The Wiener Index of Random Digital Trees

Abstract

The Wiener index has been studied for simply generated random trees, nonplane unlabeled random trees, and a huge subclass of random grid trees containing random binary search trees, random median-of-(2k+1) search trees, random $m$-ary search trees, random quadtrees, random simplex trees, etc. An important class of random grid trees for which the Wiener index was not studied so far is random digital trees. In this work, we close this gap. More precisely, we derive asymptotic expansions of moments of the Wiener index and show that a central limit law for the Wiener index holds. These results are obtained for digital search trees and bucket versions as well as tries and PATRICIA tries. Our findings answer in the affirmative two questions posed by Neininger.