The (u,v)-Calkin-Wilf Forest

TL;DR

This paper extends the Calkin-Wilf tree by exploring properties of a generalized (u,v)-Calkin-Wilf forest associated with specific matrices, revealing new symmetry and structural insights into rational number enumeration.

Contribution

It introduces a generalized (u,v)-Calkin-Wilf forest, extending known properties of the original tree to a broader setting involving matrices with nonnegative integer parameters.

Findings

Extended symmetry and successor formulas to the (u,v)-Calkin-Wilf forest

Analyzed the ancestry of rational numbers within the generalized trees

Connected the structure of these trees to matrix representations

Abstract

In this paper we consider a refinement, due to Nathanson, of the Calkin-Wilf tree. In particular, we study the properties of such trees associated with the matrices $L_u=\begin{bmatrix} 1 & 0 \\ u & 1\end{bmatrix}$ and $R_v=\begin{bmatrix} 1 & v \\ 0& 1\end{bmatrix}$, where $u$ and $v$ are nonnegative integers. We extend several known results of the original Calkin-Wilf tree, including the symmetry, numerator-denominator, and successor formulas, to this new setting. Additionally, we study the ancestry of a rational number appearing in a generalized Calkin-Wilf tree.