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

Stephan Wagner

doi:10.46298/dmtcs.2984

Abstract

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.

Highlights

A parameter F (T ) defined for rooted trees T is called an additive tree functional if it satisfies the recursion kF (T ) = F (Ti) + t(T ), (1)i=1 where T1, . . . , Tk are the branches of the tree and t(T ) is a so-called toll function, which often only depends on the size of T
I=1 where T1, . . . , Tk are the branches of the tree and t(T ) is a so-called toll function, which often only depends on the size of T
The internal path length, i.e., the sum of the distances from the root to all vertices, which can be obtained from the toll function t(T ) = |T | − 1

Summary

Introduction

A parameter F (T ) defined for rooted trees T is called an additive tree functional if it satisfies the recursion k. Let T be a random labelled tree, and let s(T ) denote the number of its subtrees (connected induced subgraphs of T , excluding the empty graph) The mean of this parameter was first studied by Meir and Moon [9] for generated families of trees (which include, amongst others, labelled trees, d-ary trees and plane trees). In the case of random labelled trees, it is asymptotically equal to (e/(e − 1))3/2en/e, and it is not much harder to determine that the variance is of asymptotic order Kn for a constant K whose numerical value is 2.15483 > e2/e, so the variance grows faster than the mean squared (K is a solution of the equation T (T (1/(eK)))2 = T (1/(eK))/e, where T (x) is the exponential generating function for rooted labelled trees) In view of this growth, one cannot expect the “usual” renormalisation (subtract the mean and divide by the standard deviation) to yield a limiting distribution.

The general central limit theorem

Simple examples

Number and size of subtrees

Final remarks