Topological entropy of transitive maps of a tree

Let T be a tree, End(T) be the number of ends of T and let L(T) be the infi mum of topological entropies of transitive maps of T. We give an elementary approach to the estimate that L(T) greater than or equal to (1/End(T)) log 2. We also divide the set of all trees (up to homeomorphisms) into pairwise disjoint subsets P(i), i is an element of {0} boolean OR N and prove that L(T) = (1/(End(T) - i))log2 if T is an element of P(i) with i = 0, i, and L (T) less than or equal to (respectively =) (1/(End(T) - i))log2 if T is an element of P(i) (respectively T is an element of P'(i)) with i greater than or equal to 2, where P'(i) is an infinite subset of P(i). Furthermore, we show that there is a tree T such that the topological entropy of each trans itive map of T is larger than L(T), and hence disprove a conjecture of Alse da et al (1997).