Digital trees or tries are a general purpose flexible data structure that implements dictionaries built on words. The present paper is focussedon the average-case analysis ofan important parameter of thistree-structure, i.e., the stack-size. The stack-size of a tree is the memory needed by a storage-optimal preorder traversal. The analysis is carried outunder a general model in which words are produced by a source (in the information-theoretic sense)that emits symbols.Under some natural assumptions that encompass all commonlyused data models (and more), we obtain a precise average-case and probabilistic analysis of stack-size. Furthermore, we studythe dependency between the stack-size and the ordering on symbols inthe alphabet:we establish that, when the source emits independent symbols, theoptimal ordering arises when the most probable symbol is the last one in this order.

It is shown by means of several examples that probability metrics are a useful tool to study the asymptotic behaviour of (stochastic) recursive algorithms. The basic idea of this approach is to find a ‘suitable' probability metric which yields contraction properties of the transformations describing the limits of the algorithm. In order to demonstrate the wide range of applicability of this contraction method we investigate examples from various fields, some of which have already been analysed in the literature.