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2017 Asymptotics of heights in random trees constructed by aggregation
Bénédicte Haas
Electron. J. Probab. 22: 1-25 (2017). DOI: 10.1214/17-EJP31

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 $\ell \in \mathbb N$.

Citation

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Bénédicte Haas. "Asymptotics of heights in random trees constructed by aggregation." Electron. J. Probab. 22 1 - 25, 2017. https://doi.org/10.1214/17-EJP31

Information

Received: 23 June 2016; Accepted: 26 January 2017; Published: 2017
First available in Project Euclid: 21 February 2017

zbMATH: 1358.05054
MathSciNet: MR3622891
Digital Object Identifier: 10.1214/17-EJP31

Subjects:
Primary: 05C05 , 60J05

Keywords: line–breaking , Random trees , uniform recursive trees

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