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

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 Sós and the bijection between it and the set of the Farey intervals due to Surányi.

💡 Research Summary

The paper establishes a precise structural equivalence between two rooted binary trees that arise from seemingly unrelated constructions. The first tree, denoted (T_{\mathcal G,\mathcal V}), is built from a generalized version of Farey sequences. For non‑negative integers (m,n) the set (G_{m,n}) consists of the symbols (0,\infty) together with all reduced fractions (p/q) where (p\in