On the locality of the Pr\"ufer code

TL;DR

This paper investigates how local changes in the Pr"ufer code affect the structure of the corresponding trees, revealing probabilistic properties of such mutations and their impact on tree edges.

Contribution

It provides asymptotic estimates for the probability that a mutation in the Pr"ufer code causes a specific number of edge changes in the tree.

Findings

Probability of a change in more than one edge is on the order of n^{-1/3+o(1)}.

Probability of a single-edge change is approximately (1 - μ/n)^2.

Probability of a 'perfect' mutation (single-edge change) is about 1/3.

Abstract

The Pr\"ufer code is a bijection between trees on the vertex set $[n]$ and strings on the set $[n]$ of length $n-2$ (Pr\"ufer strings of order $n$). In this paper we examine the `locality' properties of the Pr\"ufer code, i.e. the effect of changing an element of the Pr\"ufer string on the structure of the corresponding tree. Our measure for the distance between two trees $T,T^*$ is $\Delta(T,T^*)=n-1-| E(T)\cap E(T^*)|$. We randomly mutate the $\mu$th element of the Pr\"ufer string of the tree $T$, changing it to the tree $T^*$, and we asymptotically estimate the probability that this results in a change of $\ell$ edges, i.e. $P(\Delta=\ell | \mu).$ We find that P(\Delta=\ell | \mu)$ is on the order of $ n^{-1/3+o(1)}$ for any integer $\ell>1,$ and that $P(\Delta=1 | \mu)=(1-\mu/n)^2+o(1).$ This result implies that the probability of a `perfect' mutation in the Pr\"ufer code (one for which $\Delta(T,T^*)=1$) is $1/3.$