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.

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Correspondence to Pavel Valtr†.

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* This research was partially supported by Charles University grants No. 99/158 and 99/159 and by Czech Republic Grant GAČR 201/99/0242.

† Supported by project LN00A056 of The Ministry of Education of the Czech Republic.

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Valtr†, P. Davenport–Schinzel Trees*. Combinatorica 23, 151–184 (2003). https://doi.org/10.1007/s00493-003-0017-0

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AMS Subject Classification (2000):

  • 52C10
  • 68R15
  • 68U05
  • 05C78