The average wiener index of trees and chemical trees

Meir and Moon (J. Combin. Theory 1970, 8, 99-103) reported a combinatorial formula for the average value of the distance between a pair of vertices in the class of all labeled trees with a fixed number (= n) of vertices. From this result an expression for the average Wiener index [W-n](lab) of label ed n-vertex trees follows immediately. We show that both the average Wiener index [W-n] of nonlabeled n-vertex trees and the average Wiener index [W-n ](ch) of nonlabeled n-vertex chemical trees having n less than or equal to 20 vertices are proportional to [W-n](lab), with proportionality constants around 0.927 and 0.990, respectively. Analogous results are obtained for th e Hosoya polynomial.