A Calkin-Wilf tree is a special type of binary tree obtained by starting with the fraction  and iteratively adding  and  below each fraction . The Stern-Brocot tree is closely related, putting  and  below each fraction . Both trees generate every rational number. Writing out the terms in sequence gives 1/1, 1/2, 2/1, 1/3, 3/2, 2/3, 3/1, 1/4, 4/3, 3/5, 5/2, 2/5, 5/3, 3/4, 4/1, ...The sequence has the property that each denominator is the next numerator. This sequence, 1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, ... (OEIS A002487), is known as Stern's diatomic series, or the fusc function (Dijkstra 1982).

REFERENCES:

Bogomolny, A. "Fractions on a Binary Tree II." http://www.cut-the-knot.org/blue/Fusc.shtml.

Calkin, N. and Wilf, H. S. "Recounting the Rationals." Amer. Math. Monthly 107, 360-363, 2000.

Dijkstra, E. W. Selected Writings on Computing: A Personal Perspective. New York: Springer-Verlag, pp. 215-232, 1982.

Gibbons, L.; Lester, D.; and Bird, R. "Functional Pearl: Enumerating the Rationals." J. Func. Prog. 16, 281-291, 2006.

Schneider, K. "The Tree of All Fractions." http://demonstrations.wolfram.com/TheTreeOfAllFractions/.

Sloane, N. J. A. Sequence A002487/M0141 in "The On-Line Encyclopedia of Integer Sequences."