An Elementary Proof of the Cayley Formula Using Random Maps

TL;DR

This paper offers a simple, elementary bijective proof of Cayley's formula for counting labeled trees, using probability calculations on random maps to avoid complex structures.

Contribution

It introduces a novel, elementary bijective proof of Cayley's formula based on probability analysis of cycles in random maps.

Findings

Derived the probability of cycles in random maps from n to n+1 elements

Provided an elementary proof of Cayley's formula without complex structures

Confirmed the formula through probabilistic reasoning

Abstract

Cayley's formula states that the number of labelled trees on $n$ vertices is $n^{n-2}$, and many of the current proofs involve complex structures or rigorous computation. We present a bijective proof of the formula by providing an elementary calculation of the probability that a cycle occurs in a random map from an $n$-element set to an $n+1$-element set.