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A Combinatorial Proof of Cayley's Formula via Degree Sequences

Helia Karisani, Mohammadreza Daneshvaramoli

TL;DR

The paper tackles Cayley’s formula, which counts labeled trees on $n$ vertices as $n^{n-2}$, by presenting a new combinatorial proof centered on degree sequences. It establishes a closed-form count for trees with a fixed degree sequence, $T_{n,d_1,...,d_n} = \frac{(n-2)!}{(d_1-1)! \cdots(d_n-1)!}$, extending to zero when a degree is zero, and proves this by induction via leaf removal. By summing over all valid degree sequences and employing a double-counting perspective that partitions trees into components treated as super-vertices, the method derives Cayley’s result and offers an intuitive alternative to Prüfer codes and the Matrix-Tree Theorem. The approach provides a fresh combinatorial viewpoint that links degree sequences to enumeration problems and suggests avenues for related counting problems. Overall, it contributes a rigorous, accessible framework for understanding Cayley’s formula through structural decomposition and induction.

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.

A Combinatorial Proof of Cayley's Formula via Degree Sequences

TL;DR

The paper tackles Cayley’s formula, which counts labeled trees on vertices as , by presenting a new combinatorial proof centered on degree sequences. It establishes a closed-form count for trees with a fixed degree sequence, , extending to zero when a degree is zero, and proves this by induction via leaf removal. By summing over all valid degree sequences and employing a double-counting perspective that partitions trees into components treated as super-vertices, the method derives Cayley’s result and offers an intuitive alternative to Prüfer codes and the Matrix-Tree Theorem. The approach provides a fresh combinatorial viewpoint that links degree sequences to enumeration problems and suggests avenues for related counting problems. Overall, it contributes a rigorous, accessible framework for understanding Cayley’s formula through structural decomposition and induction.

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.
Paper Structure (3 sections, 3 theorems, 14 equations, 3 figures)

This paper contains 3 sections, 3 theorems, 14 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Cayley’s formula Proof
  3. Conclusion

Key Result

Theorem 1

Suppose $n \geq 2$ and $d_1, d_2, \dots, d_n$ are positive natural numbers with a sum of $2n-2$. In this case, the number of trees with vertices $\{1, 2, \dots, n\}$ and with vertex degrees $d_1, d_2, \dots, d_n$ is given by:

Figures (3)

  • Figure 1: A tree with $d_1 = k = 5$ and connected subtrees $\{a_i\}$.
  • Figure 2: One way of choosing $k-1 = 4$ specified edges (circled) in a tree.
  • Figure 3: $k=5$ tree components of Figure \ref{['fig:f2']}.

Theorems & Definitions (6)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Lemma 3
  • proof