Non-wandering sets of the powers of dendrite maps

Let (X, d) be a metric space and f be a continuous map from X to X. Denote by EP(f) and .(f) the sets of eventually periodic points and non-wandering points of f, respectively. It is well known that for a tree map f, the following statements hold: (1) If x . .(f) . .(f n) for some n . 2, then x . EP(f). (2) .(f) is contained in the closure of EP(f). The aim of this note is to show that the above results do not hold for maps of dendrites D with Card(End(D)) = N0 (the cardinal number of the set of positive integers).