A sign-reversing involution to count labeled lone-child-avoiding trees

TL;DR

This paper introduces a combinatorial involution technique to accurately count labeled trees on n vertices with no vertices having exactly one child, correcting a longstanding error in a major reference.

Contribution

It presents a novel sign-reversing involution method to count specific trees, correcting a misprint in the Encyclopedia of Integer Sequences.

Findings

Derived an exact formula for counting lone-child-avoiding trees

Corrected a known misprint in the literature

Demonstrated the effectiveness of involution techniques in combinatorial enumeration

Abstract

We use a sign-reversing involution to show that trees on the vertex set [n], considered to be rooted at 1, in which no vertex has exactly one child are counted by 1/n sum_{k=1}^{n} (-1)^(n-k) {n}-choose-{k} (n-1)!/(k-1)! k^(k-1). This result corrects a persistent misprint in the Encyclopedia of Integer Sequences.