Let~$T_n$ denote the set of unrooted labeled trees of size~$n$ and let~$T_n$ be a particular (finite, unlabeled) tree. Assuming that every tree of~$T_n$ is equally likely, it is shown that the limiting distribution as $n$~goes to infinity of the number of occurrences of~$M$ as an induced subtree is asymptotically normal with mean value and variance asymptotically equivalent to~$\mu n$ and~$\sigma^2n$, respectively, where the constants $\mu>0$ and~$\sigma\ge 0$ are computable.