Let X be a compact tree f:X to X be a continuous map and End(X) be the number of endpoints of X. The author proves the following: Theorem 1. Let X be a tree. Then the following holds. (i) Let n>1 be an integer with no ′ divisors less than or equal to End(X)+1. If a map f:X to X has a cycle of period n, then f has cycles of all periods greater than 2 End(X)(n-1). Moreover, h(f)>or=In2/(nEnd(X)-1). (ii) Let 10. Note that Misiurewicz's conjecture and the last result are true for graph maps; an alternative proof of the last result may be also found in a paper by Llibre and Misiurewicz (1991).