Entropy estimates for transitive maps on trees

If X is a space, define L(X) to the the infimum of all possible values h( f ), where h( f ) denotes the topological entropy of f, and f ranges over all transitive functions on X. Various lower and upper bounds are given for L(T) in the case for which T is a tree, which provides an exact value of L(T) for a large class of trees. We also examine the analogous problem when f is required to fix the endpoints of the tree.