The minimum tree for a given zero-entropy period

We answer the following question: given any n∈ℕ, which is the minimum number of endpoints en of a tree admitting a zero-entropy map f with a periodic orbit of period n? We prove that en=s1s2…sk−∑i=2ksisi+1…sk, where n=s1s2…sk is the decomposition of n into a product of primes such that si≤si+1 for 1≤i<k. As a corollary, we get a criterion to decide whether a map f defined on a tree with e endpoints has positive entropy: if f has a periodic orbit of period m with em>e, then the topological entropy of f is positive