A Note on the Asymptotic Behavior of the Depth of Tries

Abstract.

We analyze the depth distribution of digital trees called tries. Assuming that the trie is constructed from n statistically independent binary strings, we compute the probability that the depth is equal to k . We study this probability asymptotically, for n and / or k large. We obtain detailed results for $n \to \infty$ and various ranges of k . This supplements previous work, which mostly involves computing the limiting distribution as $n \to \infty$ . Our analysis also gives an accurate description of the tails of the probability distribution. If the symbols in the string are zeros and ones, we assume they occur independently with respective probabilities q and p=1-q . We study the symmetric model (p=q= $\frac12$ ), the nonsymmetric model $(p \ne q)$ , and the ``nearly symmetric'' model. In the latter we have $n \to \infty$ and simultaneously p-q $\to$ 0. Here we obtain a new limiting probability distribution, that interpolates the well-known extreme value (p=q) and normal $(p \ne q)$ distributions.