Optimal subtrees of finite metric graphs

We consider a metric graph analog of a classical extremal problem for Riemann surfaces studied by Teichmüller, Jenkins, Strebel, and others. In particular, we prove that on any finite metric graph with a marked point, there is a unique open subtree which maximizes a metric graph analog of the reduced modulus, and we also give an equivalent characterization of this optimal subtree in terms of balanced harmonic measures.