  The Wiener index W(G) of a simple connected graph G is defined as the sum of
distances over all pairs of vertices in a graph. We denote by W[T_{n}] the set
of all values of Wiener index for a graph from class T_{n} of trees on n
vertices. The largest interval of contiguous integers (contiguous even integers
in case of odd n) is denoted by W^{int}[T_{n}]. In this paper we prove that
both sets are of the cardinality (1/6)n^3+O(n^2) in the case of even n, while
in the case of odd n we prove that the cardinality of both sets equals
(1/(12))n^3+O(n^2) solving thus two conjectures posed in literature.
