Additive tree functionals with small toll functions and subtrees of random trees

Stephan Wagner (1)

1 DMS - Department of Mathematical Sciences [Matieland, Stellenbosch Uni.]

Many parameters of trees are additive in the sense that they can be computed recursively from the sum of the branches plus a certain toll function. For instance, such parameters occur very frequently in the analysis of divide-and-conquer algorithms. Here we are interested in the situation that the toll function is small (the average over all trees of a given size $n$ decreases exponentially with $n$). We prove a general central limit theorem for random labelled trees and apply it to a number of examples. The main motivation is the study of the number of subtrees in a random labelled tree, but it also applies to classical instances such as the number of leaves.

Keywords: additive tree functional; small toll function; number of subtrees; size of subtrees; random trees

Domains: Data Structures and Algorithms [cs.DS]; Discrete Mathematics [cs.DM]; Combinatorics [math.CO]; Computational Geometry [cs.CG]

Stephan Wagner. Additive tree functionals with small toll functions and subtrees of random trees. 23rd International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms (AofA'12), 2012, Montreal, Canada. pp.67-80, ⟨10.46298/dmtcs.2984⟩. ⟨hal-01197234⟩