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The Stern-Brocot or Farey Tree

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There are several versions of this tree. This one, which appears in (Graham, Knuth and Patashnik 1990, p. 117), was drawn by Alexander Bogomolny[1]. 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.
Image:A002487.gif

Contents

Extensions to the Stern-Brocot tree

One way to extend the Stern-Brocot tree to cover whole \scriptstyle \Q\,, not just the positive rationals, is to reflect it over the "Y-axis" where zero, (i.e. fraction \scriptstyle \frac{0}{1}\,) is located, and make the fractions on the left side all negative. (See A057114 for example.) (XXX - We need here an illustration like above.)

Notes

  1. Alexander Bogomolny, http://www.cut-the-knot.org/blue/Stern.shtml

References

  • Graham, R. L.; Knuth, D. E. & Patashnik, O. (1990). Concrete Mathematics. Reading, MA: Addison-Wesley. 

Authorship

The original version of this page was written by Neil Sloane. (The initial version was copied from http://oeis.org/stern_brocot.html but is intended to be filled with more information!)

Further additions by Antti Karttunen.

Cite this page as

N. J. A. Sloane, <insert your name here if you added anything essential>, et al., <a href="http://oeis.org/wiki/The_Stern-Brocot_or_Farey_Tree">The Stern-Brocot or Farey Tree</a>, OEIS Wiki.

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