  We give general theorems on asymptotic normality for additive functionals of
random tries generated by a sequence of independent strings. These theorems are
applied to show asymptotic normality of the distribution of random fringe trees
in a random trie. Formulas for asymptotic mean and variance are given. In
particular, the proportion of fringe trees of size $k$ (defined as number of
keys) is asymptotically, ignoring oscillations, $c/(k(k-1))$ for $k\ge2$, where
$c=1/(1+H)$ with $H$ the entropy of the digits. Another application gives
asymptotic normality of the number of $k$-protected nodes in a random trie. For
symmetric tries, it is shown that the asymptotic proportion of $k$-protected
nodes (ignoring oscillations) decreases geometrically as $k\to\infty$.
