Stern-Brocot Trees from Weighted Mediants

In this paper we discuss a natural generalization of the Stern Brocot tree which comes from the introduction of weighted mediants. We focus our attention on the case $k = 3$, in which $(2a + c)/(2b + d)$ and $(a + 2c)/(b + 2d)$ are the two mediants inserted between $a/b$ and $c/d$. Our main result is a determination of which rational numbers between the starting terms appear in the tree. We extend this result to arbitrary reduction schemes as well.