Free monoids and forests of rational numbers

The Calkin-Wilf tree is an infinite binary tree whose vertices are the positive rational numbers. Each such number occurs in the tree exactly once and in the form $a\mathrm{/}b$, where are $a$ and $b$ are relatively prime positive integers. This tree is associated with the matrices $L_1 = \left( \begin{matrix} 1 & 0 \ 1 & 1 \end{matrix} \right)$and$R_1 = \left( \begin{matrix} 1 & 1 \ 0 & 1 \end{matrix} \right)$, which freely generate the monoid $S{L}_2({\mathbf{N}}_0)$ of $2\times 2$ matrices with determinant 1 and nonnegative integral coordinates. For other pairs of matrices $L_u$ and $R_v$ that freely generate submonoids of $G{L}_2({\mathbf{N}}_0)$, there are forests of infinitely many rooted infinite binary trees that partition the set of positive rational numbers, and possess a remarkable symmetry property.