## The Annals of Applied Probability

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

#### 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

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