The Tree of All Fractions
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.
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