A note on counting labeled and unlabeled trees

TL;DR

This paper offers a combinatorial proof of Cayley's formula using a bijective urn model and introduces an algebraic structure on labeled trees, linking it to counting unlabeled trees via partitions.

Contribution

It presents a new combinatorial proof and an algebraic framework for understanding the enumeration of labeled and unlabeled trees.

Findings

Provides a bijective proof of Cayley's formula

Introduces an algebraic structure on labeled trees

Connects counting unlabeled trees to partition problems

Abstract

We provide a short combinatorial proof of Cayley's formula by means of a bijective map to an outcome space of an urn-drawing problem. Furthermore we introduce an algebraic structure on the set of labeled trees, which provides a more standard approach to Cayley's formula. Moreover, this algebraic structure sheds light on the problem of counting the unlabeled trees. In particular, it indicates how counting the number of unlabeled trees on $n$ vertices is connected to finding the number of partitions of $n-2$