The Tree of All Fractions

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Each node in the tree contains a rational number . The left child of each node is and the right child is . Nodes containing integers are treated as having a denominator of one. The amazing thing is that every positive rational number appears in this tree exactly once and in lowest terms.

Contributed by: Keith Schneider (March 2011)
Open content licensed under CC BY-NC-SA


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This Demonstration shows the Calkin-Wilf tree. For more information please see:

N. Calkin, H. S. Wilf, "Recounting the Rationals," The American Mathematical Monthly, 107(4), 2000 pp. 360-363. DOI: dx.doi.org/10.2307/2589182



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