A Simple Recursive Relation Characterizes a Tree Associated to Generalized Farey Sequences

TL;DR

This paper demonstrates that two distinct rooted binary trees, one related to generalized Farey sequences and the other defined by a recursive relation on pairs of positive integers, are structurally identical.

Contribution

It establishes the isomorphism between a Farey sequence-based tree and a recursively defined tree, linking number theory and combinatorial structures.

Findings

The two trees are isomorphic.

The recursive relation characterizes the tree structure.

Connections to generalized Farey sequences and permutation classes.

Abstract

This paper proves that two differently defined rooted binary trees are isomorphic. The first tree is one associated to a version of Farey sequences where the vertices correspond to the open intervals formed by two successive terms in the sequence. The other tree has the vertices consisting of pairs of positive integers whose adjacency is defined by a simple recursive relation. These trees appeared in a study of a generalization of a class of the permutations defined by S\'{o}s and the bijection between it and the set of the Farey intervals due to Sur\'{a}nyi.