The Foata-Fuchs proof of Cayley's formula, and its probabilistic uses

TL;DR

This paper provides a simple bijective proof of Cayley's formula, explores its probabilistic applications in analyzing random trees, and introduces a partial order on tree degree sequences with conjectured stochastic properties.

Contribution

It introduces a simple bijective proof of Cayley's formula and demonstrates its usefulness in probabilistic analysis of random trees, including new conjectures on tree height distributions.

Findings

A simple bijective proof of Cayley's formula is presented.

The proof is applied to derive probabilistic identities and bounds for random trees.

A new partial order on degree sequences is proposed, with conjectures on its stochastic implications.

Abstract

We present a very simple bijective proof of Cayley's formula due to Foata and Fuchs (1970). This bijection turns out to be very useful when seen through a probabilistic lens; we explain some of the ways in which it can be used to derive probabilistic identities, bounds, and growth procedures for random trees with given degrees, including random d-ary trees. We also introduce a partial order on the degree sequences of rooted trees, and conjecture that it induces a stochastic partial order on heights of random rooted trees with given degrees.