Enumerating alternating tree families
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چکیده
We study two enumeration problems for up-down alternating trees, i.e., rooted labelled trees T , where the labels v1, v2, v3, . . . on every path starting at the root of T satisfy v1 < v2 > v3 < v4 > · · · . First we consider various tree families of interest in combinatorics (such as unordered, ordered, d-ary and Motzkin trees) and study the number Tn of different up-down alternating labelled trees of size n. We obtain for all tree families considered an implicit characterization of the exponential generating function T (z) leading to asymptotic results of the coefficients Tn for various tree families. Second we consider the particular family of up-down alternating labelled ordered trees and study the influence of such an alternating labelling to the average shape of the trees by analyzing the parameters label of the root node, degree of the root node and depth of a random node in a random tree of size n. This leads to exact enumeration results and limiting distribution results. Résumé. Nous étudions deux problèmes de dénombrement d’arbres alternés haut-bas : par définition, ce sont des arbres munis d’une racine et tels que, pour tout chemin partant de la racine, les valeurs v1, v2, v3, . . . associées aux noeuds du chemin satisfont la chaı̂ne d’inégalités v1 < v2 > v3 < v4 > · · · . D’une part, nous considérons diverses familles d’arbres intéressantes du point de vue de l’analyse combinatoire (comme les arbres de Motzkin, les arbres non ordonnés, ordonnés et d-aires) et nous étudions pour chaque famille le nombre total Tn d’arbres alternés haut-bas de taille n. Nous obtenons pour toutes les familles d’arbres considérées une caractérisation implicite de la fonction génératrice exponentielle T (z). Cette caractérisation nous renseigne sur le comportement asymptotique des coefficients Tn de plusieurs familles d’arbres. D’autre part, nous examinons le cas particulier de la famille des arbres ordonnés : nous étudions l’influence de l’étiquetage alterné haut-bas sur l’allure générale de ces arbres en analysant trois paramètres dans un arbre aléatoire (valeur de la racine, degré de la racine et profondeur d’un noeud aléatoire). Nous obtenons alors des résultats en terme de distribution limite, mais aussi de dénombrement exact.
منابع مشابه
Alternating Regular Tree Grammars in the Framework of Lattice-Valued Logic
In this paper, two different ways of introducing alternation for lattice-valued (referred to as {L}valued) regular tree grammars and {L}valued top-down tree automata are compared. One is the way which defines the alternating regular tree grammar, i.e., alternation is governed by the non-terminals of the grammar and the other is the way which combines state with alternation. The first way is ta...
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تاریخ انتشار 2008