Applications of Stern-Brocot Trees
1. A Generalised Stern-Brocot Tree from Regular Diophantine Quadruples(arXiv)
Author : Philip Gibbs
Abstract : Diophantine quadruples are sets of four distinct positive integers such that the product of any two is one less than a square. All known examples belong to an infinite set which can be constructed recursively. Some observations on these regular solutions are presented. In particular we see how factors of the four numbers satisfy relations which generalise the Stern-Brocot Tree. There is also an update on the search for rational Diophantine sextuples with five new examples.
2.Stern-Brocot Trees from Weighted Mediants (arXiv)
Author : Dhroova Aiylam, Tanya Khovanova
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.
3.Cluster duality between Calkin-Wilf tree and Stern-Brocot tree (arXiv)
Author : Yasuaki Gyoda
Abstract : We find a duality between two well-known trees, the Calkin-Wilf tree and the Stern-Brocot tree, derived from cluster algebra theory. The vertex sets of these trees are the set of rational numbers, and they have cluster structures induced by one-punctured torus. In particular, the Calkin-Wilf tree is an example of the structure given by initial-seed mutations.
