CAAP 1992: CAAP '92 pp 24-48

# Varieties of increasing trees

• François Bergeron
• Philippe Flajolet
• Bruno Salvy
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 581)

## Abstract

An increasing tree is a labelled rooted tree in which labels along any branch from the root go in increasing order. Under various guises, such trees have surfaced as tree representations of permutations, as data structures in computer science, and as probabilistic models in diverse applications.

We present a unified generating function approach to the enumeration of parameters on such trees. The counting generating functions for several basic parameters are shown to be related to a simple ordinary differential equation, d/dzY(z)=φ(Y(z)), which is non linear and autonomous.

Singularity analysis applied to the intervening generating functions then permits to analyze asymptotically a number of parameters of the trees, like: root degree, number of leaves, path length, and level of nodes. In this way it is found that various models share common features: path length is O(n log n), the distribution of node levels and number of leaves are asymptotically normal, etc.

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