Yet another generalization of Postnikov's hook length formula for binary trees

TL;DR

This paper introduces a new one-parameter generalization of Postnikov's hook length formula for binary trees, where the hook length appears as an exponent, and explores special cases including a simplified formula at parameter 1/2.

Contribution

It presents a novel one-parameter extension of Postnikov's hook length formula with a unique exponent-based structure for binary trees.

Findings

Derived a new hook length formula with hook length as an exponent

Established a special case formula at parameter 1/2

Extended understanding of binary tree enumeration

Abstract

We discover another one-parameter generalization of Postnikov's hook length formula for binary trees. The particularity of our formula is that the hook length $h_v$ appears as an exponent. As an application, we derive another simple hook length formula for binary trees when the underlying parameter takes the value 1/2.