An arborist's guide to the rationals

TL;DR

This paper explores the relationships between different infinite binary trees that enumerate positive rationals, providing a new proof connecting the Farey/Stern-Brocot, Calkin-Wilf, and Conway-Fung topograph trees.

Contribution

It introduces a novel proof linking these trees through the concept of 'transpose shadows' of a matrix tree, expanding understanding of their interrelations.

Findings

Unified perspective on rational enumeration trees

New proof connecting Farey/Stern-Brocot and Calkin-Wilf trees

Enhanced understanding of the topograph's role in rational trees

Abstract

There are two well-known ways to enumerate the positive rational numbers in an infinite binary tree: the Farey/Stern-Brocot tree and the Calkin-Wilf tree. In this brief note, we describe these two trees as `transpose shadows' of a tree of matrices (a result due to Backhouse and Ferreira) via a new proof using yet another famous tree of rationals: the topograph of Conway and Fung.