Non-wandering sets of the powers of maps of a tree

Let T be a tree and let Omega (f) be the set of non-wandering points of a c ontinuous map f: T-->T. We prove that for a continuous map f: T-->T of a tr ee T: (i) if x is an element of Omega( f) has an infinite orbit, then x is an element of Omega (f(n)) for each n is an element ofN; (ii) if the topolo gical entropy of f is zero, then Omega (f) = Omega (f(n)) for each n is an element ofN. Furthermore, for each k is an element ofN we characterize thos e natural numbers n with the property that Omega (f(k)) = Omega (f(kn)) for each continuous map f of T.