A Strahler bijection between Dyck paths and planar trees

Discrete Mathematics 246 (2002) 317–329www.elsevier.com/locate/disc

A Strahler bijection between Dyck paths and planar trees

Xavier G(erard ViennotLaBRI, Universit�e Bordeaux I, 351 cours de la Liberation, 33405 Talence, France

Abstract

The Strahler number of binary trees has been introduced by hydrogeologists and rediscoveredin computer science in relation with some optimization problems. Explicit expressions have beengiven for the Strahler distribution, i.e. binary trees enumerated by number of vertices and Strahlernumber. Two other Strahler distributions have been discovered with the logarithmic height ofDyck paths and the pruning number of forests of planar trees in relation with molecular biology.Each of these three classes are enumerated by the Catalan numbers, but only two bijectionspreserving the Strahler parameters have been explicited: by Fran5con between binary trees andDyck paths, by Zeilberger between binary trees and forests of planar trees. We present here themissing bijection between forests of planar trees and Dyck paths sending the pruning numberonto the logarithmic height. A new functional equation for the Strahler generating function isdeduced. Some orthogonal polynomials appear, they are one parameter Tchebyche9 polynomials.

Resume

Le nombre de Strahler d’un arbre binaire a (et(e introduit en hydrog(eologie et red(ecouvert eninformatique en relation avec certains probl<emes d’optimisation. Des expressions explicites ont(et(e donn(ees pour la distribution du param<etre nombre de Strahler, c’est-<a-dire pour le nombred’arbres binaires (enum(er(es selon le nombre de sommets et leur nombre de Strahler. Depuis, deuxautres distributions de Strahler ont (et(e d(ecouvertes: la hauteur logarithmique des chemins de Dycket l’ordre des forets d’arbres planaires en relation avec la biologie moleculaire. Chacune de cestrois classes d’objets est (enum(er(ee par les nombres de Catalan, mais seulement deux bijectionsconservant les param<etres des distributions de Strahler ont (et(e explicit(ees: l’une de Fran5con entreles arbres binaires et les mots de Dyck, l’autre de Zeilberger entre les arbres binaires et les foretsd’arbres planaires. Nous donnons une bijection entre les forets d’arbres planaires et les mots deDyck envoyant le param<etre ordre sur le param<etre hauteur logarithmique. Une nouvelle (equationfonctionnelle est d(eduite pour la s(erie g(en(eratrice associ(ee la distribution de Strahler. Certainspolynomes orthogonaux apparaissent, ce sont des extensions <a un param<etre des polynomes deTchebyche9. c© 2002 Elsevier Science B.V. All rights reserved.

1. Strahler number of a binary tree

We use the following classical notations for binary trees. A binary tree is a tripleB=(L; r; R) or is reduced to an external vertex denoted by “ ”. Here L and R are

E-mail address: [email protected] (X.G. Viennot).

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318 X.G. Viennot /Discrete Mathematics 246 (2002) 317–329

binary trees (left and right subtree respectively) and r denotes the root of B. In Fig. 1,internal vertices are denoted by “O”. The number of binary trees with n internal vertices(and (n+ 1) external vertices) is the Catalan number

Cn=1

(n+ 1)

(2nn

):

De�nition 1. The Strahler number St(B) of the binary tree B is deNned inductively bythe relation

St( )= 1;

St(L; r; R)=

{if St(L)= St(R); then 1 + St(R);

else max{St(L); St(R)}:(1.0)

In other words, the Strahler number of a binary tree is obtained by the followingprocess: label the external vertices by 1 and each internal vertex by the recursive ruledisplayed in Fig. 2. The Strahler number is the label of the root.The Strahler number of binary trees appeared in computer science in relation with

the coding and computation of arithmetic expressions using only binary operations.Such an expression is encoded by a binary tree. Each internal vertex corresponds to abinary operation. Each external vertex corresponds to a variable (see Fig. 3).The minimum number of registers needed to compute an arithmetic expression is

exactly the Strahler number of the underlying binary tree. Computer scientists haveshown remarkable properties for the asymptotic behavior of the mean of the parameter

Fig. 1. The Strahler number of a binary tree.

X.G. Viennot /Discrete Mathematics 246 (2002) 317–329 319

Fig. 2. The Strahler labelling of a binary tree.

Fig. 3. The binary tree associated to an arithmetic expression.

Strahler number over all binary trees with n internal vertices, see Flajolet et al. [7] andKemp [11]. For that, they established the generating function for Strahlernumbers.Let Sn;k (resp. Sn;6k) be the number of binary trees B with n internal vertices and

Strahler number St(B)= k (resp: St(B)6 k). We denote the corresponding generatingfunctions by

Sk(t)=∑n¿0

Sn;k tn;

S6k(t)=∑n¿0

Sn;6k tn: (1.1)


Let Un(t) be the nth Tchebyche9 polynomial of second kind, that is the polynomialdeNned by the relation

sin(n+ 1) =(sin )Un(cos ): (1.2)

We denote by Fn(t) the polynomial

Fn(t)=Un(t=2): (1.3)

These polynomials form a sequence of monic (i.e. the coeRcient of tn is 1) orthogonalpolynomials. The moments are the Catalan numbers {Cn}n¿0. These polynomials satisfythe following 3-terms linear recurrence relation:

Fn+1(t)= tFn(t)− Fn−1(t); F0 = 1; F1 = t: (1.4)

The reciprocal polynomials F∗n (t)= tnFn(1=t) are even polynomials and we deNne Rn(t)

by the following relation:

Rn(t2)= tnFn(1=t): (1.5)

Then, Flajolet et al. [7] and Kemp [11] gave the following explicit expression for theStrahler generating function:

S6k(t)=R2k−2(t)R2k−1(t)

; (1.6a)

Sk(t)=t2

k−1−1

R2k−1(t): (1.6b)

We will call Strahler distribution the distribution given by the relation (1.6b) for aparameter deNned on a set of objects enumerated by the Catalan numbers. Here arethe Nrst values of the generating functions Sk(t):

S1(t)= 1; S2(t)=t

1− 2t; S3(t)=

t3

1− 6t + 10t2 − 4t3: (1.7)

In fact, the Strahler number of a binary tree was introduced much earlier in hydroge-ology in the morphological study of rivers networks. Horton [10] introduced a processclassifying the rivers of a Suvial network by “order”. This rule was simpliNed byStrahler [16] and corresponds to the labeling of a binary tree: a river starting at asource point has order 1, two rivers of order k joining together give rise to a river ofk + 1, while a river of order i joining a river of order k ¿ i gives rise to a river oforder k.More recently, the Horton–Stahler analysis for ramiNed patterns has been reNned with

the introduction of the “ramiNcation matrix” of a binary tree, and applied in computergraphics in order to give synthetic images of trees and landscapes, see Viennot et al.[23]. Other applications have also been given in the analysis of fractal ramiNed patternsin physics by Vannimenus and Viennot [18]. Strahler numbers also appeared in someconstructions related to the derived series of the free Lie algebra, see Reutenauer [15].A survey paper about Horton–Strahler analysis in various sciences is Viennot [22].


2. Logarithmic height of Dyck paths

Fran5con [9] gave a bijective proof of identify (1.6a) by relating the Strahler numberwith another parameter on the so called Dyck paths.Recall that a Dyck path is a sequence of points (s0; : : : ; s2n) of N × N such that

s0 = (0; 0); s2n=(2n; 0) and each elementary step is North-East (NE) or South-East(SE). An example is displayed in Fig. 4. A very classical bijection between Dyckpaths of length 2n and binary trees with n internal vertices, using the pre5x order ofbinary tree, is well known.The height H (!) of a Dyck path is the maximal height (or level) of its vertices,

i.e.

H (!)= max06i62n

{yi | si =(xi; yi)}: (2.1)

It is well known that the generating function for Dyck paths ! with bounded heightH (!)6p is given by the following relation (see [12,2]):

∑!: H (!)6p

t‖!‖=2 =Rp(t)Rp+1(t)

; (2.2)

where Rp(t) is the “modiNed” Tchebyche9 polynomial deNned by (1.5), the summationis over Dyck paths and ‖!‖ denotes the length of the path.Identity (1.6a) says that the generating function S6k(t) for binary trees having

Strahler number 6 k is the same as the generating function for Dyck paths boundedby height 2k − 2. In other words, the parameter Strahler has the same distribution thanthe following parameter deNned on Dyck paths:

LH (!)= �log2(1 + H (!))�: (2.3)

We will call the parameter LH (!) the logarithmic height of the path !.Fran5con [9] gave a bijection between binary trees and Dyck paths sending the param-

eter “Strahler number” onto the parameter “logarithmic height”. This bijection is deNned

Fig. 4. A Dyck path.


recursively and in fact proves that the double generating function S(t; x) deNned by thefollowing relation:

S(t; x)=∑k¿1

Sk(t)xk−1 =∑n;k

Sn;kxk−1tn (2.4)

satisNes the functional equation

S(t; x)= 1 +xt

(1− 2t)S

((t

1− 2t

)2; x

): (2.5)

Such an equation is a linear Read–Bajraktarevi(c equation, as considered by Bergeronet al. ([1, pp. 230,235]).

3. Order or pruning number of planar trees

Surprisingly, Strahler’s distribution reappeared in the context of molecular biology.We consider molecules of single-stranded nucleic acids, as for example RNAs. Theprimary structure is the sequence of bases linked by phosphodiester bonds. There arefour possible bases denoted by A (adenine), U (uracil), G (guanine) and C (cyto-sine). Such bases can be linked together by hydrogen bonds (A with U, and G withC). The primary structure is thus folded into a planar graph called the secondarystructure, see for example [17].The notion of order or complexity of a secondary structure has been introduced, in

relation with the computation of the energy of the secondary structure, see [14,5,24].Secondary structure of RNAs are branching structures. The important underlying struc-ture is a forest of planar trees. The order (or complexity) of the molecule is the orderof the forest, as deNned below.Recall that a planar rooted tree T (or planar tree for short) is: if T has only one

vertex, then T is reduced to that point; else T =(r;T1; : : : ; Tp), where r is a vertexcalled the root and T1; : : : ; Tp is an ordered sequence of planar trees. A forest of planar(rooted) trees is an ordered sequence of planar trees, see Fig. 5. The number of forestsof planar trees with n vertices is the Catalan number (of course, this is also the numberof planar trees with n+1 vertices). Very classical bijections are known between forestsof planar trees, Dyck paths and binary trees making a “golden triangle” of bijectionswhich commute. Let us now deNne the order of a forest.A 5lament of a forest F is a maximal sequence (s1; : : : ; sq) of vertices such that, for

i=1; : : : ; q− 1; si has only one son and this son is si+1, and moreover sq is a leaf ofthe forest (or end point, i.e. it has no son). The Nlaments are two by two disjoint. Weintroduce the operator � “deletion of Nlaments”. The forest �(F) is obtained from F bydeleting all the vertices of all Nlaments of F . The order of the forest is the minimuminteger k such that �k(F)= ∅ (see Fig. 5). Zeilberger [25] called this parameter thepruning number of the forest.


Fig. 5. Filament, operator � and pruning number of a forest of plane trees.

Waterman [24] raised the problem of Nnding the generating function for all possiblesecondary structures of order k. This was solved by Vauchaussade de Chaumont andViennot [19,20] by applying the so-called DSV-methodology due to M.P. SchVutzen-berger (using algebraic languages, see a survey in Delest [4]) and analytic calculus. Apreliminary problem was Nrst to Nnd the generating function for the pruning numberof forest. The surprise was that this parameter has indeed the Strahler distribution.In response to [13, p. 386], Zeilberger [25] constructed a bijection between binarytrees and forests of planar trees sending the Strahler number onto the pruning number.Later and independently, Bender and CanNeld also constructed a bijection. As withFran5con’s bijection quoted in Section 2, Zeilberger’s bijection is also highly recursive.A natural problem is thus to complete the “golden triangle” of Strahler bijections

between the three classes: binary trees, Dyck paths and forest of planar trees. Theremaining problem is to construct a direct bijection between forest of planar trees andDyck paths, sending the parameter pruning number onto the parameter logarithmicheight.

4. A Strahler bijection between forest of planar trees and Dyck paths

As with Fran5con’s and Zeilberger’s bijection, our bijection is deNned recursively. Wesuppose that we have constructed a bijection �k between forests of planar trees withn vertices having pruning number k, and Dyck paths of length 2n having logarithmicheight k. We then construct the bijection �k+1 for the same objects but with respectiveparameters k + 1.


First consider the case k =1. The Nlament f

of length n is associated with the following Dyck path �1(f) of length 2n:

A forest F of order 1 is a sequence of Nlaments F1; : : : ; Fp. The associated Dyck path�1(F) is obtained by concatenating the sequence of Dyck paths �1(F1); : : : ; �1(Fp).

The transition from �k to �k+1 is based on the following basic remark. Let G be aforest of order k + 1 and let F = �(G) be the forest of order k obtained by deletingthe Nlaments. Suppose that F has n vertices. Conversely, we can derive G from Fby adding some Nlaments. At the end of each leaf of F , we must add at least twoNlaments. Fig. 6 is symbolic. We have colored in red the two leftmost Nlaments. Allthe other Nlaments added are colored in green. These Nlaments are added to G bypackage of “fans of Nlaments” (sequences of Nlaments hanging from the same vertex)at each of possible positions marked by a green asterisk in Fig. 6. The number ofsuch positions is 2n + 1. Such a fan of green Nlaments may be empty, but the redNlaments have length at least 1. Remark that for the positions on the orange line (theline joining the roots of the trees of the forest; see Fig. 6), each Nlament of the fancreates a new root for the subtrees of the forest.Fig. 7 gives a precise example of a passage from a forest F of order k to a forest

G of order k +1 (here k =1). The forest F is colored black. The Nrst edge of the redNlaments is colored red, then all the other edges are colored blue. The edges of the

Fig. 6. From a forest of order k to a forest of order k + 1.


Fig. 7. From a forest of order 1 to a forest of order 2.

green Nlaments are colored in green. The forest G is obtained from F by adding thered, blue and green edges. Recall that the two red edges for each leaf are compulsory,the blue and green edges are not.If ! is the path of length 2n related to F by �k , we construct the path �=�k+1(G)

using some “rewriting rules” inside the elementary steps of ! according to the threepossible colors of the edges added to F in order to get G.

DeNne a strip as to be the portion of N × N contained between two consecutivelines with equations y=p and y=p + 1. We color alternatively the strips into twocolors yellow or white, according to the parity of p, starting with color yellow for thestrip between the line of equation y=0 and 1. The border of each strip is displayedin green, see Fig. 8(a). Assume by induction that the number of leaves in the forest Fis exactly the number of SE steps in the yellow strips (colored in red in Fig. 8). Theconstruction is as follows:Step 1: The yellow strips are “dilated” vertically by factor 3. The elementary steps

located inside the white strips (colored in black in the Ngures) are left invariant. Eachelementary step in the yellow strip (colored in red in Fig. 8(a)) is replaced accordingto the rewriting rules displayed in Fig. 8(b). We get a path �(!) as shown in Fig. 8(c).In this last Ngure, the green lines, borders of the yellow strips, have been preserved inthe dilatation. We have colored in green each of the (2n+1) intersection points of thepaths ! and �(!) with the green lines in Figs. 8(a) and (c). Intuitively, in Fig. 8(c),the red SE steps correspond to the leaves of the forest obtained from F by adding


Fig. 8. (a) Yellow strips, green border; (b) rewriting rules; (c) Step 1.

Fig. 9. Steps 2 and 3.

the red edges. The blue NE and SE steps correspond to the positions of possible blueNlaments. The green points correspond to the (2n + 1) possible positions for addingfan of green Nlaments.Step 2: Each blue step

== or \\is replaced by

==\\ ==\\ : : : == or \\==\\ : : : ==\\according to the length of the corresponding Nlaments, see the example in Fig. 9,according to the forest of Fig. 7.Step 3: At the place of each green vertex of !, a sequence (in green in Fig. 9)

of the following form is inserted. If the vertex corresponds to the ith fan Fi of greenNlaments of G, let !i be the path �1(Fi).

(a) If the vertex is on a lower border of a yellow strip then we insert !i,(b) If the vertex is an upper border of a yellow strip, then we insert the path !∗

i

obtained from !i by replacing the SE steps with NE steps, and conversely.

Steps 1–3 yield the path �=�k+1(G). An example is displayed in Fig. 9, from theforest of Fig. 7. The recurrence hypothesis about the number of leaves and SE steps


Fig. 10. Colors for the proof of (5.1).

in the yellow strips is satisNed and it is easily seen that LH(�)= k+1. Finally we cancheck that the map �k+1 is a bijection.

5. A functional equation for Strahler numbers

In fact, the recursive construction presented in the previous section gives rise to afunctional equation, di9erent from the one obtained by Fran5con [9]. We need here tointroduce a triple generating function S(t; x; y) counting forests of planar trees by theirnumber of vertices (variable t), by their pruning number (variable x) and by theirnumber of leaves (variable y).

Proposition 1. The generating function S(t; x; y) satis5es the following functionalequation:

S(t; x; y)= 1+xyt

1− (1 + y)t+

x(1− t)(1− (1 + y)t)

S

[((1− t)

1− (1 + y)t

)2; x;

t2y2

(1− t)2

]:

(5.1)

An idea of the proof is given with the formula (5.1) displayed with colors inFig. 10. Each block of terms of the same color corresponds to some constructionin the passage from the forest F to the forest G, as displayed in Fig. 7. For examplethe red term “ty” and the blue term “1=(1− t)” correspond to the blue and red edgesof the forest F . The green term is the generating function for fans of Nlaments, that is

(1− t)(1− (1 + y)t)

=1

1− ty=(1− t): (5.2)

The second term on the right-hand side corresponds to a forest of order 1. The proposi-tion follows from standard techniques in enumerative combinatorics, using for examplethe theory of species, Bergeron et al. [1], for Flajolet’s model of decomposable struc-tures [8].From Section 4, we can also deduce that the same functional equation holds for the

generating function C(t; x; y) of Dyck paths counted by their length (variable t), bytheir logarithmic height (variable x) and by their number of steps going from an oddlevel to an even level (variable y). If we write

C(t; x; y)=∑k¿1

Ck(t; y)xk ; (5.3)


we can deduce from standard techniques ([6] or [21]) the following identity:

Ck(t; y)=t2

k−1y2k−1

R2k+1−1(t; y); (5.4)

where the R(t; y) are “y-analogues” of the modiNed Tchebyche9 polynomials of thesecond kind. They are deNned as in Section 1 by replacing the recurrence (1.4) by therecurrence

Fn+1(t; y)= tFn(t; y)− �(y)Fn−1(t; y) with F0 = 1; F1 = ty; (5.5)

and �(y) is 1 or y according to whether n is odd or even. These polynomials are akind of “sieve polynomials”, as considered in Chihara [3].Combining Fran5con’s bijection with our bijection, we get a bijective proof of the

equality of the three following double distributions:

• binary trees by Strahler number and number of left “internal” edges (i.e. edgesconnecting two internal vertices),

• Dyck paths by logarithmic height and number of SE steps starting from an oddlevel,

• forests of planar trees by pruning number and number of leaves.

Recall that each of the three second parameters has individually the so-called -distribution [12], given by the Narayana numbers (1=n)( nk )(

nk−1 ). In fact the generating

polynomial (in the variable y) of these numbers are nothing but the moments of theorthogonal polynomials Fn(t; y) introduced by (5.5). Finally we can show that a slightmodiNcation of Zeilberger’s bijection preserves the corresponding double distribution.

Acknowledgements

The author is grateful to R. Pinzani for an invitation to Firenze, where this Strahlerbijection was found in the nice ambience of his combinatorics group, to one of thereferees for fruitful corrections and to E. Rassart for making some of the Ngures andthe TEX version of the paper.

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A Strahler bijection between Dyck paths and planar trees