Let T be a tree and f be a continuous map from T into itself. We show mainly in this paper that a point x of T is an w-limit point of f if and only if every open neighborhood of x in T contains at least nx + 1 points of some trajectory, where nx equals the number of connected components of T \ {x}. Then, for any open subset G w(f) in T, there exists a positive integer m = m(G) such that at most m points of any trajectory lie outside G.This result is a generalization of the related result for maps of the interval.
Let I = [0,1], 0 < a < b < 1. Let Φab ≡ {F ∈ C0(I): both F|[0,a] and F|[b, 1] are strictly increasing, and F|[a, b] is constant }. In this paper we discuss necessary and sufficient conditions for F ∈Φab to have monotone iterative roots.
