Novel characteristics of split trees by use of renewal theory

We investigate characteristics of random split trees introduced by Devroye [SIAM J Comput 28, 409–432, 1998]; split trees include e.g., binary search trees, m-ary search trees, quadtrees, median of (2k + 1)-trees, simplex trees, tries and digital search trees. More precisely: We use renewal theory in the studies of split trees, and use this theory to prove several results about split trees. A split tree of cardinality n is constructed by distributing n balls (which often represent data) to a subset of nodes of an infinite tree. One of our main results is a relation between the deterministic number of balls n and the random number of nodes N . In [5] there is a central limit law for the depth of the last inserted ball so that most nodes are close to depth lnn μ + O( √ lnn), where μ is some constant depending on the type of split tree; we sharpen this result by finding an upper bound for the expected number of nodes with depths ≥ lnn μ − ln 1 2 + n or depths ≤ lnn μ + ln 1 2 + n for any choice of > 0. We also find the first asymptotic of the variances of the depths of the balls in the tree.