A tree is locally finite if the
interval between any two points is finite. A local isomorphism of a tree with
itself is a homomorphism which is an isomorphism when restricted to any
interval. Two theorems are proved. One characterizes those locally finite
trees which have transitive automorphism groups, and those which have
transitive local-isomorphism monoids. The other theorem gives necessary and
sufficient conditions for a non-injective transformation to be centralized
by a transitive permutation group, and necessary and sufficient conditions
for it to be centralized by a transitive transformation semigroup. Also, an
example is given of a nonlocally-finite tree with transitive automorphism
group.
