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.