## Electronic Journal of Probability

### Limit laws for functions of fringe trees for binary search trees and random recursive trees

#### Abstract

We prove general limit theorems for sums of functions of subtrees of (random) binary search trees and random recursive trees. The proofs use a new version of a representation by Devroye, and Stein's method for both normal and Poisson approximation together with certain couplings.

As a consequence, we give simple new proofs of the fact that the number of fringe trees of size $k=k_n$ in the binary search tree or in the random recursive tree  (of total size $n$) has an asymptotical Poisson distribution if $k\rightarrow\infty$, and that the distribution is asymptotically normal for $k=o(\sqrt{n})$. Furthermore, we prove similar results for the number of subtrees of size $k$ with some required property $P$, e.g., the number of copies of a certain fixed subtree $T$. Using the Cramér-Wold device, we show also that these random numbers for different fixed subtrees converge jointly to a multivariate normal distribution. <br /><br />We complete the paper by giving examples of applications of the general results, e.g., we obtain a normal limit law for the number of $\ell$-protected nodes in a binary search tree or in a random recursive tree.

#### Article information

Source
Electron. J. Probab., Volume 20 (2015), paper no. 4, 51 pp.

Dates
Accepted: 13 January 2015
First available in Project Euclid: 4 June 2016

https://projecteuclid.org/euclid.ejp/1465067110

Digital Object Identifier
doi:10.1214/EJP.v20-3627

Mathematical Reviews number (MathSciNet)
MR3311217

Zentralblatt MATH identifier
1320.60026

Subjects
Primary: 60C05: Combinatorial probability
Secondary: 05C05: Trees 60F05: Central limit and other weak theorems

Rights

#### Citation

Holmgren, Cecilia; Janson, Svante. Limit laws for functions of fringe trees for binary search trees and random recursive trees. Electron. J. Probab. 20 (2015), paper no. 4, 51 pp. doi:10.1214/EJP.v20-3627. https://projecteuclid.org/euclid.ejp/1465067110

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