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/b, where a and b are relatively prime positive integers. This tree is associated with the matrices L1=(1011) and R1=(1101), which freely generate the monoid SL2(N0) of 2×2 matrices with determinant 1 and nonnegative integral coordinates. For other pairs of matrices Lu and Rv that freely generate submonoids of GL2(N0), there are forests of infinitely many rooted infinite binary trees that partition the set of positive rational numbers, and possess a remarkable symmetry property.