The asymptotic value of the Randić index for trees

Let Tn denote the set of all unrooted and unlabeled trees with n vertices, and (i,j) a double-star. By assuming that every tree of Tn is equally likely, we show that the limiting distribution of the number of occurrences of the double-star (i,j) in Tn is normal. Based on this result, we obtain the asymptotic value of the Randić index for trees. Fajtlowicz conjectured that for any connected graph G the Randić index of G is at least its average distance. Using this asymptotic value, we show that this conjecture is true not only for almost all connected graphs but also for almost all trees.