A Periodicity Theorem for Trees

In this paper we extend the notion of periodicity from words to labelled ordered trees. This is achieved by deening an equivalence relation on the support of the tree (the set of the nodes) and by requiring that equivalent nodes have the same label. Another requirement is that corresponding sons of equivalent nodes are equivalent too. Since the support of a k-ary tree may be described by a preex-closed set of words over a k-ary alphabet , this traslates to the fact that the equivalence relation on the nodes is described by a right congruence on the free monoid. The periodicity theorem considers the case of a tree having two diierent periodicities. The basic argument in the proof is the construction of the nest right congruence containing two given right congruences, presented in Section 2. By using this construction we derive in the last section a generalization of the classical Fine and Wilf theorem from words to trees. A labelled tree over an alphabet ? is a tree where the nodes are labelled with letters of ?. An ordered tree is a tree in which the sons of each node are ordered. Without loss of generality, if the tree is k-ary, we can indicate such an order by putting numbers from 1 to k on the edges leaving each node (all the edges leaving a given node have diierent labels). For the remaining part of the paper we assume that the alphabet = f1;2;:::;kg, where k 2 IN and that an ordered tree is a tree with the edges labelled by elements of. We denote by the free monoid over. A subset P of is closed under preex if for any w 2 P, every preex of w, i.e. every word v such that w = vu for some word u, belongs also to P. Given a k-ary ordered tree T, we can associate to T a set of words S over the alphabet closed under preex: each word w 2 S corresponds to a node v of the tree T in the sense that the letters of w are the labels in the path from the root to v in the tree. The empty word corresponds to the root. It is not diicult to see that, given such set S, we can retrieve the initial tree. On the other hand, every set S that is closed under preex can …