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A symbolic method to analyse patterns in plantstructure whose organogenesis is driven by a multitype

branching processCedric Loi, Paul-Henry Cournède, Jean Françon

To cite this version:Cedric Loi, Paul-Henry Cournède, Jean Françon. A symbolic method to analyse patterns in plantstructure whose organogenesis is driven by a multitype branching process. 2010. inria-00546309

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Cédric Loi et al.. JOURNAL OF COMPUTER SCIENCE AND TECHNOLOGY : 1 March 2010

A symbolic method to analyse patterns in plant structure

whose organogenesis is driven by a multitype branching pro-

cess

Cédric Loi1,2, Paul-Henry Cournède1,2, Jean Françon3

1Ecole Centrale Paris, Laboratory of Applied Mathematics and Systems, 92290 Châtenay Malabry, France2INRIA Saclay - Île-de-France, DIGIPLANTE, 91400 Orsay, France3Université de Strasbourg, 67000 Strasbourg, France

E-mail: [email protected] [email protected] [email protected]

Received March the 23th

Abstract Formal grammars like L-systems have long been used to describe plant growth dynamics.

In this article, they are used for a new purpose. The aim is to build a symbolic method derived from

computer science that enables the computation of the distribution associated to the number of complex

structures in plants whose organogenesis is driven by a multitype branching process. To that purpose,

a new combinatorial framework is set in which plant structure is coded by a Dyck word. Moreover,

the organogenesis is represented by stochastic F0L-systems. By doing so, the problem is equivalent to

determining the distribution of patterns in random words generated by stochastic F0L-system. This

method leads directly to numerous applications like parametric identication for plant growth model.

Keywords Dyck word, stochastic F0L-system, plant organogenesis model, symbolic method

1 Introduction

In botany, the organogenesis is the processduring which the buds of a plant produce neworgans. Formal grammars have long been usedto describe plant organogenesis. In particular,the parallel rewriting grammar introduced by[1] (called L-system) is well adapted to modelthe evolution of branching patterns and its al-gorithmic power has been broadly taken ad-vantage of since [2]. It has thus provided e-cient algorithms and subsequently software lan-guage for plant simulation (see [3]). For struc-tures as complex as trees, bud production rulesare inuenced by many factors and are usually

modelled by stochastic processes. In that case,the organogenesis is represented by stochastic0L-systems. This type of grammar gives in-teresting results from simulation and graphicalpoints of view by increasing the realistic aspectof geometric plants (see [4]). However, theirfull mathematical potential has not been takenadvantage of.

In this article, stochastic 0L-systems areused for a new purpose. The objective is towrite a method allowing the computation of thedistribution associated to the number of com-plex structures in plants whose organogenesis isdriven by a multitype Galton-Watson branch-ing process. This method relies on a symbolic

2 J. Comput. Sci. & Technol., March. 2010, ,

approach derived from theoretical computerscience and the analysis of algorithm (see [5]).Plants are seen as combinatorial structures. Inthis new combinatotial framework, plant topol-ogy is described by a Dyck word. The evolu-tion of the structure of the plant is given by aset of stochastic rules contained in a stochasticF0L-system. Therefore, this framework allowsthe use of powerful methods of combinatoricssuch as the symbolic method (see [5]). Includ-ing a symbolic approach in a dynamic branch-ing structure gives a more complete descrip-tion of the system. It enables the computationof the distribution of patterns in a sequenceof words whose dynamic evolution is driven bya branching process. Such results have manyapplications: comparison of stochastic models,parameter identication . . .

Some basic concepts of botany and themain features of stochastic organogenesis mod-els driven by a multitype branching process arerst recalled in Section 2. Then, a new com-binatorial framework is set up in Section 3.Plants are seen as labelled plane rooted trees.It allows the description of their structures byDyck words. The evolution rules of the organo-genesis model are represented by a stochasticF0L-system. Section 4 proposes a symbolicmethod adapted to plant growth models. Inthe last section, we show how to use the sym-bolic method for the parameter identicationof the stochastic processes during plant devel-opment.

2 Stochastic modelling of organogenesiswith a multitype branching process

Models of plant development (or organo-genesis) describe the dynamic creation of or-gans (internodes, buds, leaves, owers or fruits)and how they arrange to form plant structure.When the smallest scale of interest is that oforgans (and not cells), discrete models are gen-erally used to simulate plant structural devel-

opment.

2.1 Modelling of plant structure

In this article, only the above ground partsof plants are considered. As explained in [6],organogenesis results from the functioning ofundierentiated cells constituting the apicalmeristem and located at the tip of axes. Whenin active phase, this meristem forms buds thatwill develop into agglomerates of organs com-posed of one or several phytomers (also calledmetamers). A phytomer is a botanical entitychosen as the elementary unit used to repre-sent the plant architecture. It is composed ofan internode bearing buds (terminal and lat-eral) and a leaf (see Figure 1). Depending onthe type of plant, the internode may also bearowers and fruits.

Fig. 1. Example of phytomer

Concerning the architecture of the plant,axis and architectural units can be listed intodierent categories depending on their mor-phological parameters (length, diameter, . . .).These categories will be called Morphologi-

cal Categories (= MC) in the sequel. Othernames can be found in the literature (for exam-ple, [6] introduced the concept of physiologicalage to represent the dierent types of axes re-sulting from the meristematic dierentiation ina plant; the concept of MC is more general andis chosen to allow cases that do not correspondto the strict botanical concept of physiologicalage). By convention, the terminal bud of anaxis is thus characterized by the MC of the phy-tomer that bears it (Figure 2 gives an exampleof plant with two MCs).

Cédric Loi et al.: A Symbolic Approach to Analyse Patterns in Plants Structure 3

Fig. 2. Example of plant with two Morphological

Categories (= MC). The fruit has no MC attached.

In the sequel, an organ will be character-ized by its type (internode, bud, fruit, . . .) andby its botanical characteristics (MC, Chrono-logical Age, . . .).

2.2 Stochastic organogenesis

Plant development can be discretised intime. The time between the appearances ofnew shoots (i.e. the time step of the discretemodel) denes theGrowth Cycle (= GC). Forexample, most temperate trees grow rhythmi-cally, new shoots appearing at spring. If we donot consider polycyclism and neoformation, theGC corresponds to one year. The Chronolog-ical Age (= CA) of a plant (or of an organ) isdened as the number of GCs it has existed for.

The structure of a plant changes from oneGC to another. For example, a bud may createnew organs or the botanical characteristics ofan organ may change (such as its CA). A set ofrules called evolution rules denes the wayorgans evolve from one GC to another. In astochastic organogenesis model, an organ mayhave several possible evolutions. In that case, aprobability of occurrence is associated to eachof these. As explained in [7], the probabilitiesinvolved in the organogenesis model are the re-sult of botanical phenomena (bud survival, buddormancy, dierentiation, . . .).

In this article, we consider a class oforganogenesis models that satisfy the followinghypotheses:

1. at each GC, the organs behave indepen-dently: the evolution of an organ is notinuenced by the other organs of theplant.

2. the probabilities of evolution associatedto an organ depend only on its type andits characteristics.

Under these hypotheses, a plant structureof CA N is built recursively by using the fol-lowing algorithmic procedure:

• Initialisation: the structure of the plantat GC 0 is given by a seed (i.e. a bud ofMC 1).

• for all n = 0 · · ·N−1: the plant structureat GC n+1 is built from the plant struc-ture at GC n by replacing randomly allthe organs by one of their possible evo-lutions according to their probabilities ofoccurrence.

The underlying stochastic process associ-ated to this class of organogenesis models isa multitype Galton-Watson branching process(see [8] and [9]). As a matter of fact, the or-gans used to build plant structure can be seenas individuals in a population process. There-fore, the previous algorithmic procedure cre-ates a Markovian sequence of random plantsindexed by the GCs. The evolution rules asso-ciated to the organogenesis model are usuallyrepresented by a set of stochastic automata (see[2] and [4] and Figure 3).

Fig. 3. Example of stochastic automaton for an

organogenesis model


Figure 3 shows an example of organogene-sis model with one MC. p represents the deathprobability for a bud. In this example, a budhas a probability p to die and a probability1−p to create one internode with two lateralbuds. Note that there is no stochastic automa-ton concerning the possible evolutions of aninternode. In that case, it means that intern-odes do not evolve from one GC to another(they stay in the same state with a probabilityequal to 1). The plant on the right is one pos-sible structure occurrence after two GCs. Theassociated probability of occurrence is p(1−p)2.

In the sequel, S will denote a stochas-tic organogenesis model driven by a multi-type branching process. A `plant' refers toa branched structure generated from a seed(i.e. a bud of MC 1) according to the evo-lution rules of S. In the following section, acombinatorial framework is set up to describeplant architecture and its evolution. The struc-ture of a plant is given by a Dyck word in abijective way. Since the organogenesis is drivenby a multitype Galton-Watson branching pro-cess, the evolution rules can be represented by astochastic 0L-system (see [10] for more details).

3 Plants as combinatorial structures

3.1 Some combinatorial concepts

We recall some basic denitions and prop-erties of combinatorics (see [5] and [11] for moredetails).

Denition 1 (Plane Rooted Tree =PRT). A rooted tree is a connected and acyclicgraph with a node specically distinguishedcalled root. A plane tree is dened as a treein which subtrees coming from a common nodeare ordered and represented from left to right.

Figure 4 gives examples of PRT. The trees(2) and (3) are equivalent as rooted trees, butthey become distinct objects when regarded asPRTs.

Fig. 4. Examples of plane rooted trees

PRTs can be coded in several ways. One ofthe most classical one is the Dyck word coding(see [12]). The latter relies on a tree traversalusing the prex order (see [5], page 74):

Denition 2 (Prex order). A PRTis traversed according to the prex order if itis traversed starting from the root, proceedingdepth-rst and left-to-right, and backtrackingupwards once a subtree has been completelytraversed.

Figure 5 gives an example of preordertraversal.

Denition 3 (Dyck word coding). Ev-ery PRT is described by a Dyck word on thealphabet V1 = z, z′ as follows:

• the tree is traversed according to the pre-x order.

• an edge visited from the parent node tothe child node is represented by the letterz.

• an edge visited from the child node to theparent node is represented by the letter z′.

In the case of labelled trees (trees withcharacteristics attached to nodes and edges), a


new code deriving from Dyck word coding canbe used:

Denition 4 (Extended Dyck wordcoding). Let L = l1, . . . , lM be a set of la-bels. Every labelled PRT with labels in L isdescribed by an extended Dyck word on the al-phabet V2 = zl1 , z′l1 ,. . . , zlM , z

′lM as follows:

• the tree is traversed according to the pre-x order.

• an edge with a label li and visited fromthe parent node to the child node is rep-resented by the letter zli.

• an edge with a label li and visited fromthe child node to the parent node is rep-resented by the letter z′li.

3.2 Coding a plant structure with aDyck word

Every plant generated by the organogen-esis model S can be represented by a labelledPRT:

Denition 5 (Labelled PRT associ-ated to a plant). Let P be a plant generatedby the organogenesis model S. Let us build alabelled PRT from the plant P as follows:

• Every organ is represented by an edge andends with a node at each extremity.

• Two connected organs are represented bytwo edges having a common node.

• The root is the node below the edge rep-resenting the internode at the basis of theplant.

• The label of an edge is given by the botan-ical information of the associated organ:its nature (bud, leaf, internode, . . .) andits characteristics (MC, CA, . . .).

• The label of a node (dierent from theroot) is the label of the edge below it. Theroot has the label of the edge above it.

Such PRT is called the labelled PRT associatedto the plant P .

N.B.: the labelled PRT dened in Deni-tion 5 is unique for a given plant structure andthe converse is true.

Since a plant can be represented by a la-belled PRT, it has an associated extended Dyckword. Let OS be the minimal set of letterscoding for the type of organs needed to de-scribe plant structures generated by S. Gen-erally, the letter b codes for a bud, m foran internode, L for a leaf and F for a fruit.In the same way, let CS be the minimal setof all possible characteristics associated to S.In Figure 5, the extended Dyck word associ-ated to the plant (i.e. to its PRT) is thusw = zm,1zm,2zF z

′F zb,2z

′b,2z

′m,2zm,1zb,2z

′b,2zb,1z

′b,1zL

z′Lz′m,1z

′m,1 where the rst letter of a label rep-

resents the type of an organ and the second itsMC. Note that no MC is associated to leavesand fruits.

Fig. 5. Correspondence between plants and la-

belled plane rooted trees. For the sake of clarity,

only the labels of edges are represented.

For the sake of clarity, some conventionsof coding will be used in the sequel:

• Every organ of nature o ∈ OS having thecharacteristics c ∈ CS is represented bythe symbols oc and o

′c (instead of zo,c and

z′o,c).


• For some organs (buds, leaves, fruits, . . .),it is not necessary to represent the visitfrom the child node to the parent node ofthe associated edge. As a matter of fact,a visit from the parent node to the childnode is always immediately followed bya visit from the child node to the parentnode. Thus, only the symbol represent-ing the visit from the parent node to thechild node is used to describe that typeof organ.

By taking into account the previous con-ventions of coding, the plant of Figure 5 is thuscoded by w = m1m2Fb2m

′2m1b2b1Lm

′1m

′1.

In the sequel, the set of all labelled PRTsgenerated by organogenesis model S is denotedby T S . Let V S = oc, o′co∈OS ,c∈CS be the min-imal alphabet containing all the letters neededto describe plant structures generated by S,WV S the set of all words built on V S andDWV S

the set of all extended Dyck words generated byS (with the previous conventions of notation).Let DV S : T S → DWV S be the map which as-sociates for each labelled PRT t its correspond-ing extended Dyck word DV S (t). Then, DV S isa bijection from T S to DWV S (see [12]).

3.3 Growth dynamics and L-systems

In Section 3.2, it was proved that the struc-ture of a plant at a given GC can be representedby an extended Dyck word. We are now in-terested in the evolution of the structure withrespect to its CA starting with a seed (or abud). As for stochastic organogenesis modelsdriven by multitype branching processes, thisevolution is given by a Markovian sequence ofrandom plants indexed by the GCs (see Sec-tion 2.2). Thus, it can be described by a se-quence (tn)n∈N of T S (or the corresponding se-quence (DV (tn))n∈N of DWV S ). To completethe combinatorial framework, we need to dene

tools which describe the stochastic organogen-esis model when the structure is coded by anextended Dyck word (i.e. tools which enablethe building of DV (tn+1) from DV (tn)).

Stochastic 0L-systems are well suited toachieve this goal (see [4], [13] and [3]). They aregenerative parallel rewriting grammars whoseproduction rules are associated to a set of prob-ability distributions.

Denition 6 (Stochastic 0L-system). Let V be an alphabet and WV theset of all words built on V . A stochastic 0L-system is a construct L = 〈ωa, π〉 where:

• ωa ∈ WV is called the axiom.

• π is a transition matrix from V to WV

(i.e., ∀(u, v) ∈ V ×WV , 0 ≤ πu,v ≤ 1 and∑w∈WV

πu,w = 1) with a nite number ofnon-zero components.

A stochastic 0L-system L = 〈ωa, π〉 generatesa random sequence of words (wn)n∈N built onthe alphabet V . By denition, the axiom is theword which initiates the sequence generated byL. Then, w0 = ωa. We get wn+1 by replacingrandomly every letter x of wn by a word y witha probability πx,y (note that the evolution ofa letter is independent from the evolution ofthe other letters). By doing so, we create aMarkov chain on WV : (wn)n∈N. We can nowdene a more general class of 0L-systems calledstochastic F0L-system, extending the classicaldenition of F0L-system ([14], p. 89) to thestochastic case:

Denition 7 (Stochastic F0L-system). Let V be an alphabet and WV theset of all words built on V . A stochastic F0L-system is a construct L = 〈A, π〉 where:

• A is a non empty subset ofWV (called theset of axioms of L).


• for every ωa ∈ A, L[ωa] = 〈ωa, π〉 is astochastic 0L-system (called componentsystem of L).

The stochastic organogenesis model S can bedescribed by a stochastic F0L-system L =〈WV S , π〉. The evolution rules of S are de-scribed by a set of stochastic automata (cf Sec-tion 2.2). This set is used to ll the transi-tion matrix π. Let us take the example ofFigure 6. From the stochastic automata, wededuce πb1,b1 = p, πb1,m1b2b1m′

1= 1− p and

πb2,m2b2m′2

= 1. Note that the evolution rulesconcerning the internodes of MC 1 and MC 2are not specied. In that case, it means thatthey stay in the same state from one GC to an-other. Therefore, πm1,m1 = 1 and πm2,m2 = 1.All the other components of π are equal to 0.

Fig. 6. Stochastic automata and the associated

stochastic F0L-system

The stochastic F0L-system L = 〈WV S , π〉generates a Markov chain on DWV S , (wn)n∈N,coding for plant structures. The associatedMarkov kernel Π (called Markov kernel asso-ciated to L) can be built very easily from π(see [10] for more details). For all (u, v) ∈WV S × WV S , (Πn)u,v is the probability to getthe word v by using the stochastic 0L-systemL[u] after n steps. w0 codes for the seed (i.e.a bud of MC 1). Since the evolution rules of Sare contained in π, wn+1 is built randomly fromwn by using the same rules as those describedby S. Therefore, studying the evolution ofplant structures generated by S is completely

equivalent to studying the sequence (wn)n∈Nfrom a combinatorial point of view.

In the sequel, L = 〈WV S , π〉 will de-note a stochastic F0L-system associated to thestochastic organogenesis model S. When noconfusion arises concerning the model used,the letter S will not be specied in the corre-sponding sets. In that case, L = 〈WV , π〉. Πwill denote the Markov kernel associated to L.For all s ∈ WV , DW

π,nV [s] (⊂ DWV ) denotes

the set of all possible extended Dyck wordsgenerated by the component system L[s] aftern steps. Let T π,n[s] = D−1

V (DW π,nV [s]) be the

set of all labelled PRTs (T π,n[s] ⊂ T ) gener-ated by L[s] after n steps.

4 The symbolic method

The symbolic method is a very eectivemethod to analyse combinatorial structuresand, as a consequence, plays an important rolein analytic combinatorics (see [5] for more de-tails). As far as plants are concerned, it enablesus to compute not only the distribution of thenumber of organs (of any type) but also of spe-cic structures in the plant architecture.

4.1 Combinatorial class and Generatingfunction

In this section, basic tools of combinatoricsare recalled (see [5] for more details).

Denition 8 (Combinatorial Class). Acombinatorial class, or simply a class, is a -nite or denumerable set on which a size func-tion is dened, satisfying the following condi-tions:

• the size of an element is a non-negativeinteger.

• the number of elements of any given size


is nite.

For example, for all n ∈ N, DW π,nV [s] is a com-

binatorial class. Many size functions can bedened ( counting the number of letters codingfor internodes, . . .).

Denition 9 (Stochastic Combinato-rial Class). A stochastic combinatorial classis a set SC = (t, pt), t ∈ C such that:

• C is a combinatorial class.

• ∀t ∈ C, 0 ≤ pt ≤ 1.

•∑t∈C

pt = 1.

The set SDW π,nV [s] = (w, (Πn)s,w), w ∈

DW π,nV [s] is a stochastic combinatorial class.

Denition 10 (Generating Function(= GF) associated to a size function ina stochastic combinatorial class). Let C bea combinatorial class and SC = (t, pt), t ∈ Ca stochastic combinatorial class. Let m be asize function in C. The generating function Ψassociated to m in SC is a mapping from [0, 1]to [0, 1] dened as follows:

∀z ∈ [0, 1], Ψ(z) =∑t∈C

ptzm(t)

GF are very useful to analyse a stochastic com-binatorial class SC from a probabilistic pointof view. Suppose we are interested in gettingthe distribution of a particular characteristic cin SC. Let m be the size function (SC → N)such that, for all t ∈ SC,m(t) gives the numberof c in the element t. By reordering the termsof the GF,we get the following power series:

∀z ∈ [0, 1], ψ(z) =∑t∈C

ptzm(t) =

∑k∈N

p(k)zk

p(k) is the probability to get k characteristics cin an element of SC. Therefore, by determin-ing the coecients of the GF associated to s inSC, we get the distribution of c in SC.

4.2 Description of the method

Suppose we are interested in getting thedistribution of a particular structure (a specicsequence of phytomers, a particular element inthe plant, . . .) in a plant of CA n generatedby the stochastic organogenesis model S. Thisparticular structure can be coded on the alpha-bet V by a word u. Therefore, this problem isequivalent to determining the distribution of apattern u in the stochastic combinatorial classSDW π,n

V [s].

Denition 11 (Counting Func-tion). The counting function c is a map fromWV × WV to N such that, for all (w, u) ∈WV ×WV , c(w, u) gives the number of patternsu in the word w.

For all u ∈ WV and n ∈ N, the mappingw 7→ c(w, u), from DW π,n

V [s] to N, is a sizefunction for the combinatorial class DW π,n

V [s].Therefore, to get the distribution of a patternu in SDW π,n

V [s], we need to compute the GFassociated to the size function w 7→ c(w, u) inSDW π,n

V [s]. In the sequel, we will call GF asso-ciated to a pattern u in SDW π,n

V [s] the previousGF. It will be denoted by ψn[s](z):

∀z ∈ [0, 1], ψn[s](z) =∑

w∈DW π,nV

[s]

(Πn)s,wzc(w,u) =

∑k∈N

Pn,s(k)zk

where P n,s(k) is the probability to get k pat-terns u in a plant structure generated by L[s] =〈s, π〉 after n steps. However, this GF is rarelydetermined directly. Usually, we get it fromfunctional equations which are most of the timerecurrence relations between ψn[s] and ψn+1[s].To obtain these equations, we use a symbolicapproach as developed in [5].

Let C be a combinatorial class and SC =(t, pt), t ∈ C a stochastic combinatorial class.Suppose we are interested in a particular sizefunction m taking its argument in C. Thus, we


want to determine the GF ψ associated to min SC. The idea of the symbolic method is tobuild an equation which decomposes SC intosmaller classes either of the same type or ofsimpler types (in the sequel, we will refer tothis equation as the set equation). Then, itis transformed into an equation composed ofthe GFs of the previous combinatorial classes.When it is possible, we can solve directly thetransformed equation and we get Ψ. When thestructure of the class SC is too complex, weextract recurrence relations satised by the co-ecients of Ψ from the transformed equation.

When dealing with plants, for a givenn ∈ N, we have C = DW π,n

V [s] and SC =SDW π,n

V [s] where s is an extended Dyck wordcoding for a seed. The idea is to decomposeDW πn

V [s] into a combination of sets of typeDW πk

V [s′] with k < n and s′ ∈ WV . By doingso, we get a set equation which can be trans-formed into an equation composed of GFs.

The symbolic method can be decomposedinto the following steps:

1. Identify the structure of the plant and allstochastic behaviours.

2. Deduce the associated stochastic au-tomata.

3. Find the appropriate alphabet V to de-scribe the plant and write the stochasticF0L-system L = 〈WV , π〉.

4. Identify the pattern u ∈ WV whose dis-tribution you want to compute.

5. Write the structural property associatedto each DW π,n

V [s] for n ∈ N where s isan extended Dyck word coding for a seedand try to decompose it into a combina-tion of sets of type DW πk

V [s′] with k < nand s′ ∈ WV .

6. Write the transformed equation satis-ed by the GF ψn[s] associated to u inSDW π,n

V [s] for each n ∈ N.

7. Either solve directly the set of trans-formed equations or nd a recurrence re-lation between ψn[s] and ψn+1[s] withn ∈ N.

8. Extract from these equations the coe-cients of the GFs.

4.3 Examples

4.3.1 Example with simple elements

We want to compute the distribution as-sociated to the number of internodes in the fol-lowing plant growth model: a plant with onlyone MC and a dormancy probability p. The be-haviour of a bud is characterized by Figure 7.The alphabet is simply V = m,m′, b wherem codes for an internode and b for a bud.

Fig. 7. Plant with one morphological category and

dormancy probabilities

The transition matrix of the associatedstochastic F0L-system L = 〈WV , π〉 can be eas-ily deduced from Figure 7. We have πb,b = p,πb,mbm′ = 1 − p, πm,m = 1 and πm′,m′ = 1. Allthe other components of π are equal to zero.The pattern of interest is m. From Figure 7,we deduce that a word w ∈ DW π,n+1

V [b] can be:

- either a word v ∈ DW π,nV [b] with a prob-

ability p,

- or a word of the form mvm′ with v ∈DW π,n

V [b] with a probability 1− p.


Let ψn[b] be the GF associated to m inDW π,n

V [b]:

ψn[b](z) =∑

w∈DW πn

V[b]

(Πn)b,w zc(w,m) =∑k∈N

Pn,b(k)zk

where P n,b(k) is the probability to get k in-ternodes in a plant structure generated by theorganogenesis model described in Figure 7 aftern GCs. Then, we have:

ψn+1[b](z) =∑

v∈DW πn

V[b]

p(Πn)b,v zc(v,m)

+∑

v∈DW πn

V[b]

(1− p)(Πn)b,v zc(mvm′,m)

Given that c(mvm′,m) = c(v,m)+1, we have:

ψn+1[b](z) = p∑

v∈DW πn

V[b]

(Πn)b,v zc(v,m)

+(1− p)z∑

v∈DW πn

V[b]

(Πn)b,v zc(v,m)

and then:

ψn+1[b](z) = pψn[b](z)+(1−p)zψn[b](z) = (p+(1−p)z)ψn[b](z)(1)

Given that ψ0[b] = 1, the solution of the pre-vious geometric progression is then:

ψn[b](z) = (p+ (1− p)z)n

We deduce P n,b(k) =(nk

)pn−k(1 − p)k if

0 ≤ k ≤ n and 0 otherwise.

N.B.: as detailed in [10], the underly-ing stochastic process in this section is thatof a Galton-Watson multitype branching pro-cess (see [8] and [9]). As a matter of fact,let Bn and Mn be two random variables onsome probability space (Ω,F ,P) (where P isa probability measure) such that Bn and Mn

give respectively the number of letters b andm in a word generated randomly by either thecomponent system L[b] or the component sys-tem L[m] after n steps. Then, the sequence

of random vectors((Bn,Mn)

)n∈N

is a Galton-

Watson multitype branching process. Let Φn

be the probability generating function associ-ated to (Bn,Mn) for all n ∈ N. In that case,Equation 1 is simply the translation of thecomposition formula for branching processes:Φn+1 = Φ1(Φn) ([15]).

4.3.2 Example with a complex structure

In this section, we introduce an exam-ple which illustrates well the benet of thesymbolic approach. We want to compute thedistribution associated to the number of `Y-structures' (see Figure 8).

Fig. 8. Example of Y-structure

The plant development model is given byFigure 9.

Fig. 9. Leeuwenberg type growth model with death

probability

The alphabet is V = m,m′, b, d where drepresents a dead bud and the stochastic F0L-system can be easily deduced from Figure 9.We have πb,d = p, πb,mbbm′=1−p, πm,m = 1,πm′,m′ = 1 and πd,d = 1. All the other compo-nents of π are equal to zero. Counting the num-ber of `Y-structures' is equivalent to countingthe number of patterns m′m. Thus, the pat-tern of interest is m′m. As for the structuralproperty of DW π,n

V [s], we have to break down


the structure of a PRT in a way which high-lights how Y-structures appear in the topologyand how they are connected to substructures(see Figure 9). From Figure 9, we deduce thata word w ∈ DW π,n+1

V [b] can be:

- either a dead bud d with a probability p,

- or a word of the form mvm′ with v ∈DW π,n

V [bb] with a probability 1 − p. Inthat case, v represents a branched struc-ture.

Let ψn[b] be the GF associated to m′m inDW π,n

V [b]:

ψn[b](z) =∑

w∈DW πn

V[b]

(Πn)b,w zc(w,m′m) =∑k∈N

Pn,b(k)zk

where P n,b(k) is the probability to get k Y-structures in a plant generated by the organo-genesis model described in Figure 9 after nGCs. Then, we have:

ψn+1[b](z) = pzc(d,m′m) +∑

v∈DW πn

V[bb]

(1− p)(Πn)bb,vzc(mvm′,m′m)

= p+ (1− p)ψn[bb](z) (2)

Now, we need to nd a recurrence relationfor ψn[bb](z). It can be deduced straightfor-ward by the decomposition of Figure 10:

Fig. 10. Decomposition of branched structures of

CA n+ 1

We deduce that a word w ∈ DW π,n+1V [bb]

(i.e. a branched structure) can be:

- two dead buds dd with a probability p2.

- one branched structure mvm′ with v ∈DW π,n

V [bb] and one dead bud d (either vdor dv) with a probability p(1−p) for eachcase.

- two branched structures mv1m′ and

mv2m′ with (v1, v2) ∈ DW π,n

V [bb] ×DW π,n

V [bb] with a probability (1− p)2.

Therefore:

ψn+1[bb](z) = p2zc(dd,m′m)

+∑

v∈DW πn

V[bb]

(1− p)p(Πn)bb,vzc(mvm′d,m′m)

+∑

v∈DW πn

V[bb]

(1− p)p(Πn)bb,vzc(dmvm′,m′m)

+∑

(v1,v2)∈(

DW πn

V[bb]

)2

(1− p)2(Πn)2bb,vzc(mvm′mvm′,m′m)

Since c(mvm′d,m′m) = c(dmvm′,m′m) =c(v,m′m) and c(mvm′mvm′,m′m) =2c(v,m′m) + 1, we get:

ψn+1[bb](z) = p2+2(1−p)pψn[bb](z)+(1−p)2z (ψn[bb](z))2

(3)

By identifying the coecients of the powerseries involved in Equations 2 and 3 , we get thedistribution of Y-structures. The same methodwould work to compute, for example, the dis-tributions associated to the number of apexes(i.e. terminal nodes in a tree).

5 Application to parameter identica-tion

In plant stochastic organogenesis model,the parameters identication of the automatamay be delicate and necessitates an importantsampling work. For stochastic organogenesismodels driven by a multitype branching pro-cess, classical methods based on branching pro-cesses have already been established (see [16]and [17]). These methods rely on the calibra-tion of the parameters so that the theoreticalmean and variance associated to the numberof phytomers (computed from the model) arethe closest to the experimental ones. However,phytomers are not always easily identiable in


a plant. Therefore, the idea is to use a botani-cal structure that is more easy to identify andto count (for example apices or Y-structures, cfSection 4.3.2). By confronting the theoreticaldistribution of this structure to the experimen-tal one, we are able to nd the best set of pa-rameters that will give to the model the closestbehaviour to real plants. However, this the-oretical distribution can rarely be determinedwith classical branching process methods. Inthat case, the symbolic method of Section 4provides a good alternative.

Let S be a stochastic organogenesis modeldriven by a multitype branching process andL = 〈WV , π〉 the associated stochastic F0L-system. Suppose we are interested in comput-ing the theoretical distribution of a particularstructure in random plant architectures gener-ated by S. This particular structure is codedon the alphabet V by the word u. Let ψn[s]be the GF associated to u in SDW π,n

V [s] wheres ∈ WV is the word coding for a seed. As men-tioned in Section 4.2, the theoretical distribu-tion is given by the coecient of ψn[s] seen aspower series:

∀z ∈ [0, 1], ψn[s](z) =∑

w∈DW π,nV

[s]

(Πn)s,wzc(w,u) =

∑k∈N

pn,s(k)zk

Since card(DW π,nV [s]) < ∞ (the stochas-

tic automata can only generate a nite num-ber of structures), then, for all n ∈ N,maxc(w, u)|w ∈ DW π,n

V [s] exists and is -nite. In that case, we set:

∀n ∈ N, Ln = maxc(w, u)|w ∈ DW π,nV [s]

Thus, for all l > Ln, pn,s(l) = 0. Let φn be avector in [0, 1]L

n+1 such that:

∀n ∈ N, φn = (pn,s(0), pn,s(1), . . . , pn,s(Ln))

Generally, the symbolic method leads to a re-cursive equation between ψn[s] and ψk[r] with

k ∈ K and r ∈ R where K and R are respec-tively nite subsets of 0, . . . , n − 1 and WV .For the sake of clarity, we will suppose that themethod gives us a recursive equation betweenψn[s] and ψn−1[s] (the extension to the generalcase is straightforward). Therefore, by identify-ing the coecients of the power series, we get aset of recurrence relations between the compo-nents of φn and φn−1 which enables the build-ing of φn from φn−1. The stochastic automatadepend on a set P of parameters which havea botanical meaning (survival probability,. . .).As a consequence, the set of recurrence rela-tions between the components of φn and φn−1

depends also on P and, thus, φn = φn(P ).Suppose we have a plot of plants of CA

N ∈ N r 0. For each of these plants, wemeasure the number of characteristics u. Bydoing so, we get the experimental distribu-tion of the particular structure: pexp(l)l∈N.Suppose that there exists l > LN such thatpexp(l) 6= 0, then the organogenesis model is notwell dened. As a matter of fact, for l > LN ,pN,s(l) = 0. In that case, φN cannot get as closeas we want to the experimental distribution. Inthat case, the organogenesis model (i.e. the au-tomata) needs to be modied. Then, the modelS is said to be well dened if:

minl ∈ N|P exp(l) = 0 > LN

Let φexp be a vector in [0, 1]LN+1 such that:

φexp = (pexp(0), pexp(1), . . . , pexp(LN))

The set of parameters P is estimated by theleast square estimator P :

P = argminP∈[0,1]card(P )

∥∥∥φexp − φN(P )∥∥∥2

2

where, for all α = (α0, . . . , αLN ) ∈ [0, 1]LN+1,

‖α‖2 =(∑LN

i=0(αi)2)1/2

.

Several optimisation algorithms can beused to nd P . One of the most appropriate al-gorithms to solve the minimisation problem isthe Levenberg-Marquardt algorithm (see [18]).


6 Conclusion

In this article, a symbolic method was setto analyse complex structures in plants whoseorganogenesis is driven by a multitype branch-ing process. To that purpose, a new combinato-rial framework was introduced. Plant structureis represented by a plane rooted tree and, as aconsequence, can be coded by a Dyck word.The evolution rules of the organogenesis modelare given by a stochastic 0L-system. By doingso, the evolution of plant structure is coded bya Markovian sequence of Dyck words. There-fore, studying plant structure and its develop-ment is completely equivalent to studying theMarkovian sequence from a combinatorial pointof view. A symbolic method was then estab-lished and enables the computation of the dis-tribution associated to the number of complexstructures in plant topology.

This result has numerous applications. Forinstance, in this article, we have shown thatsuch a method can be used for parameter iden-tication. Moreover, it can also be used tocompare stochastic organogenesis models. Asa matter of fact, by confronting the theoreti-cal distribution of a structure of a given type,we are able to choose the model which has theclosest behaviour to that of real plants.

References

[1] A. Lindenmayer, Mathematical modelsfor cellular interactions in development. i.laments with one-sided inputs, Journalof Theoretical Biology, vol. 18, pp. 280289, 1968.

[2] A. Smith, Plants, fractals and formal lan-guages, Computer Graphics (SIGGRAPH84 Conference Proceedings), vol. 18, no. 3,pp. 110, 1984.

[3] W. Kurth, Growth grammar interpreterGROGRA 2.4: A software tool for the

3-dimensional interpretation of stochas-tic, sensitive growth grammars in thecontext of plant modelling. Introductionand Reference Manual. Berichte desForschungszentrums Waldokosysteme derUniversitat Gottingen, Ser. B, Vol. 38,1994.

[4] P. Prusinkiewicz and A. Lindenmayer, TheAlgorithmic Beauty of Plants. Springer-Verlag, New-York, 1990.

[5] P. Flajolet and R. Sedgewick, AnalyticCombinatorics. Cambridge UniversityPress, 2009.

[6] D. Barthélémy and Y. Caraglio, Plantarchitecture: a dynamic, multileveland comprehensive approach to plantform, structure and ontogeny, Annals ofBotany, vol. 99, no. 3, pp. 375407, 2007.

[7] M.-Z. Kang, P.-H. Cournède, P. de Reye,D. Auclair, and B.-G. Hu, Analyti-cal study of a stochastic plant growthmodel: application to the GreenLabmodel, Mathematics and Computers inSimulation, vol. 78, no. 1, pp. 5775, 2008.

[8] C. Mode, Multitype branching processes:Theory and applications. American Else-vier Publishing Co. Inc, New York, 1971.

[9] K. Athreya and P. Ney, Branching Pro-cesses. Dover Publications, 2004.

[10] C. Loi and P.-H. Cournède, Generat-ing Functions of Stochastic L-Systems andApplication to Models of Plant Develop-ment, Discrete Mathematics and The-oretical Computer Science Proceedings,vol. AI, pp. 325338, 2008.

[11] J. Riordan, An Introduction to Combina-torial Analysis. Courier Dover Publica-tions, 2002.


[12] D. Knuth, The Art of Computer Program-ming. Addison-Wesley Professional, 1997(3rd edition), vol. 1.

[13] J. Françon, Sur la modélisation informa-tique de l'architecture et du développe-ment des végétaux. in 2ème Col-loque International: L'Arbre. Institut deBotanique, Montpellier, France, 1990.

[14] G. Rozenberg and A. Salomaa, The Math-ematical Theory of L-systems. AcademicPress, New York, 1980.

[15] T. Harris, The theory of branching pro-cesses. Springer, Berlin, 1963.

[16] Y. Guédon, D. Barthélémy, Y. Caraglio,and E. Costes, Pattern analysis in branch-ing and axillary owering sequences,Journal of Theoretical Biology, vol. 212,pp. 481520, 2001.

[17] M. Kang, P.-H. Cournède, J.-P. Quadrat,and P. de Reye, A stochastic languagefor plant topology, in Plant growth Mod-eling, simulation, visualization and theirApplications., T. Fourcaud and X. Zhang,Eds. IEEE Computer Society (Los Alami-tos, California), 2007.

[18] J. Moré, The Levenberg-Marquardt Al-gorithm: Implementation and Theory,Springer Berlin / Heidelberg ed. Springer,2006, pp. 105116, ISBN: 978-3-540-08538-6.

Cédric Loi graduatedfrom the French engineerschool Ecole Centrale Parisin 2007 where he studied ap-plied mathematics. Then, hereceived a master degree instatistics from the universityParis XI. He is currently a

PhD student in the labora-tory of applied mathematics

at Ecole Centrale Paris. His thesis is about thestudy of plant organogenesis models driven bymultitype branching processes and their para-metric identication.

Paul-Henry Cournèdegraduated from Ecole Cen-trale Paris and the Univer-sity of Cambridge in 1997,and received his PhD inApplied Mathematics fromEcole Centrale Paris in 2001.He is currently full professorat the department of Math-ematics of Ecole Centrale

Paris. His research activity in the laboratory ofApplied Mathematics and Systems focuses onthe mathematical modeling of plant growth.He belongs to the research group Digiplante(INRIA, Ecole Centrale Paris, CIRAD).

Jean Françon is a re-tired professor of the Univer-sity of Strasbourg. In the70's and 80's his research atthe research institute for ad-vanced mathematics (IRMA)and at the centre for nu-clear studies (CEN) in Stras-bourg led to pioneering worksin combinatorics and in the

analysis of algorithms. He then participatedto the creation of the "Equipe d'InformatiqueGéométrique et Graphique" in the laboratoryof Image, Computer and Remote Sensing Sci-ences at the University of Strasbourg (LouisPasteur University) and also started workingon geometrical modelling and discrete geome-try. His interest for the potential applicationsof combinatorics to models of plant growthdates back to the 80's but has recently foundnew echoes.

a symbolic method to analyse patterns in plant structure ...keywords dyck word, stochastic...

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