Continuous maps of trees and nonwandering sets

Let ƒ be a continuous map of a tree X into itself. Let Ω(ƒ) denote the set of nonwandering points for ƒ. In this paper we show the following two results: (1) if Ω(ƒ) is finite then Ω(ƒ) is the set of periodic points of ƒ, (2) Ω(ƒ) is contained in the closure of the set of eventually periodic points of ƒ. Also we give some examples which imply that these results are not true for the case that X is a dendrite or a graph.