## Electronic Journal of Probability

### Novel characteristics of split trees by use of renewal theory

Cecilia Holmgren

#### Abstract

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 Devroye [SIAM J Comput 28, 409-432, 1998] there is a central limit law for the depth of the last inserted ball so that most nodes are close to depth $\ln n/\mu+O(\ln n)^{1/2})$, where $\mu$ 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 $\geq \mu^{-1}\ln n-(\ln n)^{1/2+\epsilon}$ or depths $\leq\mu^{-1}\ln n+(\ln n)^{1/2+\epsilon}$ for any choice of $\epsilon>0$. We also find the first asymptotic of the variances of the depths of the balls in the tree.

#### Article information

Source
Electron. J. Probab., Volume 17 (2012), paper no. 5, 27 pp.

Dates
Accepted: 16 January 2012
First available in Project Euclid: 4 June 2016

https://projecteuclid.org/euclid.ejp/1465062327

Digital Object Identifier
doi:10.1214/EJP.v17-1723

Mathematical Reviews number (MathSciNet)
MR2878784

Zentralblatt MATH identifier
1244.05058

Rights

#### Citation

Holmgren, Cecilia. Novel characteristics of split trees by use of renewal theory. Electron. J. Probab. 17 (2012), paper no. 5, 27 pp. doi:10.1214/EJP.v17-1723. https://projecteuclid.org/euclid.ejp/1465062327

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