Stern-Brocot tree

The root of the Stern-Brocot tree corresponds to the number 1.

The Stern-Brocot tree was discovered independently by and .

The Stern-Brocot tree forms an infinite binary search tree with respect to the usual ordering of the rational numbers.

In addition, it maps rational numbers to dyadic rationals, as can be seen by a recursive definition closely related to the Stern-Brocot tree.

He is also known for the Stern-Brocot tree which he wrote about in 1858 and which Brocot independently discovered in 1861.

It uses Furstenberg's formula for the Lyapunov exponent of a random matrix product and integration over a certain fractal measure on the Stern-Brocot tree.

The sequence of values M computed by this search is exactly the sequence of values on the path from the root to q in the Stern-Brocot tree.

The Stern-Brocot tree provides an enumeration of all positive rational numbers, in lowest terms, obtained purely by iterative computation of the mediant according to a simple algorithm.

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.

These paths correspond by an order preserving bijection to the points of the Cantor set, or (through the example of the Stern-Brocot tree) to the set of positive irrational numbers.

The path from the root 1 to a number q in the Stern-Brocot tree may be found by a binary search algorithm, which may be expressed in a simple way using mediants.

Along with the definitions in terms of continued fractions and mediants described above, the Stern-Brocot tree may also be defined as a Cartesian tree for the rational numbers, prioritized by their denominators.

The Stern-Brocot sequence of order i consists of all values at the first i levels of the Stern-Brocot tree, together with the boundary values 0/1 and 1/0, in numerical order.

A similar process of mediant insertion, starting with a different pair of interval endpoints [0/1,1/0], may also be seen to describe the construction of the vertices at each level of the Stern-Brocot tree.

In number theory, the Stern-Brocot tree is an infinite complete binary tree in which the vertices correspond precisely to the positive rational numbers, whose values are ordered from left to right as in a search tree.

However, it is closely related to a different binary search tree on the same set of vertices, the Stern-Brocot tree: the vertices at each level of the two trees coincide, and are related to each other by a bit-reversal permutation.

Stern was a German number theorist; Brocot was a French clockmaker who used the Stern-Brocot tree to design systems of gears with a gear ratio close to some desired value by finding a ratio of smooth numbers near that value.

This problem is fairly distinct from that of rounding a value to a fixed number of decimal or binary digits, or to a multiple of a given unit m. This problem is related to Farey sequences, the Stern-Brocot tree, and continued fractions.

Each open interval (L,H) occurring at some step in the search is the interval (L,H)representing the descendants of the mediant M. The parent of q in the Stern-Brocot tree is the last mediant found that is not equal to q.

It follows from the theory of Cartesian trees that the lowest common ancestor of any two numbers q and r in the Stern-Brocot tree is the rational number in the closed interval [q, r] that has the smallest denominator among all numbers in this interval.

The Stern-Brocot tree may itself be defined directly in terms of mediants: the left child of any number q is the mediant of q with its closest smaller ancestor, and the right child of q is the mediant of q with its closest larger ancestor.

He is known for his discovery (independently of and contemporaneously with German number theorist Moritz Stern) of the Stern-Brocot tree, a mathematical structure useful in approximating real numbers by rational numbers; this sort of approximation is an important part of the design of gear ratios for clocks.