  We prove that it is always possible to color online nonrepetitively any
(partial) $k$-tree (that is, graphs with tree-width at most $k$) with $4^k$
colors. This implies that it is always possible to color online nonrepetitively
cycles, trees and series-parallel graphs with $16$ colors. Our results
generalize the respective (offline) nonrepetitive coloring results.
