A Topological Characterization of End Sets of a Twinning of a Tree

We give two topological characterizations of which subsets of ends of a tree T can be the twinned ends of a twinning of T. We then use this characterization to prove that in a semi-homogeneous tree with countable degrees, a subset S of ends of T can be the set of twinned ends if and only if S is countable and dense in the topology on the set of ends of T.