Davenport–Schinzel Trees*

In this paper we study labeled–tree analogues of (generalized) Davenport–Schinzel sequences.

We say that two sequences a ₁ ... a _k , b ₁ ... b _k of equal length k are isomorphic, if a _i = a _j iff b _i = b _j (for all i, j). For example, the sequences 11232, 33141 are isomorphic. We investigate the maximum size of a labeled (rooted) tree with each vertex labeled by one of n labels in such a way that, besides some technical conditions, the sequence of labels along any path (starting from the root) contains no subsequence isomorphic to a fixed “forbidden” sequence u.

We study two models of such labeled trees. Each of the models is known to be essentially equivalent also to other models. The labeled paths in a special case of one of our models correspond to classical Davenport–Schinzel sequences.

We investigate, in particular, for which sequences u the labeled tree has at most O(n) vertices. In both models, we answer this question for any forbidden sequence u over a two-element alphabet and also for a large class of other sequences u.