Extremal problems for colored trees and Davenport-Schinzel sequences

In the theory of generalized Davenport–Schinzel sequences one estimates the maximum lengths of finite sequences containing no subsequence of a given pattern. Here we investigate a further generalization, in which the class of sequences is extended to the class of colored trees. We determine exactly the extremal functions associated with the properly 2-colored path of four vertices and with the monochromatic path of any length. We prove that the extremal function of any colored path grows almost linearly (this is a characteristic feature of DS sequences). Three problems are posed.