Labeled trees, maps, and an algebraic identity

TL;DR

This paper presents a concise proof of a notable algebraic identity related to labeled trees and their indegree sequences, and extends it to count rooted spanning forests with specified indegree configurations.

Contribution

It offers a direct proof of a key identity in labeled tree enumeration and generalizes it to rooted spanning forests, addressing a problem posed by Shin and Zeng.

Findings

Proved a remarkable identity in labeled tree enumeration

Generalized the identity to rooted spanning forests

Provided a solution to an open problem by Shin and Zeng

Abstract

We give a short and direct proof of a remarkable identity that arises in the enumeration of labeled trees with respect to their indegree sequence, where all edges are oriented from the vertex with lower label towards the vertex with higher label. This solves a problem posed by Shin and Zeng in a recent article. We also provide a generalization of this identity that translates to a formula for the number of rooted spanning forests with given indegree sequence.