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The Stern-Brocot or Farey Tree

There are several versions of this tree. This one, which appears in R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 117, was drawn by Alexander Bogomolny. For another version see J. C. Lagarias, Number Theory and Dynamical Systems, pp. 35-72 of S. A. Burr, ed., The Unreasonable Effectiveness of Number Theory, Proc. Sympos. Appl. Math., 46 (1992). Amer. Math. Soc.

The nth order Farey series is the set of reduced fractions between 0 and 1 whose denominators are n or less, arranged in increasing order, and corresponds to a subtree of the Stern-Brocot tree.

There are also many associated sequences:

  • The numerators and denominators of the fractions in the full tree give A007305/A047679.
  • The numerators and denominators of the fractions in the left-hand subtree give A007305/A007306.
  • The numerators and denominators of the triangle whose nth row consists of the Farey series of order n give A006842/A006843.
  • See also A049455/A049456, A002487 and A057431.

Stern-Brocot Tree

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Last modified November 30 00:53 EST 2015. Contains 264663 sequences.