On the variance of the external path length in a symmetric digital trie

Digital searching is a well-known technique for storing and retrieving information using lexicographical (digital) structure of words. A radix trie (in short: trie) is such a digital search tree that edges are labelled by elements from aa alphabet (e.g., binary alphabet consisting of O’s and l’s) and leaves (external nodes) contain keys [ 1,4,9]. More precisely, in a binary case a key is a (possible infinite) sequence of O’s and l’s, where 0 means “go left” and 1 means “go right”. The keys are stored in external nodes and the access path from the root to a leaf is the minimal prefix information contained in an external node (see Fig. I for an example of a trie). There are a number of applications of tries in computer science and telecommunications, e.g., dynamic hashing, radix exchange sort [4,9], partial match retrieval of muhidimensional data, lexicographical sorting (111, tree-type conflict resolution algorithm for broadcast communications [lo, 141, etc. Two quantities of a digital trie are of special interest: the depth of a leaf and the external path length. The average depth of a leaf has been studied in [3,9], the variance in [6] (binary case) and 1141 (general tries) and higher moments of the depth in [14]. The average value of the external path length is closely related to the average depth of a leaf, but not the variance. The variance of the external path length was never determined up to now, although the external path length finds important applications in practice, e.g., for modified lexicographical sorting [I 11 and for conflict