Some remarks on inverse Wiener index problem

In 1995, Gutman and Yeh (1995) [3] conjectured that for every large enough integer ww there exists a tree with Wiener index equal to ww. The conjecture has been solved by Wang and Yu (2006) [7] and independently by Wagner (2006) [6]. We present an algorithm with a constant number of operations to construct a tree with a given Wiener index. Moreover, we show that there exist 2Ω(w4) non-isomorphic trees with Wiener index ww.