A special type of binary tree obtained by starting with the fractions  and  and iteratively inserting  between each two adjacent fractions  and . The result can be arranged in tree form as illustrated above. The Farey sequence  defines a subtree of the Stern-Brocot tree obtained by pruning off unwanted branches (Vardi 1991, Graham et al. 1994).

REFERENCES:

Bogomolny, A. "Stern-Brocot Tree." http://www.cut-the-knot.org/blue/Stern.shtml.

Brocot, A. "Calcul des rouages par approximation, nouvelle méthode." Revue Chonométrique 3, 186-194, 1861.

Graham, R. L.; Knuth, D. E.; and Patashnik, O. Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, pp. 116-117, 1994.

Haynes, B. "On the Teeth of Wheels." American Scientist 88, No. 4, July-August 2000. http://www.americanscientist.org/template/AssetDetail/assetid/20826.

Stern, M. A. "Über eine zahlentheoretische Funktion." J. reine angew. Math. 55, 193-220, 1858.

Vardi, I. Computational Recreations in Mathematica. Redwood City, CA: Addison-Wesley, p. 253, 1991.

Viswanath, D. "Random Fibonacci Sequences and the Number 1.13198824...." Math. Comput. 69, 1131-1155, 2000.