  We investigate \Delta_n, the distance between randomly selected pairs of
nodes among n keys in a random trie, which is a kind of digital tree.
Analytical techniques, such as the Mellin transform and an excursion between
poissonization and depoissonization, capture small fluctuations in the mean and
variance of these random distances. The mean increases logarithmically in the
number of keys, but curiously enough the variance remains O(1), as n\to\infty.
It is demonstrated that the centered random variable
\Delta_n^*=\Delta_n-\lfloor2\log_2n\rfloor does not have a limit distribution,
but rather oscillates between two distributions.
