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 − ∑k i=2 sisi+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 periodm with em > e, then the topological entropy of f is positive.