Nonrepetitive choice number of trees

A nonrepetitive coloring of a path is a coloring of its vertices such that the sequence of colors along the path does not contain two identical, consecutive blocks. The remarkable construction of Thue asserts that 3 colors are enough to color nonrepetitively paths of any length. A nonrepetitive coloring of a graph is a coloring of its vertices such that all simple paths are nonrepetitively colored. Assume that each vertex $v$ of a graph $G$ has assigned a set (list) of colors $Lv$. A coloring is chosen from ${Lv}{v\in V(G)}$ if the color of each $v$ belongs to $Lv$. The Thue choice number of $G$, denoted by $\pil(G)$, is the minimum $k$ such that for any list assignment $\set{Lv}$ of $G$ with each $|Lv|\geq k$ there is a nonrepetitive coloring of $G$ chosen from ${Lv}$. Alon et al. (2002) proved that $\pil(G)=O(\Delta^2)$ for every graph $G$ with maximum degree at most $\Delta$. We propose an almost linear bound in $\Delta$ for trees, namely for any $\epsi>0$ there is a constant $c$ such that $\pil(T)\leq c\Delta^{1+\epsi}$ for every tree $T$ with maximum degree $\Delta$. The only lower bound for trees is given by a recent result of Fiorenzi et al. (2011) that for any $\Delta$ there is a tree $T$ such that $\pi_l(T)=\Omega(\frac{\log\Delta}{\log\log\Delta})$. We also show that if one allows repetitions in a coloring but still forbid 3 identical consecutive blocks of colors on any simple path, then a constant size of the lists allows to color any tree.