## Electronic Journal of Probability

### Asymptotics of heights in random trees constructed by aggregation

Bénédicte Haas

#### Abstract

To each sequence $(a_n)$ of positive real numbers we associate a growing sequence $(T_n)$ of continuous trees built recursively by gluing at step $n$ a segment of length $a_n$ on a uniform point of the pre–existing tree, starting from a segment $T_1$ of length $a_1$. Previous works [5, 10] on that model focus on the influence of $(a_n)$ on the compactness and Hausdorff dimension of the limiting tree. Here we consider the cases where the sequence $(a_n)$ is regularly varying with a non–negative index, so that the sequence $(T_n)$ explodes. We determine the asymptotics of the height of $T_n$ and of the subtrees of $T_n$ spanned by the root and $\ell$ points picked uniformly at random and independently in $T_n$, for all $\ell \in \mathbb N$.

#### Article information

Source
Electron. J. Probab., Volume 22 (2017), paper no. 21, 25 pp.

Dates
Accepted: 26 January 2017
First available in Project Euclid: 21 February 2017

https://projecteuclid.org/euclid.ejp/1487646307

Digital Object Identifier
doi:10.1214/17-EJP31

Mathematical Reviews number (MathSciNet)
MR3622891

Zentralblatt MATH identifier
1358.05054

#### Citation

Haas, Bénédicte. Asymptotics of heights in random trees constructed by aggregation. Electron. J. Probab. 22 (2017), paper no. 21, 25 pp. doi:10.1214/17-EJP31. https://projecteuclid.org/euclid.ejp/1487646307

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