A Combinatorial Proof of Cayley's Formula via Degree Sequences

TL;DR

This paper presents a new combinatorial proof of Cayley's formula, emphasizing degree sequences and structural properties of labeled trees to offer an accessible perspective and connect to related enumeration problems.

Contribution

It introduces a novel combinatorial proof of Cayley's formula based on degree sequences, differing from traditional methods like Prufer sequences.

Findings

Provides a combinatorial proof emphasizing degree sequences

Highlights structural properties of labeled trees

Suggests connections to related enumeration problems

Abstract

Cayley's formula is a fundamental result in combinatorics that counts the number of labeled trees on n vertices. While existing proofs use approaches such as Prufer sequences and the Matrix-Tree Theorem, we give a combinatorial proof that highlights the role of degree sequences and structural properties of labeled trees. Our goal is to provide an accessible perspective and suggest connections to related enumeration problems.