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For positive integers u and v, let Lu = [1 & 0 \\ u & 1] and Rv = [1 & v \\ 0 & 1 ]. Let Su,v be the monoid generated by Lu and Rv, and let Gu,v be the group generated by Lu and Rv. In this paper we expand on a characterization of matrices M = [a & b \\ c & d] in Sk,k and Gk,k when k ≥ 2 given by Esbelin and Gutan to Su,v when u,v ≥ 2 and Gu,v when u,v ≥ 3. We give a simple algorithmic way of determining if M is in Gu,v using a recursive function and the short continued fraction representation of b/d. [ABSTRACT FROM AUTHOR]

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 and , where and are non-negative 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. [ABSTRACT FROM AUTHOR]