Two trees enumerating the positive rationals

TL;DR

This paper introduces two novel tree structures that systematically enumerate all positive rational numbers, extending the Calkin-Wilf tree concept with new recurrence formulas and uniqueness properties.

Contribution

The paper presents two new trees, as ternary and quinary analogues of the Calkin-Wilf tree, with recurrence formulas and a uniqueness characterization for enumerating positive rationals.

Findings

Two new trees represent all positive rationals.

Recurrence formulas for node-to-rational mapping are provided.

The sequences are unique under certain relation constraints.

Abstract

We give two trees allowing to represent all positive rational numbers. These trees can be seen as ternary and quinary analogues of the Calkin-Wilf tree. For each of these two trees, we give recurrence formulas allowing to compute the rational number corresponding to the node n. These are analogues of the formulas given by Donald Knuth and Moshe Newman for the Calkin-Wilf tree. Finally, we show that the two sequences we have obtained, together with Calkin-Wilf sequence, are the only ones which satisfy a relation analogue to Newman's relation and enumerate the positive rationals.