The (u,v)-Calkin-Wilf Forest

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 $L_u=\left[\begin{array}{cc}1& 0\\ u& 1\end{array}\right]$ and $R_v=\left[\begin{array}{cc}1& v\\ 0& 1\end{array}\right]$, 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.

18 pages

References in corpus (4)

• Enumerating Trees
• Free monoids and forests of rational numbers
• Enumerating the rationals from left to right
• Combinatorial and Additive Number Theory Problem Sessions: '09--'25

Cited by in corpus (6)

• Free monoids and forests of rational numbers
• Subgroups of $S{L}_2(\mathbb{Z})$ characterized by certain continued fraction representations
• Mean Row Values in $(u,v)$-Calkin-Wilf Trees
• Orphans in Forests of Linear Fractional Transformations
• Hyperbinary partitions and q-deformed rationals
• Maximal entries of elements in certain matrix monoids