Average profiles, from tries to suffix-trees

We build upon previous work of Fayolle (2004) and Park and Szpankowski (2005) to study asymptotically the average internal profile of tries and of suffix-trees. The binary keys and the strings are built from a Bernoulli source (p, q). We consider the average number pk,P(ν) of internal nodes at depth k of a trie whose number of input keys follows a Poisson law of parameter ν. The Mellin transform of the corresponding bivariate generating function has a major singularity at the origin, which implies a phase reversal for the saturation rate pk,P(ν)/2 as k reaches the value 2 log(ν)/(log(1/p) + log(1/q)). We prove that the asymptotic average profiles of random tries and suffix-trees are mostly similar, up to second order terms, a fact that has been experimentally observed in Nicodème (2003); the proof follows from comparisons to the profile of tries in the Poisson model.