Cluster duality between Calkin-Wilf tree and Stern-Brocot tree

TL;DR

This paper uncovers a duality between the Calkin-Wilf and Stern-Brocot trees, linking their structures through cluster algebra theory and revealing new insights into their rational number representations.

Contribution

It introduces a novel duality between two classical trees using cluster algebra structures derived from one-punctured torus configurations.

Findings

Identifies a duality between Calkin-Wilf and Stern-Brocot trees

Shows both trees have cluster structures from one-punctured torus

Relates the trees' vertex sets to rational numbers

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.