  A vertex coloring of a graph is nonrepetitive if there is no path in the
graph whose first half receives the same sequence of colors as the second half.
While every tree can be nonrepetitively colored with a bounded number of colors
(4 colors is enough), Fiorenzi, Ochem, Ossona de Mendez, and Zhu recently
showed that this does not extend to the list version of the problem, that is,
for every $\ell \geq 1$ there is a tree that is not nonrepetitively
$\ell$-choosable. In this paper we prove the following positive result, which
complements the result of Fiorenzi et al.: There exists a function $f$ such
that every tree of pathwidth $k$ is nonrepetitively $f(k)$-choosable. We also
show that such a property is specific to trees by constructing a family of
pathwidth-2 graphs that are not nonrepetitively $\ell$-choosable for any fixed
$\ell$.
