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Let $\mathcal{T}_n$ denote the set of all unrooted and unlabeled trees with$n$ vertices, and $(i,j)$ a double-star. By assuming that every tree of$\mathcal{T}_n$ is equally likely, we show that the limiting distribution ofthe number of occurrences of the double-star $(i,j)$ in $\mathcal{T}_n$ isnormal. Based on this result, we obtain the asymptotic value of Randi\'c indexfor trees. Fajtlowicz conjectured that for any connected graph the Randi\'cindex is at least the average distance. Using this asymptotic value, we showthat this conjecture is true not only for almost all connected graphs but alsofor almost all trees.

The asymptotic value of Randic index for trees