Labeled Fibonacci Trees

TL;DR

This paper introduces a class of labeled Fibonacci trees where Fibonacci sequences appear along branches, labels at each level are consecutive, and the set forms a group isomorphic to a7^2, generalizing the Wythoff array.

Contribution

It defines a new class of labeled Fibonacci trees with algebraic and combinatorial properties, linking them to the Wythoff array and establishing their group structure.

Findings

Labels at each level are consecutive integers.

The set of labeled trees forms a a7^2 group.

Properties of the Wythoff array are recovered as a special case.

Abstract

The study describes a class of integer labelings of the Fibonacci tree, the tree of descent introduced by Fibonacci. In these labelings, Fibonacci sequences appear along ascending branches of the tree, and it is shown that the labels at any level are consecutive integers. The set of labeled trees is a commutative group isomorphic to $\mathbb{Z}^2$, and is endowed with an order relation. Properties of the Wythoff array are recovered as a special instance, and further properties of the labeled Fibonacci trees are described. These trees can be viewed as generalizations of the Wythoff array.