Limit Laws for Heights in Generalized Tries and PATRICIA Tries

We consider digital trees such as (generalized) tries and PATRICIA tries, built from n random strings generated by an unbiased memoryless source (i.e., all symbols are equally likely). We study limit laws of the height which is defined as the longest path in such trees. It turns out that this height also represents the number of random questions required to recognize n distinct objects. We shall identify three natural regions of the height distributions. For tries, in the region where most of the probability mass is concentrated, the asymptotic distribution is of extreme value type (i.e., double exponential distribution). Surprisingly enough, the height of the PATRICIA trie behaves quite differently in this region: It exhibits an exponential of a Gaussian distribution (with an oscillating term) around the most probable value k 1 = blog 2 n + p 2 log 2 n \Gamma 3 2 c+1. In fact, the asymptotic distribution of PATRICIA height concentrates on one or two points. For most n all the mass..