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

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 positive rational numbers, and they have cluster structures induced by a one-punctured torus. In particular, the Calkin–Wilf tree is an example of the structure given by initial-seed mutations.