A forest of linear fractional transformations

The Calkin-Wilf tree is an infinite binary tree whose vertices are the positive rational numbers. Each number occurs in the tree exactly once and in the form $a/b$, where are $a$ and $b$ are relatively prime positive integers. For every $2\times 2$ matrix with nonnegative integral coordinates and nonzero determinant, it is possible to construct an analogous tree with this root. If the root is the identity matrix, then the tree consists all matrices with determinant 1, and this tree possesses the basic properties of the Calkin-Wilf tree of positive rational numbers. The set of all matrices with nonzero determinant decomposes into a forest of rooted infinite binary trees.