Subtrees of a random tree

TL;DR

This paper investigates the asymptotic number of subtrees in a uniformly random labeled tree, providing bounds and computational evidence suggesting a precise exponential growth rate.

Contribution

It establishes bounds for the number of subtrees in random labeled trees and offers computational evidence for a tight asymptotic estimate.

Findings

Bounds for c(n) between 1.41805386^n and 1.41959881^n

Strong computational indication that c(n) ≤ 1.41806183^n

Asymptotic exponential growth rate of subtrees in random trees

Abstract

Let $T$ be a random tree taken uniformly at random from the family of labelled trees on $n$ vertices. In this note, we provide bounds for $c(n)$, the number of sub-trees of $T$ that hold asymptotically almost surely. With computer support we show that $1.41805386^n \le c(n) \le 1.41959881^n$. Moreover, there is a strong indication that, in fact, $c(n) \le 1.41806183^n$.