Section: New Results

Discrete Structures

Participants : Yves Denneulin, Bruno Gaujal.

Distributing Labels on Infinite Trees

Sturmian words are infinite binary words with many equivalent definitions. They have a minimal factor complexity among all aperiodic sequences. Also, they are balanced sequences (the labels 0 and 1 are as evenly distributed as possible) and they can be constructed using a mechanical definition. All these properties make them good candidates for being extremal points in scheduling problems over two processors.

In [11] , we study infinite unordered d-ary trees with nodes labeled by 0, 1. We introduce the notions of rational and Sturmian trees along with the definitions of (strongly) balanced trees and mechanical trees, and study the relations among them. In particular, we show that (strongly) balanced trees exist and coincide with mechanical trees in the irrational case, providing an effective construction. Such trees also have a minimal factor complexity, hence are Sturmian. We also give several examples illustrating the inclusion relations between these classes of trees.