Generating functions for generating trees

Abstract

Certain families of combinatorial objects admit recursive descriptions in terms of generating trees: each node of the tree corresponds to an object, and the branch leading to the node encodes the choices made in the construction of the object. Generating trees lead to a fast computation of enumeration sequences (sometimes, to explicit formulae as well) and provide efficient random generation algorithms. We investigate the links between the structural properties of the rewriting rules defining such trees and the rationality, algebraicity, or transcendence of the corresponding generating function.Comment: This article corresponds, up to minor typo corrections, to the article submitted to Discrete Mathematics (Elsevier) in Nov. 1999, and published in its vol. 246(1-3), March 2002, pp. 29-5

Analysis of three graph parameters for random trees

“…For the path node-covering number P (T ) it has been given in [13]. Hence, we will only briefly discuss these two cases, and give a more detailed recursive description of the size of the kernel K(T ) of a directed tree, extending the considerations of [2].…”

“…For labelled trees and planted plane trees the constants µ [N ] have been computed in [10] and [11], whereas the constants µ [P ] have been computed in [13]. Finally for labelled trees the constant µ [K] already appears in [2].…”

“…For these tree families analogous results for the path node-covering number of random size-n trees are shown in [13]. The expected size of the kernel has been determined for randomly chosen trees of size n, asymptotically for n → ∞, for the family of directed rooted labelled trees in [2]. However, as far as we know, the only limiting distribution result has been obtained in [15] for the node-independence number of random rooted labelled trees: it was shown for this tree family that the node-independence number of a randomly chosen tree of size n is asymptotically normal, with mean µn and variance νn,…”

On recurrence formulas for computing the stochastic complexity

“…Respectively, a formal power series is holonomic (D-finite) if and only if its coefficient sequence is holonomic (P-recursive). The following proposition defines holonomicity condition between ordinary and exponential generating functions [1].…”

“…The model selection task is to search from a fixed model family (a set of parametric models with varying complexity) the model minimizing the stochastic complexity (1).…”