  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 of
the number of occurrences of the double-star $(i,j)$ in $\mathcal{T}_n$ is
normal. Based on this result, we obtain the asymptotic value of Randi\'c index
for trees. Fajtlowicz conjectured that for any connected graph the Randi\'c
index is at least the 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.
