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2018 Cost functionals for large (uniform and simply generated) random trees
Jean-François Delmas, Jean-Stéphane Dhersin, Marion Sciauveau
Electron. J. Probab. 23: 1-36 (2018). DOI: 10.1214/18-EJP213

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.

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Jean-François Delmas. Jean-Stéphane Dhersin. Marion Sciauveau. "Cost functionals for large (uniform and simply generated) random trees." Electron. J. Probab. 23 1 - 36, 2018. https://doi.org/10.1214/18-EJP213

Information

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

zbMATH: 1398.05061
MathSciNet: MR3858915
Digital Object Identifier: 10.1214/18-EJP213

Subjects:
Primary: 05C05 , 60F17 , 60J80

Keywords: Brownian excursion , Continuum random tree , cost functional , Random binary tree , toll function

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