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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.
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Contents
Extensions to the Stern-Brocot tree
One way to extend the Stern-Brocot tree to cover whole , not just the positive rationals, is to reflect it over the "Y-axis" where zero, (i.e. fraction ) 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
- ↑ 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.
