An arborist's guide to the rationals

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.

7 pages, 4 trees; this version is updated with more references and historical info; also the result turns out to be previously known, so this paper provides a new proof, not a new theorem