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

Abstract

We fix integers $u,v\ge 1$, 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 $u,v\ge 1$. 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 $z$.