The n-dimensional Stern–Brocot tree

Abstract

This paper generalizes the Stern–Brocot tree to a tree that consists of all sequences of [Formula: see text] coprime positive integers. As for [Formula: see text] each sequence [Formula: see text] is the sum of a specific set of other coprime sequences, its Stern–Brocot set [Formula: see text], where [Formula: see text] is the degree of [Formula: see text] With an orthonormal base as the root, the tree defines a fast iterative structure on the set of distinct directions in [Formula: see text] and a multiresolution partition of [Formula: see text]. Basic proofs rely on a matrix representation of each coprime sequence, where the Stern–Brocot set forms the matrix columns. This induces a finitely generated submonoid [Formula: see text] of [Formula: see text], and a unimodular multidimensional continued fraction algorithm, also generalizing [Formula: see text]. It turns out that the [Formula: see text]-dimensional subtree starting with a sequence [Formula: see text] is isomorphic to the entire [Formula: see text]-dimensional tree. This allows basic combinatorial properties to be established. It turns out that also in this multidimensional version, Fibonacci-type sequences have maximal sequence sum in each generation.