Mean Row Values in $(u,v)$-Calkin-Wilf Trees

TL;DR

This paper studies the average values at each level of a family of infinite binary trees called $(u,v)$-Calkin-Wilf trees, showing that these averages converge to a specific value depending on $u$ and $v$, extending previous results.

Contribution

It generalizes the known convergence of mean row values from the case $u=v=1$ to all $u,v ext{ with } u,v o ext{integers} ext{ and } u,v eq 0$ in the $(u,v)$-Calkin-Wilf trees.

Findings

Mean row value converges to $v + rac{ ext{log} 2}{u}$ for all $z$ in the specified interval.

Convergence is uniform across the initial rational values $z$.

The result extends the classical $3/2$ limit for the standard Calkin-Wilf tree to a broader family.

Abstract

We fix integers $u,v \geq 1$, and consider an infinite binary tree $\mathcal{T}^{(u,v)}(z)$ with a root node whose value is a positive rational number $z$. For every vertex $a/b$, we label the left child as $a/(ua+b)$ and right child as $(a+vb)/b$. The resulting tree is known as the $(u,v)$-Calkin-Wilf tree. As $z$ runs over $[1/u,v]\cap \mathbb{Q}$, the vertex sets of $\mathcal{T}^{(u,v)}(z)$ form a partition of $\mathbb{Q}^+$. When $u=v=1$, the mean row value converges to $3/2$ as the row depth increases. Our goal is to extend this result for any $u,v\geq 1$. We show that, when $z\in [1/u,v]\cap \mathbb{Q}$, the mean row value in $\mathcal{T}^{(u,v)}(z)$ converges to a value close to $v+\log 2/u$ uniformly on $z$.