Balance in trees

We show that the set of vertices in a tree T of smallest weight balance is the (branch weight) centroid. We characterize the set of edges in T with smallest weight-edge difference in terms of the vertices of smallest weight balance. A similar characterization is obtained for the set of edges in T with smallest distance-edge difference in terms of the vertices of smallest distance balance. This yields a new proof that the set of vertices in T of smallest distance balance consists of a single vertex or two adjacent vertices (possibly disjoint from the center and the branch weight centroid).