Electronic Journal of Probability

Cost functionals for large (uniform and simply generated) random trees

Jean-François Delmas, Jean-Stéphane Dhersin, and Marion Sciauveau

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Abstract

Additive tree functionals allow to represent the cost of many divide-and-conquer algorithms. We give an invariance principle for such tree functionals for the Catalan model (random tree uniformly distributed among the full binary ordered trees with given number of nodes) and for simply generated trees (including random tree uniformly distributed among the ordered trees with given number of nodes). In the Catalan model, this relies on the natural embedding of binary trees into the Brownian excursion and then on elementary $ L^2$ computations. We recover results first given by Fill and Kapur (2004) and then by Fill and Janson (2009). In the simply generated case, we use convergence of conditioned Galton-Watson trees towards stable Lévy trees, which provides less precise results but leads us to conjecture a different phase transition value between “global” and “local” regimes. We also recover results first given by Janson (2003 and 2016) in the Brownian case and give a generalization to the stable case.

Article information

Source
Electron. J. Probab., Volume 23 (2018), paper no. 87, 36 pp.

Dates
Received: 21 September 2017
Accepted: 15 August 2018
First available in Project Euclid: 12 September 2018

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1536717746

Digital Object Identifier
doi:10.1214/18-EJP213

Mathematical Reviews number (MathSciNet)
MR3858915

Zentralblatt MATH identifier
1398.05061

Subjects
Primary: 05C05: Trees 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60F17: Functional limit theorems; invariance principles

Keywords
random binary tree cost functional toll function Brownian excursion continuum random tree

Rights
Creative Commons Attribution 4.0 International License.

Citation

Delmas, Jean-François; Dhersin, Jean-Stéphane; Sciauveau, Marion. Cost functionals for large (uniform and simply generated) random trees. Electron. J. Probab. 23 (2018), paper no. 87, 36 pp. doi:10.1214/18-EJP213. https://projecteuclid.org/euclid.ejp/1536717746


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