Electronic Journal of Probability
- Electron. J. Probab.
- Volume 22 (2017), paper no. 21, 25 pp.
Asymptotics of heights in random trees constructed by aggregation
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 $\ell \in \mathbb N$.
Electron. J. Probab., Volume 22 (2017), paper no. 21, 25 pp.
Received: 23 June 2016
Accepted: 26 January 2017
First available in Project Euclid: 21 February 2017
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 05C05: Trees 60J05: Discrete-time Markov processes on general state spaces
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