Variation of the Wiener index under tree transformations

The Wiener index W(T) is defined as the sum of distances between all pairs of vertices of the tree T. In this paper we find the variation of the Wiener index under certain tree transformations, which can be described in terms of coalescence of trees. As a consequence, conditions for nonisomorphic trees having equal Wiener index are presented. Also, a partial order on the collection of trees (with a fixed number of vertices) is introduced, providing structural information about the behavior of W.