Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Scaling limits of $k$-ary growing trees

Bénédicte Haas and Robin Stephenson

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For each integer $k\geq2$, we introduce a sequence of $k$-ary discrete trees constructed recursively by choosing at each step an edge uniformly among the present edges and grafting on “its middle” $k-1$ new edges. When $k=2$, this corresponds to a well-known algorithm which was first introduced by Rémy. Our main result concerns the asymptotic behavior of these trees as the number of steps $n$ of the algorithm becomes large: for all $k$, the sequence of $k$-ary trees grows at speed $n^{1/k}$ towards a $k$-ary random real tree that belongs to the family of self-similar fragmentation trees. This convergence is proved with respect to the Gromov–Hausdorff–Prokhorov topology. We also study embeddings of the limiting trees when $k$ varies.


Pour chaque entier $k\geq2$, on introduit une suite d’arbres discrets $k$-aires construite récursivement en choisissant à chaque étape une arête uniformément parmi les arêtes de l’arbre pré-existant et greffant sur son « milieu » $k-1$ nouvelles arêtes. Lorsque $k=2$, cette procédure correspond à un algorithme introduit par Rémy. Pour chaque entier $k\geq2$, nous décrivons la limite d’échelle de ces arbres lorsque le nombre d’étapes $n$ tend vers l’infini : ils grandissent à la vitesse $n^{1/k}$ vers un arbre réel aléatoire $k$-aire qui appartient à la famille des arbres de fragmentation auto-similaires. Cette convergence a lieu en probabilité, pour la topologie de Gromov–Hausdorff–Prokhorov. Nous étudions également l’emboîtement des arbres limites quand $k$ varie.

Article information

Ann. Inst. H. Poincaré Probab. Statist. Volume 51, Number 4 (2015), 1314-1341.

Received: 5 February 2014
Revised: 23 April 2014
Accepted: 24 April 2014
First available in Project Euclid: 21 October 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Random growing trees Scaling limits Self-similar fragmentation trees Gromov–Hausdorff–Prokhorov topology


Haas, Bénédicte; Stephenson, Robin. Scaling limits of $k$-ary growing trees. Ann. Inst. H. Poincaré Probab. Statist. 51 (2015), no. 4, 1314--1341. doi:10.1214/14-AIHP622.

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