  We study a model of multi-excited random walk on a regular tree which
generalizes the models of the once excited random walk and the digging random
walk introduced by Volkov (2003). We show the existence of a phase transition
of the recurrence/transience property of the walk. In particular, we prove that
the asymptotic behavior of the walk depends on the order of the excitations,
which contrasts with the one dimensional setting studied by Zerner (2005). We
also consider the limiting speed of the walk in the transient regime and
conjecture that it is not a monotonic function of the environment.
