Benjamini-Schramm convergence and subtrees of trees

TL;DR

This paper investigates the asymptotic behavior of subtrees in sequences of trees that converge locally, showing that subtree entropy and density stabilize or are dense in the unit interval under certain conditions.

Contribution

It establishes the convergence of subtree entropy and density for Benjamini-Schramm convergent trees and explores their distribution in the unit interval.

Findings

Subtree entropy converges to a constant depending on the limit tree.

Subtree density converges if long paths are excluded.

Density of subtree density and entropy spans the entire interval [0,1].

Abstract

In this paper, we study the asymptotic behaviour of the number of subtrees and the subtree density for a sequence of trees that converges in the Benjamini-Schramm sense. Benjamini-Schramm convergence, also called local weak convergence, describes the local behaviour of a sequence of graphs. Here we show that for a Benjamini-Schramm-convergent sequence of trees, the subtree entropy, i.e., the logarithm of the number of subtrees divided by the order, converges to a constant depending only on the limit. The same holds true for the subtree density, i.e., the probability of a uniformly random vertex being contained in a uniformly random subtree, provided that long paths are ruled out in the limit. Related to this, we show that the subtree density and the average subtree entropy are dense in different parts of the unit interval $[0,1]$ for both general trees and series-reduced trees.