Modified Stern-Brocot Sequences

TL;DR

This paper revisits the classical Stern-Brocot tree, offers a new proof of its properties, and generalizes it to arbitrary starting terms, ensuring all rationals between them appear exactly once.

Contribution

It introduces a generalized Stern-Brocot tree with arbitrary initial terms and proves its completeness and uniqueness properties.

Findings

Every rational between 0 and 1 appears in the classical tree.

Generalized trees with arbitrary starting points contain all rationals between them exactly once.

The paper provides a new proof of the classical properties of the Stern-Brocot tree.

Abstract

We present the classical Stern-Brocot tree and provide a new proof of the fact that every rational number between 0 and 1 appears in the tree. We then generalize theStern-Brocot tree to allow for arbitrary choice of starting terms, and prove that in all cases the tree maintains the property that every rational number between the two starting terms appears exactly once.