## Electronic Journal of Probability

### Asymptotic distribution of two-protected nodes in ternary search trees

#### Abstract

We study protected nodes in $m-$ary search trees, by putting them in context of generalized Pólya urns. We show that the number of two-protected nodes (the nodes that are neither leaves nor parents of leaves) in a random ternary search tree is asymptotically normal. The methods apply in principle to $m-$ary search trees with larger $m$ as well, although the size of the matrices used in the calculations grow rapidly with $m$; we conjecture that the method yields an asymptotically normal distribution for all $m \leq 26$.

The one-protected nodes, and their complement, i.e., the leaves, are easier to analyze. By using a simpler urn (that is similar to the one that has earlier been used to study the total number of nodes in $m-$ary search trees), we prove normal limit laws for the number of one-protected nodes and the number of leaves for all $m \leq 26$.

#### Article information

Source
Electron. J. Probab., Volume 20 (2015), paper no. 9, 20 pp.

Dates
Accepted: 5 February 2015
First available in Project Euclid: 4 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465067115

Digital Object Identifier
doi:10.1214/EJP.v20-3577

Mathematical Reviews number (MathSciNet)
MR3311222

Zentralblatt MATH identifier
1327.60032

Rights

#### Citation

Holmgren, Cecilia; Janson, Svante. Asymptotic distribution of two-protected nodes in ternary search trees. Electron. J. Probab. 20 (2015), paper no. 9, 20 pp. doi:10.1214/EJP.v20-3577. https://projecteuclid.org/euclid.ejp/1465067115

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