  We show that the Euler walk on a Cayley tree exhibits two regimes (dynamic
phases): a condensed phase and a low-density phase. In the condensed phase the
self-organized area grows as a compact domain. In the low-density phase the
proportion of self-organized (visited) nodes decreases rapidly from one
generation of the tree to the next. We describe in detail returns of the Euler
walk to the root and growth of the self-organized domain in the condensed
phase. We also investigate the critical behaviour of the Euler walk at the
point separating the two regimes.
