學(xué)士學(xué)位論文-leavesclassificationandleafmassestimation數(shù)模競賽二等獎

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Leaves_Classification_and_Leaf_Mass_Estimation

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LeavesClassificationandLeafMassEstimation

Summary

Forthefirstproblem,weestablishourneuralnetworkmodeltoclassifyleavesoftreesbytakingeightcharacteristicsofleafintoconsideration.Theeightcharacteristicsconsistofsawtoothnumber,petiolelength,bladelength,bladewidth,bladethickness,leafareaandcirculardegree.Ourresultsaresummarizedinaconclusionthatweclassifyleavesintofourteentypesincludinglinear,lanceolate,oblanceolate,spatulate,ovat,obovate,elliptic,oblong,deltoid,reniform,orbicular,peltate,perfoliateandconnate.Ourneuralnetworkimplementtheclassificationtaskreliablyandcorrectly.

Forthesecondproblem,wesetupourAHPmodeltofigureoutthereasonswhyleaveshavethevariousshapesandcometoaconclusionthatgene,auxin,climateanddiseasearethemainreasonswhichleadtovariousshapes.

Forthethirdproblem,wediscussthisissuefromtheperspectiveofgrowthevolutionaryandhormones,buildcellsmechanicmodeltosolvethisproblemandsumuptheconclusionthattheshapesareinclinedtominimizeoverlappingindividualshadowsthatarecastsoastomaximizeexposure.Theshapeiseffectedbythedistributionofleaveswithinthevolumeofthetreeanditsbranches.

Forthefourthproblem,weusestatisticalanalysisknowledgetoanalysethedataamongtreeprofiles,branchingstructureandleafshapes,aftermathematicallyanalyzing,finallyfindthatleavesshapeshaveadirectrelationwiththetreeprofileandbranchingstructure,

Forthefifthproblem,weformulateourvolumetricmethodforleafmassestimationandlinearregressionmodelforseekingandcomparingthecorrelationbetweentheleafmassandtreeheight,treemassandcrownvolume.Weobtainthatcrownvolumehasthehighestcorrelationwithtreeleafmass.Sowemakeuseofthecrownvolumetoestimatetheleafmass.

Atlast,wewriteonepagesummarysheetofourkeyfindings.

Keywords:neuralnetwork,leafclassification,leafmassestimation,AHP,leafshape,volumetricmethod,linearregressionmodel

Contents

Contents

0

Ⅰ.Introduction

1

Ⅱ.SomeDefinitions

1

Ⅲ.GeneralAssumptions

1

Ⅳ.Symbols

2

Ⅴ.Problemanalysis

2

Ⅵ.Models

3

6.1Neuralnetworkmodeltoclassifytreeleaves

3

6.1.1Neuromime

3

6.1.2Multi-layerperceptronnetwork

4

6.1.3Back-propogation

5

6.1.4NN’susetoclassifyleaves

6

6.2Studyingthereasonsofthevariousshapesthatleaveshave.

6

6.2.1SetupaAHPmodeltovaluethesebasefactors

6

6.2.2Pairedcomparisonmatrixstructure

7

6.2.3Calculationoftheweightvectorandtheconsistencytest

8

6.3Optimizeleavesshapeformaximizeexposure

9

6.3.1Explainandanswerrequirment

9

6.3.2SetupaElasticmechanicsmodel

9

6.4Treeprofileandbranchingstructure’sinfluenceonleafshape.

10

6.4.1Analysisabouttheimpactoftreeprofiletoleafshape

10

6.4.2Electrictreebranchangle’simpactanalysis

13

6.5Estimationoftheleafmass

14

6.5.1Buildupavolumetricmodel

14

6.5.2Thecorrelationofleafmassvs.meancrownradius’scubic

15

6.5.3Thecorrelationbetweentheleafmassandtheheightofthetree

16

6.5.4Thedryleafmassvs.thevolumeofthetree

17

6.5.5Therelationshipbetweentheleafmassandmeancrownradius

18

Ⅶ.Conclusions

19

Ⅷ.StrengthsandWeaknessoftheModel

19

Ⅸ.FutureWork

20

Ⅹ.References

20

KeyFindings

21

Ⅰ.Introduction

Asisknowntoall，therearenottwoleavesexactlyalike.Plantleaveshavediverseandelaborateshapesandvenationpatterns.Thebeautyofthemhasattractedcuriosityofmanypeopleinvolvingbiologists,physicists,mathematician,artists,computerscientists,etc.foralongtime.Theleafstudyofforestsandofindividualtreeisimportanttounderstandresourceallocationoftrees,atmosphere—biosphereexchangeprocesses,andtheenergybudget,itwouldalsobevaluableforindividualtreegrowth.

Theaimofthisarticleistodevelopmodelsforleafshapesclassificationandtofigureoutthemainfactorswhichleadtothevariousleafshapes.Atthesametime,wefindouttheinteractionbetweentree(It’sprofile/branchingstructure)andtreeleaf.Thoughtherearesomanymethodstoestimatetheleafmass.Wesolvethisproblemthroughacorrelationbetweentheleafmassandthesizecharacteristicsofthetree.

Ⅱ.SomeDefinitions

Leaf

Toaplant,leavesarefoodproducingorgans.Leaves"absorb"someoftheenergyinthesunlightthatstrikestheirsurfacesandalsotakeincarbondioxidefromthesurroundingairinordertorunthemetabolicprocessofphotosynthesis.

Phototropism[1]

Phototropismisdirectionalgrowthinwhichthedirectionofgrowthisdeterminedbythedirectionofthelightsource.Itcausestheplanttohaveelongatedcellsonthefarthestsidefromthelight.Phototropismisoneofthemanyplanttropismsormovementswhichrespondtoexternalstimuli.

PolarAuxinTransport(PAT)[2]

PATistheregulatedtransportoftheplanthormoneauxininplants.Itisanactiveprocess,thehormoneistransportedincell-to-cellmannerandoneofthemainfeaturesofthetransportisitsdirectionality(polarity).Thepolarauxintransporthascoordinativefunctioninplantdevelopment,thefollowingspatialauxindistributionunderpinsmostofplantgrowthresponsestoitsenvironmentandplantgrowthanddevelopmentalchangesingeneral.

ApicalDominance[3]

Itisthephenomenonwherebythemaincentralstemoftheplantisdominantoverothersidestems;onabranchthemainstemofthebranchisfurtherdominantoveritsownsidebranch.

Ⅲ.GeneralAssumptions

Theinfluenceofvariationinthicknessofleavescanbeneglect.

Wedonottaketheinfluenceoftheartificialfactorintoconsideration.

Regardlessoftheinfluenceofdeformationofcell.

Weregardthecrownofthetreeasahalfsphere.

Theleavesinthecrownareeventlydistributed.

Neglectgenicmutationinfluence.

Ⅳ.Symbols

symbol

Instructions

climate,disease,auxin,gene

thelargesteigenvalue

eigenvectors

consistencyratio

consistencyindex

thepointaleaflocateoncoordinatesystem

acoefficientrelatedonleafshape

Treebranchangle

theleafmass

（Mark：Othersymbolswillbegiveninthespecificmodel）

Ⅴ.Problemanalysis

Thefirstquestionrequiresustobuildamathematicalmodeltodescribeandclassifyleaves.Wethinkthatthestandardofclassificationistheshapeofleaf.Soweneedtostudythecharacteristicsofleafandtoensurethathowtodefineatypeofleafbythecombinationofsomecharacteristics.Inaddition,weshouldfigureouthowandhowmuchthesecharacteristicshaveinfluenceondefiningatypeofleaf.Sowetakeeightcharacteristicsintoconsiderationincludingmastersawtoothnumber,petiolelength,bladelength,bladewidth,bladethickness,leafareaandcirculardegree.Wefindthatneuralnetworksholdthecapacitytoprocesshugedataandcanbeusedtodescribecognition,classificationandsomeotherintelligentbehaviors.Sowemakeadecisiontousetheneuralnetworkstomakeaclassificationoftreeleaves.

Thesecondquestionrequiresustofigureoutthereasonsthatwhytheleaveshavevariousshapes.Itiseasytoknowthattheshapeofaleafmainlydecidedbythegeneofthetree.Butweknowthattheleavesofthesametreealwayshavedifferentshapeswiththesamegenes.Sowecandrawaconclusionthattheshapeofleafisnotonlydecidedbythegeneofthetreeaswellasinfluencedbyenvironmentalfactors.WechoosethesefactorstoanalyzethespecificinfluenceontheformingprocessoftheshapeofleafbyusinganAHPmodel.

Thethirdquestionwantsustogetknowofthatwhethertheleafhavea“hobby”tokeepastatetomaximizeexposureandminimizeoverlappingindividualshadowsthatarecast.Inaddition,iftheshapeofleafiseffectedbythedistributionofbranchesandthevolumeofthetree.Soweshouldmakeasurveytomakeitclearthattherelationshipbetweencrown’ssurfaceareaandtheleafareaofatotaltree.Thenweneedtostudythesunshine’sinfluenceontheformationofleaf.

Wethinkthefourthquestion’saimistoresearchthatwhetherthetreeprofileorthebranchingstructurehasinfluenceonleafshape.Inthisquestionwethinkthatthe“profile”ofatreeisthecrown,andthereisapossibilitythatdifferentcrownhasdifferentinfluenceontheleafshape.

Thelastquestionisrequireustofindacorrelationbetweentheleafmassandthesizecharacteristicsofthetree(height,mass,volumedefinedbytheprofile),andthenmakeuseofoneormoreofthischaracteristicstoestimatetheleafmassofatree.

Ⅵ.Models

6.1Neuralnetworkmodeltoclassifytreeleaves

Ourdutyistofindanapproachtohowtoclassifyleaves.WeuseNeuralnetworkmodeltoclassifytreeleaves

Asforclassification,Neuralnetworkmodelisgreatlyabletogetafairlyidealconclusion.Todistinguishoneleafshapepatternsfromeachother,Neuralnetworkmodelisoptimal.Throughastudysampleprogressonandoff,inwhichweadjustaccordingly.Eventuallyourmodelisso“smart”astoidentifydifferentleafshapes.Aleafsamplecharacterize8featuresasmentioned-above.Anditisnecessaryforustoexplainthemodelandweseparateasthreepartstoexpatiate.

6.1.1Neuromime

Thefollowgraphisabasepartof.

Figure6.1-1：neuromime

Solutiontoinputsignal:

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Whereistheweight,istheinputnodevalue:

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isThresholdvalue:

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isactivationfunction,istheoutputofaneuroninthesuccessivelayer.Theactivationfunctionisanonlinearfunctionandisgivenby:

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6.1.2Multi-layerperceptronnetwork

Thisisthemainstructureof.

Figure6.1-2：Multi-layerperceptronnetwork

ThestructureoftheArtificialNeuralNetworkANNinthisworkcontainsthreelayers:input,hiddenandoutputlayersasshowninfigure6.1-2.Weuseinputlayertoinputthecharacteristicsoftheleaves.Eachlayercontainsandnodes.Thenodeisalsocalledneuronorunit.ThisstudysummarizedeightfactorsforANNinput,thatistosay.Theeightinputunitsaresawtoothnumber,petiolelength,bladelength,bladewidth,bladethickness,leafareaandcirculardegree.

Forthehiddenlayerwemake.Thefunctionoftheoutputlayeristooutputclassifiedinformationcorrespondingtotheinputdata.Thevalueofrangesfromthetypesofleavesweneedtoidentify.Theisdenotedasnumericalweightsbetweeninputandhiddenlayers,betweenhiddenandoutputlayersasalsoshowninfigure6.1-2.

Infact,asforasampleof“”,theinputofthehiddenlayeris:

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Thecorrespondingoutputstate:

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Therefore,thesuperimposedsignalreceivedis:

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Thefinaloutputofthenetworkis:

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Wehopethefinaloutputisidealization.Forexample.Forexample,afterlearningmapleleaf‘sfeatures,iftheoutputisliketheformof,wecalledtheoutputlikethistheidealoutput,theidealoutputisnotedfor.

Figure6.1-2：Differenttypesofshapes

Linear.Lanceolate.Oblanceolate.Spatulate.Ovate.Obovate.Elliptic.Oblong.Deltoid.Reniform.Orbicular.Peltate.PerfoliateConnate.

6.1.3Back-propogation

Inordertominimizingthedifferencesbetweenactualoutputanddesiredoutput，wechooseBPalgorithm,whichisonepartof.

Assetforth,theerrorobtainedwhentrainingapair(pattern)consistingofbothinputandoutputgiventotheinputlayerofthenetworkisgivenby:

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Whereisthethcomponentofthedesiredoutputvectorandisthecalculatedoutputofthneuronintheoutputlayer.

Combine

GOTOBUTTONZEqnNum106698

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(8)

with

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(9)

,wecandraw:

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Thisisanonlinearfunctionwhichiscontinuouslydifferentiable.Inordertoobtaintheminimumpointandthevalue,themostconvenientistousethesteepestdescentmethodtogettheminimalvalueof,when,wegettheidealvalueofthevariablesand.

6.1.4NN’susetoclassifyleaves

Through,singleseveralmodelsleavesandgroupingandnumberofthem.Then,learningeachgroup,isacquaintanceeachmodels.Ifwanttoclassifyoneleaf.Weareabletolettosolvethisproblem,eventually,weclassifytheleafaslike-model.

6.2Studyingthereasonsofthevariousshapesthatleaveshave.

Leaveshaveavarietyofforms.Therearelotsofreasonsaccountforleavesvaryinginshapesandsize,listedasfollows:Overall,thereasonscanbedividedintoexternalandinternalfactors.

Externalfactors:

Seasonsandclimate(includingwind,sunlight,moisture,temperature);

Plantdiseasesandinsectpests;

Artificialfactor;

Internalfactors:

Deformationofcells,moisturelossofMesophyllcellsmaycausevolumedecrease;

Phytohormoneauxin;

Differencegene.

webelievethatthereexits4basefactorsthatleadtothevarietyofleavesshape.Theyareclimate,disease,phytohormoneandgene.Andweendeavorfindoutreasonstothem.

climate:thechangeofsunshine，water，temperature，humiditywhichaltersleavesshape.

disease:througheffectingtheactivityofanenzyme,sothatinfluenceleavesshape.

Phytohormoneauxin:haveinfluenceongeneexpression

gene:throughDNAdeterminethegeneralleafshape

6.2.1SetupaAHPmodeltovaluethesebasefactors

Wesolvethisproblembasedonthereasonslistedabove.Afteranalyzingallofthem,weholdanopinionthathumanattemptisusuallyfairlyhaphazard.Sinceweviewalltheleaves’livingenvironmentisstable,wedon’ttakeartificialfactorintoconsideration.Wedefinite"totalimpact"as"targetlayer”,andclimate,disease,phytohormone,geneasthe"criterionlayer".Asshowninthefollowingfigure6.2-1:

Totalimpact

Climate

Disease

Auxin

Gene

Figure6.2-1：reasonsforthevariousshapes

6.2.2Pairedcomparisonmatrixstructure

Toanalyzetheeffectsofelectricvehicles’widespreaduseontheenvironment,social,economicandhealth,weeachtaketwofactors:

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Theyareusedtorepresentenvironmental,economic,socialandhealthbyturns.Allresultsareavailablethefollowingpairwisecomparisonmatrix:

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Obviously:

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Theresultweusedpairedcomparisonofthepairedcomparisonmatrixis:

Whenwetakecomparisonofthemqualitatively,therearefiveclearhierarchyinpeople'smindsusually,whichisexpressedas:

Table6.1-1:themeaningoftheMeasure1-9

Meaning

1

andhavethesameinfluence

3

hasaslightlystrongerinfluencethan

5

hasstrongerinfluencethan

7

hassignificantlystrongerinfluencethan

9

hasAbsolutelystrongerinfluencethan

2,4,6,8

heratiooftheinfluenceof

to

locatesbetweenthetwonearclasses

1,1/2，，1/9

heratioof

is

the

reciprocal

of

6.2.3Calculationoftheweightvectorandtheconsistencytest

UseMATLABsoftwaretocalculatethepairwisecomparisonmatrixforthelargesteigenvalueandthemaximumeigenvectorthattheeigenvaluecorresponding.Thenwewillnormalizationtheabove-mentionedvector,thenormalizedresultsastheweightvectorofthecomparisonfactor.Followingtheresults:

Usually,thepairscomparisonmatrixisnotthesamearray.Butifitsfeaturevectorsofeigenvaluecorrespondingcanbeusedasaweightvectoroffactorstobecompared,theextentofitsinconsistencyshouldbewithintherangeof:

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Selecting0.1inthistypehasacertainsubjectivewishes.

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representsconsistencyindex,representsitsRandomConsistencyIndex,representsitsconsistencyratio.

Table6.1-2:

1

2

3

4

5

6

7

8

9

10

0

0

0.58

0.90

1.12

1.24

1.32

1.41

1.45

1.49

Fromtheequationabovewecandraw:

Fromthetableabove,wecanget:

Thecorrespondingis

Thismeansourmodelhaspassedtheconsistencytest,canbeusedasaweightvector.

Fromtheanalysisweelicitaconclusionthatgeneismaximallyimpacted,andthengene,phytohormone,climate,deaseisfollowing.

6.3Optimizeleavesshapeformaximizeexposure

6.3.1Explainandanswerrequirment

Fromthemodel2’sconclusionwegetabove,weacquaintthatsunlightisacriticalfactorforplants.Plantsarephotoautotrophs,obtaintheirownenergythroughphotosynthesisandproduceoxygeninthemeantime.Fromevolutionaryconsiderations,itseemsthattheleavesalwaysinafavorabledirection,sothattheycanmaximizetheirexposuretothesun.Asisconsideredabove,sunlightchangesthebladeshapethroughinfluencingthedistributionofgrowthhormones.Thuswediscussthisissuefromtheperspectiveofgrowthhormones.Firstofall,weneedtoknowmorespecificallyhowgrowthhormonesaffectleaves.Growthhormonesisadirectionaltransport,butsometimesittransportstothebacklight.Byinhibitingthegrowthoftheshadedsidetoeffectsthephototropismmovementofplants.Duetothisphenomenonleaves“dotheirbestefforts”makethemselvesexposed.Inotherwords,Itistominimizemutualshadingimpact.Wetrytobuildcellsmechanicmodeltosolvethisproblem,meanwhile,explainreasonsforthisphenomenon.

Conclusion:plantsalways“optimize”theirleavesshapeformaximizeexposure.Putanotherway,theyare“minimize”overlappingindividualshadowsthatarecast.Theresultcanprimelyexplainthereasons.

6.3.2SetupaElasticmechanicsmodel

WechooseElasticmechanicsmodeltosimplifyandimitatePhysicalforceofmesophyllcells.Weassumethateachcellofleafissubjecttotwoforces,oneistheexpansiveforcegeneratedbycytoplasmofcellsinside;anotheroneisexternaltensiongeneratedbycellwall,asshowninfigure6.3-1.

Figure6.3-1：Elasticrigidmodel

Inordertodescribethetwophysicalforcesleafcellssufferedmoreaccurately.Wecanbuildcontractivespringtopresenttheexpansiveforceofthecell,similarly,tensionspringscanbeusedtoexpressthetensionbetweenthecells.Therefore,onlythesetwoforcesbalanceeachother,acellcanstayinstablegeometrywhichcanusethefollowingequationtoexpress.

Whereiscells’originallengthinthesaturatedstate,presentsthelengthafterpowerexpansionandisexternalimpulse,representsspringstiffness.

Thismodeldescribebecauseofthephotosynthesis,cellsaffectedbygrowthhormones,leadingtoaresultthattheshapes“minimize”overlappingindividualshadowsthatarecast,soastomaximizeexposure.

Thus,aleaftendstoincreasethesurfaceareaaslargeaspossibletomaximizemetaboliccapacity,becausemetabolismproducestheenergyandmaterialsrequiredtosustainandreproducelife.

6.4Treeprofileandbranchingstructure’sinfluenceonleafshape.

Westrivetoexploretherelationshipsbetweendistributionofleaveswithinthe“volume”ofthetreeandleavesshape.

6.4.1Analysisabouttheimpactoftreeprofiletoleafshape

Weanalyzethisproblembasedonbiology.Wetaketheinfluenceofwindintoconsiderationinadditiontothosefourfactorsabove.Especiallyforthosehugetrees,spatialdistributionwouldinfluencetheleafshape,namelytheanswertothisquestionispositive.Becauseofthecomplexityofgeneticmutation,wesolvethisproblembasedonenvironmentandauxinwithoutregardtogenemutation.Weusetheimpactofwindinsteadofenvironmentalinfluence.

Auxinsarenotsynthesizedinallcells(evenifcellsretainthepotentialabilitytodoso,onlyunderspecificconditionswillauxinsynthesisbeactivatedinthem).Forthatpurpose,auxinshavetobetranslocatedtowardthosesiteswheretheyareneeded.Translocationisdriventhroughouttheplantbody,primarilyfrompeaksofshootstopeaksofroots.Polarauxintransportwouldleadtophyllotaxisdisorderandleavesofdifferentsize.(fromdevelopmentmechanismoftheleaves).Astheresult,someleaveswellbeenrichedresponsetounevendistributionoftheauxins.[8]Leavesawayfromsunmayhavebiggerleafareatogetmoresunlight.Duetotheinfluenceofwind,thedownwindleavesweresignificantlybetterthanthoseinupwinddirection.

Conclusion:leavesshapeareinfluencedbyleaves’three-dimensionaleffectatthetreeanditsbranches.

Setupmathematicfunctions

Spaceanalysisonatreeasisshowninfigure.Nowwechoosealeaflocatedintostudyits’shape.Determineleafshapethroughintegratedimpactsofauxin,sunlightandwind.Weprovideacoefficienttoindicatetheleafshape:

Figure6.4-1：Treespacecoordinatesystem

Weprovideacoefficienttoindicatetheleafshape:

Wheresisleafarea,isthelengthofleaf,isthewidthofleaf.

Findoutspacefunctionexpressionsaboutthethreefactors:

WhereNisAuxinconcentrations,isLightflux,isleafsurface,iswindforce,iswindspeedandisaconstantcoefficient,whichusedtoquantifytheinfluenceofwind

Thenwecanget:

Auxinconcentrations:

Lightflux:

Weintroduce

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Thenwecanget:

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whereisweedspeed,isLuminousflux,whichusedtoquantifytheinfluenceofwind,isthecorresponingweightcoefficient.

Whethertreebranchinginfluenceleavesshape.

Firstly,webelievethatthemainfeatureoftreeprofilewouldbetreeshape.duetothesubtledifferencesoflight,temperature,humidityandvelocityofwindamongdifferenttreeshapes,sothatleafshapesarevarious.Inaword,leafshapesarerelativetotreeshapes.Thereisagraphwecanoffer.

Table6.4-1:someexplanation

:

crowntypeparameter

leafareaindex

meantiltangle

scatteredlightsitefigure

directlightsitefigure

generallightsitefigure

Table6.4-2:Thecanopycharacteristicsindexesunderdifferenttreeshape

Treeshape

LAI

ELADP

MLA

ISF

DSF

CSF

Opencebtershape

Spindleshape

Disperselaminationshape

Note:Differentlettersincolumnsofthetableshowthesignificantdifferences()

Treeswithopencebtershapehavetheadvantagesofreceivingmoresunlight.It’slowerthantheothertwo,lowerthanspindleshape,lowerthandisperselaminationshape.

haslargerlightsitecoefficientthantheothertwoshapes.Its’,andare、、largerthanandis、、largerthan.

TreeswithSpindleshapehavegrowingweakness,noapparentspacingbetweenlayersandpoorlighting.Its’LAIisthebiggestamongthethreeshapes.Its’increasingrangofISFis25%higherthan,its’andis,lowerthan.

Soitisobviouslythatleafshaperelatestotreeshape.

6.4.2Electrictreebranchangle’simpactanalysis

Inadditiontotreeprofile,treebranchanglesalsoinfluenceleafshapes.Figureshowsdifferenttreebranchangleeffectonthetreearea.

Whentwonewbranchunits(unit3andunit5)arisefromthedistalendofapreviousunit(motherunit)thereisaregularasymmetryinthebranchangles(θ1andθ2,respectively)aprevioustheoreticalmodelfortreelikebodiestodevelopareliablecomputersimulationoftreegeometryforthisspecies.Thetreelikebodiesintheoriginal,theoreticalmodelweredevelopedwithonlytwoparameters,theratioofmothertodaughterbranchunitlengthsandtheasymmetryofforking.

Figure6.4-2:Treebranchangle’simpactonleafarea.

Figure6.4-2isvariationoftheeffectiveleafareaofabranchtierdependingonthebranchangles,and.ThetiterofthefivelateralbranchcomplexesissimulatedwiththreeordersofbifurcationaccordingtotherulesofTerminalia-branching.Thedivergenceangleofthefirstbranchunitofeachbranchcomplexequals.Thesignofthebranchanglesofthefirstbranchunitineachsuccessivebranchcomplexalternate.Theratiosofbranchlengthsofunits3and5tothatoftheirmotherunitare0.94and0.87,respectively.Theradiusoftheleafdiskapproximationis0.8wherethelengthofthelongestdistalbranchunitisunity.Simulationsofthebranchtiersareprojectedonahorizontalplane.and,respectively,areasfollows:(a)and-;(b)and;(c)and;(d)and;(e)and.Maximumeffectiveleafareais(c)

Figureshowsthattheobservedbranchanglesresultinthemaximumeffectiveleafsurfacepossibleforabranchsystemthatfollowsthispatternofbranching.Inaddition,thenaturalconstraintsonbifurcationoflateralbranchesresultinagreatereffectiveleafareaperleafclusterthanwhenbranchingisunrestrained.

Conclusion:WegettheconclusionthatbranchingpatterninTerminaliaiscorrelatedwithefficientpresentationofleafsurfacetodirectsunlightratherthansimplywithmaximumtotalleafsurface.

6.5Estimationoftheleafmass

Asisknowntoall,atreeusuallyhastensofthousandsofleaveswhichwecannotaccuratelycountevenbymoderninstruments.Soit’shardtocalculatetheleafmassofatreeprecisely.Butwecantrytolookingforsomech