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; 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 this properties make them good candidates for being extremal points in scheduling problems over two processors. In this paper, we consider the problem of generalizing Sturmian words to trees. The problem is to evenly distribute labels 0 and 1 over infinite trees. We show that (strongly) balanced trees exist and can also be constructed using a mechanical process as long as the tree is irrational. Such trees also have a minimal factor complexity. Therefore they bring the hope that extremal scheduling properties of Sturmian words can be extended to such trees, as least partially. Such possible extensions are illustrated by one such example.Comment: 30 pages, use pgf/tik

In the Maze of Data Languages

In data languages the positions of strings and trees carry a label from a finite alphabet and a data value from an infinite alphabet. Extensions of automata and logics over finite alphabets have been defined to recognize data languages, both in the string and tree cases. In this paper we describe and compare the complexity and expressiveness of such models to understand which ones are better candidates as regular models

Weighted spanning trees on some self-similar graphs

We compute the complexity of two infinite families of finite graphs: the Sierpi\'{n}ski graphs, which are finite approximations of the well-known Sierpi\'nsky gasket, and the Schreier graphs of the Hanoi Towers group $H^{(3)}$ acting on the rooted ternary tree. For both of them, we study the weighted generating functions of the spanning trees, associated with several natural labellings of the edge sets.Comment: 21 page