Balance Vertices in Trees

A new notion of balanced bipartitions of the vertices in a tree T is introduced and studied. It gives rise to a new central set of vertices in T, each of which can be considered to be a discrete version of the center of gravity of T. We seek vertices x, called balance vertices, such that the two sums of the distances from x to all the vertices in each of two subtrees of T are as equal as possible, where the two subtrees have only x in common, but, together, contain all the vertices of T. We discuss some of the computation involved in a first step in determining the balance vertices. We prove that the median vertices, the center vertices, and the balance vertices may be arbitrarily far apart. We also prove that the set of balance vertices of a tree T consists of a single vertex or two adjacent vertices; the proof involves a new type of "double orientation" of the edges of T. © 1999 John Wiley & Sons, Inc. Networks 34: 264-271, 1999