An arborist's guide to the rationals

• Published in 2014
• Added on 2026-06-23

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.

Links

• https://arxiv.org/abs/1403.2928v2
• https://arxiv.org/pdf/1403.2928v2

Other information

BibTeX entry

@article{Anarboristsguidetotherationals,
	key = {Anarboristsguidetotherationals},
	type = {article},
	title = {An arborist's guide to the rationals},
	author = {Katherine E. Stange},
	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.},
	comment = {},
	date_added = {2026-06-23},
	date_published = {2014-06-23},
	urls = {https://arxiv.org/abs/1403.2928v2,https://arxiv.org/pdf/1403.2928v2},
	collections = {attention-grabbing-titles,fun-maths-facts},
	url = {https://arxiv.org/abs/1403.2928v2 https://arxiv.org/pdf/1403.2928v2},
	year = 2014,
	urldate = {2026-06-23},
	archivePrefix = {arXiv},
	eprint = {1403.2928},
	primaryClass = {math.NT}
}