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 π ch (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 π ch (G) ≤ c(∆, z) for some constant c(∆, d) depending only on ∆ and z.
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