The \((u,v)\)-Calkin-Wilf forest (Q2810676)

The Calkin-Wilf tree [\textit{N. Calkin} and \textit{H. S. Wilf}, Am. Math. Mon. 107, No. 4, 360--363 (2000; Zbl 0983.11009)] is an infinite binary tree generated by two rules. The number 1, represented as 1/1, is the root of the tree and each vertex a/b has a left child \(a/(a + b)\) and a right child \((a + b)/b\).NEWLINE \par In this paper, the authors consider a Nathanson's refinement (see [\textit{M. B. Nathanson}, ``Free monoids and forest of rational numbers'', Preprint, \url{arXiv:1406.2054}]) of the Calkin-Wilf tree. They study several properties of such trees associated with the matricesNEWLINENEWLINE\(L_u =\begin{pmatrix}1&0\\u&1\end{pmatrix}\)NEWLINE and \(R_v =\begin{pmatrix}1&v\\0&1\end{pmatrix}\),NEWLINE where \(u\) and \(v\) are non-negative integers. In particular, they extend several known results of the original Calkin-Wilf tree, including the successor and numerator-denominator formulas.