Neil Calkin

In number theory, the Calkin–Wilf tree is a tree in which the vertices correspond 1-for-1 to the positive rational numbers. The tree is rooted at the number 1, and any rational number expressed in simplest terms as the fraction a/b has as its two children the numbers a/(a + b) and (a + b)/b. Every positive rational number appears exactly once in the tree.

The sequence of rational numbers in a breadth-first traversal of the Calkin–Wilf tree is known as the Calkin–Wilf sequence. Its sequence of numerators (or, offset by one, denominators) is Stern's diatomic series, and can be computed by the fusc function.

The Calkin–Wilf tree is named after Neil Calkin and Herbert Wilf, who considered it in their 2000 paper. The tree was introduced earlier by Jean Berstel and Aldo de Luca as Raney tree, since they drew some ideas from a paper by George N. Raney. Stern's diatomic series was formulated much earlier by Moritz Abraham Stern, a 19th-century German mathematician who also invented the closely related Stern–Brocot tree. Even earlier, a similar tree appears in Kepler's Harmonices Mundi (1619).

The Calkin–Wilf tree may be defined as a directed graph in which each positive rational number a/b occurs as a vertex and has one outgoing edge to another vertex, its parent. We assume that a/b is in simplest terms; that is, the greatest common divisor of a and b is 1. If a/b < 1, the parent of a/b is a/(b − a); if a/b is greater than one, the parent of a/b is (a − b)/b. Thus, in either case, the parent is a fraction with a smaller sum of numerator and denominator, so repeated reduction of this type must eventually reach the number 1. As a graph with one outgoing edge per vertex and one root reachable by all other vertices, the Calkin–Wilf tree must indeed be a tree.

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