  We study fringe subtrees of random $ m $-ary search trees and of preferential
attachment trees, by putting them in the context of generalised P\'olya urns.
In particular we show that for the random $ m $-ary search trees with $ m\leq
26 $ and for the linear preferential attachment trees, the number of fringe
subtrees that are isomorphic to an arbitrary fixed tree $ T $ converges to a
normal distribution; more generally, we also prove multivariate normal
distribution results for random vectors of such numbers for different fringe
subtrees. Furthermore, we show that the number of protected nodes in random
$m$-ary search trees for $ m\leq 26 $ has asymptotically a normal distribution.

Probability