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May 2024 A recursive distributional equation for the stable tree
Nicholas Chee, Franz Rembart, Matthias Winkel
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Bernoulli 30(2): 1029-1054 (May 2024). DOI: 10.3150/23-BEJ1623


We provide a new characterisation of Duquesne and Le Gall’s α-stable tree, α(1,2], as the solution of a recursive distributional equation (RDE) of the form T=dgξ,Ti,i0, where g is a concatenation operator, ξ=ξi,i0 a sequence of scaling factors, Ti, i0, and T are i.i.d. trees independent of ξ. This generalises the characterisation of the Brownian Continuum Random Tree proved by Albenque and Goldschmidt, based on self-similarity observed by Aldous. By relating to previous results on a rather different class of RDE, we explore the present RDE and obtain for a large class of similar RDEs that the fixpoint is unique (up to multiplication by a constant) and attractive.


We’d like to thank the referee for valuable suggestions that improved the legibility of the paper.


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Nicholas Chee. Franz Rembart. Matthias Winkel. "A recursive distributional equation for the stable tree." Bernoulli 30 (2) 1029 - 1054, May 2024.


Received: 1 July 2022; Published: May 2024
First available in Project Euclid: 31 January 2024

MathSciNet: MR4699544
Digital Object Identifier: 10.3150/23-BEJ1623

Keywords: Gromov–Hausdorff distance , recursive distributional equation , R-tree , stable tree

Rights: This research was funded, in whole or in part, by [UKRI]. A CC BY 4.0 license is applied to this article arising from this submission, in accordance with the grant’s open access conditions

Vol.30 • No. 2 • May 2024
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