a b s t r a c tThe 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 ∆ are given, and the corresponding extremal trees are determined, where 2 ≤ ∆ ≤ n − 1. Moreover, the trees minimizing W P (T ) among all trees T of order n and k leaves are characterized, where 2 ≤ k ≤ n − 1.
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