A remark on the enumeration of rooted labeled trees

TL;DR

This paper provides a simpler proof for a known enumeration formula of rooted labeled trees with a specific child configuration, enhancing understanding of tree enumeration in combinatorics.

Contribution

The paper introduces a more straightforward proof of a classical enumeration result for rooted labeled trees with a fixed number of lower-numbered children of the root.

Findings

Simplified proof of the enumeration formula

Clarification of the combinatorial structure of rooted trees

Enhanced understanding of labeled tree enumeration

Abstract

Two decades ago, Chauve, Dulucq and Guibert showed that the number of rooted trees on the vertex set $[n+1]$ in which exactly $k$ children of the root are lower-numbered than the root is $\binom{n}{k} \, n^{n-k}$. Here I give a simpler proof of this result.