On the Wiener polarity index of trees with maximum degree or given number of leaves

The Wiener polarity index W"P(G) of a graph G=(V,E) is the number of unordered pairs of vertices {u,v} of G such that the distance d"G(u,v)=3. In this paper, the minimum (resp. maximum) Wiener polarity index of trees with n vertices and maximum degree @D are given, and the corresponding extremal trees are determined, where [email protected][email protected]@?n-1. Moreover, the trees minimizing W"P(T) among all trees T of order n and k leaves are characterized, where [email protected][email protected]?n-1.