## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 1, Number 3 (1991), 406-418.

### On the Distribution of Leaves in Rooted Subtrees of Recursive Trees

Hosam M. Mahmoud and R. T. Smythe

#### Abstract

We study the structure of $T^{(k)}_n$, the subtree rooted at $k$ in a random recursive tree of order $n$, under the assumption that $k$ is fixed and $n \rightarrow \infty$. Employing generalized Polya urn models, exact and limiting distributions are derived for the size, the number of leaves and the number of internal nodes of $T^{(k)}_n$. The exact distributions are given by intricate formulas involving Eulerian numbers, but a recursive argument based on the urn model suffices for establishing the first two moments of the above-mentioned random variables. Known results show that the limiting distribution of the size of $T^{(k)}_n$, normalized by dividing by $n$ is $\operatorname{Beta}(1, k - 1)$. A martingale central limit argument is used to show that the difference between the number of leaves and the number of internal nodes of $T^{(k)}_n$, suitably normalized, converges to a mixture of normals with a $\operatorname{Beta}(1, k - 1)$ as the mixing density. The last result allows an easy determination of limiting distributions of suitably normalized versions of the number of leaves and the number of internal nodes of $T^{(k)}_n$.

#### Article information

**Source**

Ann. Appl. Probab., Volume 1, Number 3 (1991), 406-418.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1177005874

**Digital Object Identifier**

doi:10.1214/aoap/1177005874

**Mathematical Reviews number (MathSciNet)**

MR1111525

**Zentralblatt MATH identifier**

0738.05034

**JSTOR**

links.jstor.org

**Subjects**

Primary: 05C05: Trees

Secondary: 60G42: Martingales with discrete parameter 68E05

**Keywords**

Recursive trees rooted subtrees generalized Polya urn models martingale central limit theorem

#### Citation

Mahmoud, Hosam M.; Smythe, R. T. On the Distribution of Leaves in Rooted Subtrees of Recursive Trees. Ann. Appl. Probab. 1 (1991), no. 3, 406--418. doi:10.1214/aoap/1177005874. https://projecteuclid.org/euclid.aoap/1177005874