamic model derived from this premise would be expressingthe net changein sapwoodareaas a functionof the net changein foliage area. Net change,as pointed out above, would be the difference betweengrowth and mortality of both sapwood area and foliage area.Neither foliage nor sapwood area mortality were examined in this study. The pipe model theory merely postulatesa functional balancebetweensapwoodareaandfoliage area,andvariability in the annualamountof stemgrowthversusfoliage growthmay be allowed as long as the overall balance is maintained (Mfikelfi 1986, Mfikelfi et al. 1995).
Teble3. Parameter estimates, standard errors(inparentheses), R2s,andFurnival's (1961)Indexformodelsdescribing the relationship between foliage area above (CURFOL or TOTFOL) and cross-sectional area increment (CSAI) fitted
to the aggregated balsamfir data.TheR2spresented arethe pseudo nonlinear adjusted R2s(Neteret al. 1988). Equationno.--
Parameterestimates
foliage type
R2
Fumival's Index
(4)--Current
0.86
1.330
(4)--Total
0.89
1.455
(5)--Current
0.91
1.233
130
0.6711 (0.01210) 0.5499
(0.001292)
(0.01129)
0.009840
(0.001203) (5)•Total
0.83
1.412
13•
0.005685 (0.0007010) 0.01004
0.6072
(0.01262)
0.01521
0.5062
(0.002078)
(0.01271)
132
0.01058
(0.001052) 0.006909
(0.001119)
ForestScience 46(1)2000 91
Table 4. Number of trees by rank of Furnival's Index and model formulation for equations predicting CSAI from foliage.
(Equationno.)-foliagetype
Rankof Fumival'sIndex 2 3
1
4
Mean Fumival'sIndex
Western hemlock
(4)•Current (4)•Total (5)•Current (5)---Total
8 0 20 7
12 1 14 8
13 4 1 17
2 30 0 3
2.66 3.83 1.46 2.46
0 0 25 0
4 16 0 5
3 8 0 14
18 1 0 6
3.56 2.40 1.00 3.04
Balsam fir
(4)•Current (4)•Total (5)---Current (5)•Total
One importantdifferencebetweenthe resultspresented here,and the resultsof Chiba et al. (1988) and Osawaet al.
(1991)isthescaleatwhichthepipemodelwasapplied.Here, thepipemodeltheorywasappliedwithinindividualtreesto predict CSAI at any point along the stem over a single growing season.Both Chiba et al. (1988) and Osawa et al. (1991) appliedthepipemodeltheorybetweentreestopredict CSAI at the base of the live crown over severalgrowing CurrentFoliage
Total Foliage
A. Western Hemlock d
seasons. Basedon regressions of CSAI at crownbaseversus total CURFOL andTOTFOL usingEquation(4) for western hemlockandbalsamfir (Table6), theirassumption of proportionalgrowthat crownbasewasconfirmed.When TOTFOL was regressedagainstCSAL both specieshad parameter
estimates for[51, thepower coefficient, thatwerenotsignificantlydifferentfrom 1; however,their models(particularly Osawaet al.'s) do notextrapolatewell to theentirebole. In orderto make their extrapolation,Osawaet al. (1991) explicitlyassumed thatgrowthperunitfoliagewasconstant overtheentirestem.Theresultsfromthisstudysuggest that thisis not the case.FigureslB, 1D, and 1F illustratecrosssectionalarea incrementper unit foliage trendspredicted
fromEquations (4)and(5)forvarious values of[5! and[52. As can be seen from Figure lB, growth per unit foliage is
constant if andonlyif J5 ! = 1.Forvalues of0 < [51< 1,aswas observed here,growthperunitfoliageis maximumat treetip anddeclineswith increasingdepthinto crown(i.e., increasingfoliagearea).Figure5 confirmsthattheobserved growth perunitfoliagefor westernhemlockwassimilarto thetrends predictedin Figure 1. Similar trendswere observedfor balsam fir as well.
c5 0.05
0.35
0,65
0.95
0.05
0.35
0.65
0.95
B. Balsam Fir
The high growthper unit foliageat the tip of the stemis most likely the result of the currentleaderbeing a net importerof carbonfromlowerbranches (Sprugelet al. 1991), while the relativelyconstantefficiencybelow the point of maximumcrowndevelopmentreflectsallocationof carbon A. Western Hemlock
0.05
0.35
0.65
0.95
0.05
0.35
0.65
0.95
b1 Parameter Figure3, Relativefrequency of the Jt1parameterestimatesfrom Equation(4) for cross-area increment(CSAI,cm2 yr-1) fittedto datafrom individualtreesbyfoliagetype andspecies:(A)western hemlock; and (B) balsam fir.
92
ForestScience 46(1)2000
B. Balsam Fir
5
10
15
20
25
5
10
15
20
Cross-Sectional AreaIncrement (cm2yr'l)
Figure 4. Comparisonof observed (1) and predicted crosssectionalarea increment(CSAI)from Equations(4) (-- --) and (5)
(
) by species:(A) western hemlock;and (B) balsamfir.
Currentfoliage Bz0
Species Western hemlock Balsam fir
Totalfoliage Bz0
14
19
2
28
7
0
0
6
19
1
9
15
for mechanical andphysiological support(Longet al. 1981). Thedecrease in growthperunitfoliagewithincreasing depth into crownmay be the resultof severalinteractingfactors including:(1) decreased photosynthetic rateswithincreased foliageage(Schultzeet al. 1977,JarvisandLeverenz1983), and(2) decreased photosynthetic rateswith decreasing light intensity(Woodman1971). Even thoughrespirationrates decreasewith increasingdepthinto crown (Brookset al. 1991),
the
net
result
is
an
B. Total Foliage
A. Current Foliage
Table 5. Number of trees exhibiting significant positive or negative effects of DISTQFOLin Equation (5).
increase
in
Diala•c,• Blow •adralic bt•an Foliag• HI, (m)
Fi9ure 5. Vertical trtnndsin cross-st-•tionalarea increment per
unitfoliage{cml½m'l) relativeto quadraticmeanfoliageheight for western hemlockby foliage age class:(A) Currentfoliage area
the
(cmz) above;and (B)Totalfoliagearea(cm2) above.
respiration:photosynthesis ratio (Sprugelet al. 1991). The commonlyheldassumption thatcross-sectional area
(Larson1963, Long et al. 1981, Potbieret al. 1989). The primaryfactorcontrollingthesechangesin growthandcarbon allocationmay be the ratio of photosynthetic tissueto respirationtissue(Lavigne1991,JackandLong 1992). The allocationof growth and the resultingstem form represents a balancebetweenmechanicalandphysiological
increment remains constant below base of the live crown
(Mitchell 1975, Valentine 1985, 1988, Osawa et al. 1991)
alsois not justified basedon the resultsof this study.For cross-sectional area increment, the coefficients associated
withquadratic meanfoliage height ([•2),atthespecies level, weresignificantly lessthanzerofor westernhemlock(Table 2) andsignificantlygreaterthanzerofor balsamfir (Table3).
supportof thestemsandcrowns(Longet al. 1981).A basic premiseof the growthmodelsderivedfrom thepipemodel theoryis that increasingfoliagecorresponds to increasing growth.Whilethispremiseisgenerallyappropriate, thestrict interpretationof constantproportionalitybetweenfoliage andgrowthis not.Allocatingcarbonbetweensitesof photosyntheticproduction(sources)and utilization (sinks) is a major factor controllingtree growth (Dixon 1990). The allocationcoefficientfor stemwoodproductionhas been shownto vary continuouslyalongthe stemin responseto crown structureand the associated physiologicaland envi-
Resultsat theindividuallevel werelessconclusive(Table 4).
The negativecoefficientsin westernhemlocksuggestthat, belowthepointof maximumcrowndevelopment, growthis no longerproportionalto eithercurrentor totalfoliagearea, butratherdecreases withincreasing distancefromthefoliage (Figures1E and 4A). Growthmay actuallyincreasefrom quadratic meanfoliageheightto baseof thelive crowndue to addedfoliage,butmaydecrease slightlybelowbaseof the live crown.It is interesting to notethatfor 31 of 35 western hemlocktrees,cross-sectional areaincrementatbreastheight wasapproximately 5% lessthantheincrementatcrownbase, and was, at the time of destructivesampling,the slowest growingpointbelowbaseof thelive crownon thesetrees.
ronmental
modifications.
These coefficients
also are ex-
pectedto changeover time as prioritiesfor photosynthate allocationchange(Bassowet al. 1990).Constantallocation coefficientsare merely estimatesof averagegrowth rates (Bassowet al. 1990), and may not reflect the vertical and temporaldistributionof carbon,as demonstrated here for stemwoodgrowth..
Thepositive [•2coefficients associated withthebalsam fir indicatean increasein CSAIbelowcrownbase(Figures1E and4B). Theprimaryreasonfor theincreasing ratein balsam fir andtheconstantor decliningratein westernis mostlikely thedifferences in treedevelopment stage(e.g.,Figure4). The balsamfir treesaveraged6.3 m in heightwhile thewestern hemlocktreesaveragedalmost16 m. The averageheightto
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Table 6. Parameterestimates and standarderrors {in parentheses)for a variation of model (4) describingcrosssectional area increment (CSAI) at crown base.
Currentfoliage Species Western hemlock
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(0.1844)
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