Thue choosability of trees

A vertex colouring of a graph G is nonrepetitive if for any path P = (v1, v2, . . . , v2r) in G, the first half is coloured differently from the second half. The Thue choice number of G is the least integer l such that for every l-list assignment L of G, there exists a nonrepetitive L-colouring of G. We prove that for any positive integer l, there is a tree T with πch(T ) > l. On the other hand, it is proved that if G′ is a graph of maximum degree ∆, and G is obtained from G′ by attaching to each vertex v of G′ a connected graph of tree-depth at most z rooted at v, then πch(G) ≤ c(∆, z) for some constant c(∆, d) depending only on ∆ and z.