  We prove limit theorems for sums of functions of subtrees of binary search
trees and random recursive trees. In particular, we give simple new proofs of
the fact that the number of fringe trees of size $ k=k_n $ in the binary search
tree and the random recursive tree (of total size $ n $) asymptotically has a
Poisson distribution if $ k\rightarrow\infty $, and that the distribution is
asymptotically normal for $ k=o(\sqrt{n}) $. Furthermore, we prove similar
results for the number of subtrees of size $ k $ with some required property $
P $, for example the number of copies of a certain fixed subtree $ T $. Using
the Cram\'er-Wold device, we show also that these random numbers for different
fixed subtrees converge jointly to a multivariate normal distribution. As an
application of the general results, we obtain a normal limit law for the number
of $\ell$-protected nodes in a binary search tree or random recursive tree.
  The proofs use a new version of a representation by Devroye, and Stein's
method (for both normal and Poisson approximation) together with certain
couplings.
