Stern-Brocot Trees from Weighted Mediants

TL;DR

This paper generalizes Stern-Brocot trees using weighted mediants, specifically analyzing the case k=3 to determine which rationals appear, and extends results to various reduction schemes.

Contribution

It introduces a natural generalization of Stern-Brocot trees with weighted mediants and characterizes the rationals that appear in these generalized trees.

Findings

Characterization of rationals in the k=3 weighted mediant tree

Extension of results to arbitrary reduction schemes

Determination of which rationals between starting terms appear

Abstract

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.