 Stern-Brocot tree

A binary tree enumerating every positive reduced fraction exactly once. Begin with boundary fractions $0/1$ and $1/0$; between neighboring fractions $a/b<c/d$, insert their mediant $(a+c)/(b+d)$ and recurse into the two new intervals. The determinant relation $bc-ad=1$ is preserved, and the new denominator is $b+d$.