Acta Hortic. 1160. ISHS 2017. DOI 10.17660/ActaHortic.2017.1160.8 Proc. X Int. Symp. on Modelling in Fruit Research and Orchard Management Ed.: E. Costes

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IMapple: a source-sink developmental model for ‘Golden Delicious’ apple trees M.Fiser1,J.Ravi1,B.Benes1,B.Shi2andP.Hirst21DepartmentofComputerGraphics,Technology,PurdueWestLafayetteUniversity,IN47907,USA;2DepartmentofHorticultureandLandscapeArchitecture,PurdueWestLafayetteUniversity,IN47907,USA.Abstract

Functionalstructuralbotanicaltreemodelsareoneofthemostdifficultmodelsinbiology. We introduce IMapple, a model for ‘Golden Delicious’ apple (Malus × domestica)treesbasedonsource-sinkdescriptionsofplantresources.Ourmodelusesprecisegeometricalrepresentationswithhighqualitypolygonalmeshesandgeometricdetails.Weintroduceanovelsimulationalgorithmthatprovidesdetailedinformationabout leaf irradiance by simulating direct illumination and self-shadowing. Thisprovidesinformationaboutincominglightandenergythatisdistributedbyasource-sinkmathematicalmodel.Ourmodel aims to be parametric, highly interactive, andusable. Information about the qualitative and quantitative relationships of differentparts of themodel isprovided interactively via a simple graphicaluser interface ordirectlyfrommeasureddata.Thesimulationrunsin3D,atinteractiveframerates,andallowsforfastexperimentationandvisualevaluation.

Keywords:functional-structuralplantmodels,computergraphics,geometricrepresentation,Malus×domestica

INTRODUCTIONModels that describe and predict tree growth can have a number of potentialapplications.First,theyareusefultoholisticallyintegrateknowledgefromanumberofareasincludingtreephysiology,environmentalphysiologyandorchardmanagement.Secondlytheycan be useful predictive tools under changing environmental, cultural or managementscenarios, and thirdly they offer the potential to develop useful, real-world educationalopportunitiesforstudents,growersandscientists.STUDY

GenerativemethodsforvegetationmodelingOneofthefirstproceduralmodelsofplantswasdescribedbyHonda(1971).Theauthordefinedatreeasarecursivestructureofbranchessimilartoeachother(self-similarity).Everybranchwasastraightlinesegmentin3Dthatforkedtotwodaughterbranchesattheend.Theonlyparameterstothemodelweretherelativeratioofbranchlengthsandbranchingangles.Themodelstartedwithasinglebranchasthemaintrunkanditerativelyforkedbranchesfrompreviousiterationsusingthegivenparameters.Theresulting3Dmodelswereverysimplebutatree-likestructurewasapparent.Smith (1984) introduced the term graftal parallel graph grammar language, whichshares properties with fractals and can be used for procedural plant modeling. A plant'sstructureexhibitsfractalbehavior.Atthattime,fractalswerewellknownandreceivedmuchattention. Generalization of a plant structure to fractal rules was a very natural extensionbecause plants in real life do have self-similar parts in different scales, which is thefundamentalpropertyoffractals.Inhiswork,theauthoralsotalksaboutL-systems,aparallelgrammarinventedbyLindemayer(1968)forcellularorganismmodelingbutusedwithgreatsuccess for plant modeling as well. An important extension to L-systems called Open L-systemswaspresentedbyMechandPrusinkiewicz(1996)whointroduceanenvironmentalmodule that serves as a feedback loop for an environment state and transfers the

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environmentalinformationtoanL-systemstring;standardL-systemtechniqueslikecontext-sensitive rules can be used to take advantage of this additional information and affectrewriting. The authors demonstrated their system on trees growing close to each other,competingforspace.Biologically-basedmodelsOne of the first biologically-based models in computer graphics was the work of deReffyeetal.(1988)whointroducedvariousplantmodelsinsimpletermsofproceduralandgenerativemodeling. Chiba et al. (1994) presented a growthmodel that takes into accountseveralbiology-basedgrowthregulationssuchaswithering,heliotropism(attractiontolight)and geotropism (tendency to grow against gravity). Their model simulates withering andheliotropismbyestimationoftheamountofsunlightreachingindividualplantparts.Growthregulationsaresimulatedwithimaginaryhormonesproducedbytreebuds.Ourworkbuilds on theprevious research of Forshey andElfving (1989). Theirworkprovided the necessary data describing apple tree vegetative growth and croppingcharacteristics. Trees do not grow constantly over time. The growth of most tree organsoccursinflushesthatoftenrespondtodifferentenvironmentalconditionsatvarioustimesofthe growing season (Rom, 1996). Year-to-year seasonal variations and other factors alsoinfluence tree growth (Forshey and Elfving, 1989). These authors also provide detaileddescriptionsofmanyprocessesinthetreeincludingshootgrowth, leafdevelopment, flowerbudinitiation,fruitgrowth,andbranchsecondarygrowth.More recently, various approaches have been used to describe the impact of externalinfluencesongrowthprocessesofplantssuchaslight(Runionsetal.,2007),wind(Pirketal.,2014),andplantmechanicalbehavior(Pirketal.,2012;Michaelrajetal.,2003).TheMAppleTmodel by Costes et al. (2008) is a simulation of apple tree development that uses bothbiomechanical and stochastic approaches to achieve realistic results. The underlyingsimulationsystemusedL-systemsforthetreegenerationandhiddensemi-Markovchainsforrandomizationof tree topologyandgeometry.Similarly,agrowthsimulationofpeachtrees,calledL-PEACH,wasdescribedbyLopezetal.(2010).Theirmodelislargelybasedoncarbonassimilation, distribution, and utilization. They simulate tree resources using an electricalcircuit analogy. The system also allowed simulating pruning and its impact on treeproductivity.TheEduAPPLEmodel(Koheketal.,2015)simulatesresponsesofappletreestovariouspruningandtreemanagementinterventionsbasedonprioritizingflowratesbetweenmetamerswithinthetree.In the work described here, we also use an electrical circuit analogy to simulateresourceswithin the tree similar to the approachusedbyMAppleT andL-PEACH.However,theses previous studies are narrowly specialized for apples and peaches respectively. Incontrast,ourmodel canbeapplied toanykindof trees.Also, IMapple incorporates tree-to-tree interaction therefore thegrowthofanorchardcanbesimulated,andnot just thatofasingle, isolatedtree.Thereforetheobjectiveofthisstudywastodevelopanovel, interactivetree growth model using apple as an example. Furthermore, we sought to incorporate adetailedlightsimulationmoduleandtousedetailedandrealistic3Dmeshestosimulatetreeresponsestoenvironmentalfactors.Systemoverview:biologicaltreeabstractionWerepresentatreeusinganabstractionshowninFigure1.Thebasicbuildingblockofa tree isan internodewhich isanuninterruptedpartof thestembetweentwonodes.Otherorganscanonlybeattachedatnodes(theendsofinternodes).Thetermbranchusuallyrefersto a groupof consecutive internodes.Agrowingunit is a set of consecutive internodes thatgrewinthesameyear.Leavesare locatedatnodesandalwaysoccur togetherwitha lateralbud in the leaf axil. A terminalbud is always located at the distal end of the branch.Otherorganssuchasflowersandfruitcanbealsorepresentedbasedontheneedsoftheuser.Budsare important parts of the tree structure because all growth emanates from them. Forexample,brancheshaveterminalbudsthatmaydevelopterminalshootsand/orfruit.

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Figure1. An example of a tree structure and corresponding electrical circuit. In areinternodes,Lnareleaves,Bnarebuds,Mnareterminalmeristems,andAnarefruit.Mathematically, the system represents a biological tree using a rooted tree datastructure.Thisrepresentationwillbecalledagraph.Everynodeofthegraphrepresentsatreeorgan.Nodescanbedefinedbytheuserbuttherearesomecommonnodespredefinedinthesystem,suchasinternode,leaf,fruit,bud,orflower.Therearetwomaintypesofnodes:logicand spatial. Thedifference is that the spatial nodedoes have somephysical representationwhereas the logic node does not. The position of every node in 3D space is relative to itsparentnode.Theonlynodewithabsolutepositionistheroot.Everyspatialnodehaslengthanddirectionattributes.Spatialnodesmayhaveextrainformationbasedontheuser'sneedsuchas thicknessor shape.However, the lengthanddirectionare enough todetermine thepositionofchildnodes inthegraph.Theabsolutepositionofanynodecanbecomputedbyevaluating the position of all spatial nodes on the path from the root to the node. Forperformance reasons, thisdistance calculation is cached in everynode.When the lengthordistanceattributesof anodeare changed, all cacheddistances in a sub-treeof theaffectednodeareinvalidated.Thisensuresvalidresults.GrowthsimulationThe tree growth simulation is performed iteratively by discrete time stepsΔ Thenumber of time steps per day is a user specified parameter that controls precision of thesimulation.Inourexperimentsthetimestepwasintherangeofhours1-8ofsimulationtime.The shorter the time step is, themoreprecise the simulationbecomes, but processing andsimulation becomemore time consuming. The system overview is shown in Figure 2. Thegrowth model represents the logic necessary to perform the tree growth simulation. Thesimulationisperformedbyinvokinguserdefinedactions.Threemainactionsareperformedat every simulation time step: time-step-start, resourcetransport, and time-step-end. Time-step-start and time-step-end actions perform the growth logic required before and afterresource transport, respectively.Forexample,a time-step-startactioncomputes theamountofresourcesproducedbya leafbasedontheamountof incoming lightbeing intercepted inthe environmental simulation. The time-step-end action can then use the transportedresourcesforgrowthsimulation.

Figure2. Thegrowthsimulationsystemoverview.Thesystemismodularandtheindividualstepsandtheirorderisdeterminedbytheuser.

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Thegrowthsimulationisamodularsystemdividedintopartscalledmodules.Thewholegrowth model could be theoretically developed as a single module but it would make themodeldifficulttomaintainandextend.Asinglemoduleusuallyhasasingleresponsibility.Forexample,aleafmodulesimulatesleaves,afruitmodulesimulatesfruit,etc.Modulescaneasilycommunicatebetweeneachothersothattheremaybemodulesthatservequiteagenericrolesuchasatemperaturemodulethatprovidesenvironmentalinformationtoallothermodules.Actions are invoked serially over allmodules - one action is invokedon allmodules beforeinvoking the next action. Serial processing is important to ensure the determinism of theprocess.Theorderof action invocationdependson the typeof action.For start actions theorder is from the beginning to the end of the list ofmodules. For end actions the order isreversed-fromendtobeginning.Thisallowsmodulesearlierinthepipelinetopreparedataforlatermodulesin*-startactions.In*-endactionstheearliermodulesareinvokedafterthelateronessotheycancollectthedata.LightsimulationSinceinterceptedlightrepresentsthemainsourceofenergyintothetreeandorchardsystem, our algorithm evaluates illumination of each leaf. A physically correct simulationwoulduseMonteCarlopathtracingthatwouldallowforcalculationofmultiplelightbounces.In apple trees, themost important illumination isdirect illumination that canbe efficientlyconvertedintoavisibilityproblem(Benes,1997).Weviewthetreefrommultipledirectionson a hemisphere and check the visibility of each leaf. A visible leafmeans it is illuminatedfrom the given position. Each point on the celestial hemisphere has an irradiance valuecorresponding to the amount of light emitted from that point of sky over the day.We thensimplyrenderthetreecanopyfrommultipledirectionsbyusingZ-buffervisibilityandcountthevisiblepixelsforeachleaf.Inordertodistinguishtheleaves,eachleafiscalculatedwithadifferentanduniquecolor.Thefinalvalueneedstobecosine-corrected.ThelightsimulationisveryfastevenforlargertreesbecausethedrawingisacceleratedbytheGraphicalProcessingUnit(GPU).Moreover,thetaskisparallelizablesowhiletheGPUisrenderingthenextsampleimage the CPU is counting pixels from the previous iteration. In our implementation onesample isusuallycompleted in5-10msand100samplescanbecomputedin lessthanonesecond.ResourcetransportationsimulationThegrowthsimulationsystemcontainsresourcesimulationandatransportationsub-system.Thisismainlytoallowrealisticsimulationbyallowingtheusertoimplementgrowthmechanisms that are driven by resources. Resourcesmay include water, carbohydrates, ornutritionanditisuptotheusertodefine.Theresourcetransportsystemismotivatedbytheresearchof(Lopezetal.,2010)whosimulatedtheflowofcarbohydrates inatreeusingtheanalogy of an electrical circuit. This is possible due to analogies between hydraulic andelectric current flow as shown in Table 1. An electrical circuit is built from two logicalcomponents:connectionsandsource/sinks.Aconnectionisaresistorandasource/sinkisaresistorwithavoltagesourceconnectedinseries.Everynodeofatreemaybeeitherofthetwo logical parts or nothing (ignored). For example, an internode is both a connection andsink,aleafisasource,andafruitisasink.Thereisnoneedfortheexplicitconstructionoftheelectrical circuit. Figure 1 shows amodel of a tree and the corresponding electrical circuit.Notethatinthisexamplebudshavenoelectricalproperties,thus,arenotrepresentedintheelectriccircuit.Table1.Analogybetweenhydraulicandelectricterms.

Resource analogy Hydraulic entity Electric entity SymbolAmount of resources Mass or volume Charge q Resource flow Flow rate Current i Resource concentration Pressure concentration Voltage v(e) Resource flow resistance Hydraulic resistance Resistance r

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CircuitsolvingAllcurrentsinanormalizedcircuitaresolvedbythetechniqueoffoldingandunfoldingdescribed by Lopez et al. (2010). Currents are computed iteratively because electricalcomponents may be non-linear their voltage/resistance may depend on the current orpotential.First,allelectricpropertiesofallcomponentsareupdatedbasedoncurrentsfromthepreviousiteration.Thisisdonebycallingaspecialupdateactionthatmaybeimplementedbyanymodule.Then,currentsarecomputedusingfold-unfoldoperations.Ifthenewcurrentsnot converged, another iteration is performed. Otherwise resources are transported usingcomputedcurrentvalueandsimulationtimestep.3DgeometricmodelrepresentationThe 3D model is generated as a triangular mesh and as a single solid object with asmooth surface, without holes, self-intersections, or other defects. The mesh has a goodtopologyfortexturingandediting.Itisalso3Dprintable.Weusesubdivisionsurfacestodoso.Thegenerationalgorithmhastwomainsteps:initialmeshcreationandsubdivision.Figure3showsseveralstepsofthesubdivisionin3D.Thetreegraphissimplifiedandcubesareplacedatthecenterofeachnode.Thesizeofeachcubeisbasedonthebranchdiameterofeachnode.Thecubesarethenalignedbasedontheincomingandoutgoingbranchestominimizeerrors.Finallythestraightsegmentsbetweentheboxesarefilled.

Figure3. A 2D example of steps of the reconstruction algorithm. a) Initial tree graph. b)Simplificationof the graph. c) Initial cubes creation. d)Cube alignmentbasedonthebranches.e)Connectionofthecubes.IMapple‘GoldenDelicious’appletreemodelWeuselightasaprimaryresource.Waterisnotsimulatedbecauseweassumefarmershave access to enough water to ensure water is not limiting. A module that simulates theeffectsofwaterdeficitsonresourceaccumulation(photosynthesis)couldbeaddedatalaterdate. Temperature affects growth and this is considered in the growthmodel. Temperaturedatawere available from historical data from aweather station located at the farmwherestudieswere conducted.Resource generation is simulatedby twomodules: lightsimulationmoduleand leafmodule.Theformercalculateslightusingasimulationbyestimationofper-leaf illumination and the latter uses this information to generate resources that can betransported to other tree organs. The light simulation framework described above wasadaptedtoreflecttheshapeandcharacteristicsofleavesof‘GoldenDelicious’appletrees.Itiswellknown thatapple leavesorient towards light (Clayton-Greenetal.,1993)and thiswasalsoobserved inourstudy.Thisphenomena is simulatedby shaping the leaves inaV-formandmeasuring their illumination sectors independently. The leaves can rotate towards thedirectionofthestrongestilluminationdirection.Notethatthismaynotnecessarilybeupasotherleavesmaycreateshadeandtheoptimaldirectionmaybesidewaysinthedirectionofthe brightest illumination. The leaf illumination is computed as a sum of four partialilluminationsfrontright,frontleft,backrightandbackleft.The performance of a leaf is the ratio between current captured illumination andmaximal possible captured illumination. The performance of every leaf ismonitored everyiteration. Leaveswith low performance have a high probability of senescing and abscising.

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Every leaf converts received illumination to resources, represented as electric charge. Thisconversionisnotlinearanditdependsonseveralfactors.Weuserelationshipsdescribedby(Lopez et al., 2010). A leaf has charge capacity proportional to its size. Both resourceassimilationrateandsourcevoltagedependontheamountofaccumulatedcharge(Figure4).Themotivationbehindthefunctionsisthatthemoreunusedresourcestheleafhas,thelowertheresourceassimilationratebutthehigherthevoltageofitssource.Highersourcevoltagewill causehigher current flowing from the leaf.Resourceassimilation ratealsodependsoncurrent temperature. The temperature is converted togrowingdegreedays usingmaximumandminimumtemperaturesfrommeasureddata.

Figure4. Resource assimilation rate as a function of accumulated charge (left) and leafsourcestrengthasafunctionofaccumulatedcharge(right).Thegrowth ofeachorganrequiresresources (electriccharge)andeveryorganhasanassociatedcostpervolume.Therateofgrowthisaffectedbytheresistanceofthesinkandtheamountofavailable resources.Weassigned thosevaluesexperimentallyand foreachorganwe used assumed functions. Similarly, functions (graphs) were created for internodeelongation and secondary (thickening) growth, internode sink resistance as a function ofvigor,internodesinkresistanceasafunctionofage,internodesinkstrengthasafunctionofaccumulated charge, terminal branch growth, and fruit growth. More details and exactfunctionvaluesaredescribedbyFiser(2015).Graphsshowingtreeresponsesweredevelopedusing data collected from seven ‘Golden Delicious’ trees growing on G.16 rootstock at thePurdue Meigs Farm, Lafayette, Indiana, USA. Measurements were made on a weekly tomonthly schedule throughout the 2014 growing season. More details are described by Shi(2015).

RESULTSANDDISCUSSIONThesimulationsystemdescribedherewas implemented in .NET framework4.5usingC# programming language. All libraries were compiled to DLLs and user interfaces wereexecutables. The system was developed in Visual Studio 2013 under Windows operatingsystem.OpenGL4.0 andOpenTK frameworkwere used for real-time 3D visualizations andlight simulation. The entire application has around23,000 lines of C# code.Weperformedexperimentstoanalyzethebehaviorof theIMapplemodelof ‘GoldenDelicious’apple trees.Theemphasiswasgiventotheilluminationsimulationwhichisthemostimportantfactorofthe model, since illumination is the main determinant of vegetative growth and fruitdevelopmentandquality(Warringtonetal.,1996).Usinghistoricaltemperaturedataandtheillumination model of a sunny day, the first four years of growth of an apple tree weresimulated(Figure5).Thetreeisnotpruned,however,andduetoself-shadowingitdevelopsarelatively symmetric structure. The figure shows leaves that senesced and abscised due toinsufficientillumination.

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Figure5. Full featuredmodelofGoldenDeliciousapple tree.Temperaturedata is from theyear2014andthe illuminationmodelwasset toasunnyday.Notethattreesarenotpruned.Our framework is capable of simulatingmore than one tree at a time. Simulation ofmultiple trees allows tree to tree interactions to be studied. This is obviously of greatimportance since orchard performance rather than individual tree performance is of theprimaryconcerninrealworldcommercialproduction.Theilluminationsimulationprocessesalltreesatoncetoaccountforshadowingofthetreesbetweeneachother.Figure6showsasmall orchard of unpruned ‘Golden Delicious’ trees. Note that trees in the middle of theplantingaregenerallysmallerandhavefewerlateralbranches.

Figure6. Asimulatedorchardshowstheeffectofmutualshadowingandilluminationascanbeseenonthesmallertrees inthemiddle.Thissimulationwasperformedunder20min.

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Themajorityoffigurespresentedhereweresimulatedwith4-8timestepsperday,360daysperyear.Lightsimulationwasperformedwith7-13samplespertimestepandresolutionofthepixelbufferwas512×512pixelsand1024×1024pixelsforlargertrees.Forsmalltrees(younger than twoyears) the light simulation takes themajorityof theprocessing time.Astreesgrowandbecomelarger,thesimulationstepstakelonger.AllfiguresweregeneratedonaPCwithIntelCorei7-2600KCPUclockedat3.4GHzandNVIDIAGTX580GPU.Simulationof single treeswascompleted inunder twominuteswithmaximummemoryconsumptionofseveralhundredmegabytes.TheexampleinFigure6wasgeneratedin23minutes.Allimagesweregeneratedin7200iterations.CONCLUSIONSWehavedevelopeda3DsimulationframeworkinspiredbyL-peach(Lopezetal.,2010)and MAppleT (Costes et al., 2008) that use the analogy of an electrical circuit. While ourframeworkdescribedhereisappliedtoGoldenDeliciousapples,ithastheflexibilitytoenableit to be adapted to simulate the growth of other tree species. It uses data that is eithermeasuredorassumedusinginteractivegraphs.Preciselightsimulationiscarriedouttakinginto account leaf area, distribution, shape and orientation. Not only is this light simulationimportantbecauseitdeterminestheprovisionofresourcesviaphotosynthesis,butalsoplaysaprimaryroleinfloweringandfruitquality.Calculationsandprocessingareperformedusingthegraphicscardforrapidresults.ACKNOWLEDGEMENTSThe authors gratefully acknowledge support from USDA/SCRI and NVIDIA PurdueCUDA Research Center. Additionally we thank Tristand Tucker and other personnel at thePurdueMeigsfarmformaintainingresearchplots.LiteraturecitedBenes,B.(1997).Visualsimulationofplantdevelopmentwithrespecttoinfluenceoflight.InComputerAnimationandSimulation’97,ThalmannanddePanneeds.(Wien/NewYork:Springer–Verlag),pp.125–136.Chiba,N.,Ohshida,K.,Muraoka,K.,Miura,M.,andSaito,N.(1994).Agrowthmodelhavingtheabilitiesofgrowth-regulations for simulating visual nature of botanical trees. Comput. Graph. 18 (4), 469–479http://dx.doi.org/10.1016/0097-8493(94)90059-0.Clayton-Green,K.,Reynolds,J.,andFerree,D.(1993).Influenceoforchardmanagementsystemonleafangleandleaf orientation in apple trees. J. Hortic. Sci. 68 (5), 679–687 http://dx.doi.org/10.1080/00221589.1993.11516400.Costes,E.,Smith,C.,Renton,M.,Guedon,Y.,Prusinkiewicz,P.,andGodin,C.(2008).MAppleT:simulationofappletree development using mixed stochastic and biomechanical models. Funct. Plant Biol. 35 (10), 936–950http://dx.doi.org/10.1071/FP08081.deReffye,P.,Edelin,C.,Françon,J.,Jaeger,M.,andPuech,C.(1988).Plantmodelsfaithfultobotanicalstructureanddevelopment. Paper presented at: 15th Annual Conference on Computer Graphics and Interactive Techniques(SIGGRAPH1988)(Atlanta,Georgia:ACM,SIGGRAPH),p.151–158.Fiser, M. (2015). Framework for functional tree simulation applied to Golden Delicious apple trees. MS Thesis(PurdueUniversity).Forshey, C.G., and Elfving, D.C. (1989). The relationship between vegetative growth and fruiting in apple trees.Hortic.Rev.(Am.Soc.Hortic.Sci.)11,229–287.Honda,H.(1971).Descriptionoftheformoftreesbytheparametersofthetree-likebody:effectsofthebranchingangle and the branch length on the shape of the tree-like body. J. Theor. Biol. 31 (2), 331–338http://dx.doi.org/10.1016/0022-5193(71)90191-3.Kohek,S.,Guid,N.,Tojnko,S.,Unuk,T.,andKolmanic,S.(2015).EduAPPLE:interactiveteachingtoolforappletreecrownformation.Horttechnology25(2),238–246.Lindemayer,A.(1968).MathematicalmodelsforcellularinteractionindevelopmentIandII.J.Theor.Biol.18,280–315http://dx.doi.org/10.1016/0022-5193(68)90079-9.Lopez,G.,Favreau,R.R.,Smith,C.,andDeJong,T.M.(2010).L-PEACH:acomputer-basedmodeltounderstandhowpeachtreesgrow.Horttechnology20(6),983–990.

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Mech,R.,andPrusinkiewicz,P.(1996).Visualmodelsofplantsinteractingwiththeirenvironment.Paperpresentedat: 23rd Annual Conference on Computer Graphics and Interactive Techniques (SIGGRAPH 1996) (NewOrleans,Louisiana:ACMSIGGRAPH).Michaelraj,J.,Baker,B.,andHart,J.C.(2003).Structuralsimulationoftreegrowth.Vis.Comput.19(2-3),151–163.Pirk, S., Stava,O., Kratt, J., Said,M. A.M., Neubert, B.,Mech, R., Benes, B., andDeussen, O. (2012). Plastic trees:Interactiveself-adaptingbotanicaltreemodels.ACMTransactionsonGraphics31(4),50:1–50:10.Pirk, S., Niese, T., Hadrich, T., Benes, B., and Deussen, O. (2014) Windy Trees: Computing Stress Response forDevelopmentalTreeModels(ACMTrans.Graph),33:6,11.Rom,C.R.(1996).CoordinationofRootandShootGrowth:RootsandRootstocks.InTreeFruitPhysiology:Growthanddevelopment,K.Maib,P.K.Andrews,G.A.Lang,andK.Mullinixeds. (Yakima,WA:GoodFruitGrowerPress).p.53–67.Runions, A., Lane, B., and Prusinkiewicz, P. (2007). Modeling trees with a space colonization algorithm. Paperpresented at: Third Eurographics Conference on Natural Phenomena (Aire-la-Ville, Switzerland: EurographicsAssociation),p.63–70.Shi,B.(2015).Growthparametersfor’GoldenDelicious’appletrees.MSThesis(PurdueUniversity).Smith,A.R.(1984).Plants,fractals,andformallanguages.Comput.Graph.18(3),1–10http://dx.doi.org/10.1145/964965.808571.Warrington, I.J., Stanley, C.J., Tustin, D.S., Hirst, P.M., and Cashmore, W.M. (1996). Light transmission, yielddistribution, and fruit quality in six tree canopy forms of ‘Granny Smith’ apple. J. Tree Fruit Prod.1 (1), 27–54http://dx.doi.org/10.1300/J072v01n01_03.

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