Tries are data structures for storing sets where each element is represented by a key that can be viewed as a string of characters over a finite alphabet. These structures have been extensively studied and analyzed under several probability models. All of these models, however, preclude the occurrence of sets in which the key of one element is a prefix of that of anotherαsuch a key is called a prefixing-key. This paper presents an average case analysis of several trie varieties, which we generically called prefixing-tries, for representing sets with 'unrestricted' keys, that is, sets in which the key of one element may be a prefix of that of another. The underlying probability model, which we call the prefix model, P P h, n, m assumes as equally likely all n-element sets whose keys are composed of at most h characters from a fixed alphabet of size m. For each of the trie varieties analyzed, we derive exact formulas for the expected space required to store such a set, and the average time required to retrieve an element given its key, as functions of h, n, and m. Our approach to the analysis is of interest in its own right. It provides a unifying framework for computing the expectations of a wide class of random variables with respect to the prefix model. This class includes the cost functions of the trie varieties analyzed here.