  Tries are general purpose data structures for information retrieval. The most
significant parameter of a trie is its height $H$ which equals the length of
the longest common prefix of any two string in the set $A$ over which the trie
is built. Analytical investigations of random tries suggest that ${\bf E}(H)\in
O(\log(\|A\|))$, although $H$ is unbounded in the worst case. Moreover, sharp
results on the distribution function of $H$ are known for many different random
string sources. But because of the inherent weakness of the modeling behind
average-case analysis---analyses being dominated by random data---these results
can utterly explain the fact that in many practical situations the trie height
is logarithmic. We propose a new semi-random string model and perform a
smoothed analysis in order to give a mathematically more rigorous explanation
for the practical findings. The perturbation functions which we consider are
based on probabilistic finite automata (PFA) and we show that the transition
probabilities of the representing PFA completely characterize the asymptotic
growth of the smoothed trie height. Our main result is of dichotomous
nature---logarithmic or unbounded---and is certainly not surprising at first
glance, but we also give quantitative upper and lower bounds, which are derived
using multivariate generating function in order to express the computations of
the perturbing PFA. A direct consequence is the logarithmic trie height for
edit perturbations(i.e., random insertions, deletions and substitutions).
