The closeness or the distance of a vertex u in a graph G, denoted by δ G(u), is the sum of distances between u and all other vertices of G . The Wiener dimension of a connected graph is defined as the number of different distances of its vertices. In this paper we prove that any tree has Wiener dimension 2 if and only if it is isomorphic to a star graph or a bi-star graph. We also identify certain classes of trees with Wiener dimension 3 and 4.