The Annals of Probability

Continuum tree asymptotics of discrete fragmentations and applications to phylogenetic models

Bénédicte Haas, Grégory Miermont, Jim Pitman, and Matthias Winkel

Full-text: Open access


Given any regularly varying dislocation measure, we identify a natural self-similar fragmentation tree as scaling limit of discrete fragmentation trees with unit edge lengths. As an application, we obtain continuum random tree limits of Aldous’s beta-splitting models and Ford’s alpha models for phylogenetic trees. This confirms in a strong way that the whole trees grow at the same speed as the mean height of a randomly chosen leaf.

Article information

Ann. Probab. Volume 36, Number 5 (2008), 1790-1837.

First available in Project Euclid: 11 September 2008

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Markov branching model self-similar fragmentation continuum random tree ℝ-tree phylogenetic tree


Haas, Bénédicte; Miermont, Grégory; Pitman, Jim; Winkel, Matthias. Continuum tree asymptotics of discrete fragmentations and applications to phylogenetic models. Ann. Probab. 36 (2008), no. 5, 1790--1837. doi:10.1214/07-AOP377.

Export citation


  • [1] Aldous, D. (1996). Probability distributions on cladograms. In Random Discrete Structures (Minneapolis, MN, 1993). IMA Vol. Math. Appl. 76 1–18. Springer, New York.
  • [2] Aldous, D. J. (1991). The continuum random tree. I. Ann. Probab. 19 1–28.
  • [3] Aldous, D. J. (1991). The continuum random tree. II. An overview. In Stochastic Analysis (M. T. Barlow and N. M. Bingham, eds.) 23–70. Cambridge Univ. Press.
  • [4] Aldous, D. J. (1993). The continuum random tree. III. Ann. Probab. 21 248–289.
  • [5] Aldous, D. J. (2001). Stochastic models and descriptive statistics for phylogenetic trees, from Yule to today. Statist. Sci. 16 23–34.
  • [6] Barlow, M., Pitman, J. and Yor, M. (1989). Une extension multidimensionnelle de la loi de l’arc sinus. Séminaire de Probabilités XXIII. Lecture Notes in Math. 1372 294–314. Springer, Berlin.
  • [7] Berestycki, J. (2002). Ranked fragmentations. ESAIM Probab. Statist. 6 157–175 (electronic).
  • [8] Bertoin, J. (2001). Homogeneous fragmentation processes. Probab. Theory Related Fields 121 301–318.
  • [9] Bertoin, J. (2002). Self-similar fragmentations. Ann. Inst. H. Poincaré Probab. Statist. 38 319–340.
  • [10] Bertoin, J. (2006). Random Fragmentation and Coagulation Processes. Cambridge Univ. Press.
  • [11] Bingham, N. H., Goldie, C. M. andTeugels, J. L. (1989). Regular Variation. Cambridge Univ. Press.
  • [12] Devroye, L. (2002). Laws of large numbers and tail inequalities for random tries and PATRICIA trees. J. Comput. Appl. Math. 142 27–37.
  • [13] Dong, R. (2007). Partition structures derived from Markovian coalescents with simultaneous multiple collisions. arXiv:0707.1606.
  • [14] Dong, R., Gnedin, A. and Pitman, J. (2007). Exchangeable partitions derived from Markovian coalescents. Ann. Appl. Probab. 17 1172–1201.
  • [15] Dudley, R. M. (2002). Real Analysis and Probability. Cambridge Univ. Press.
  • [16] Duquesne, T. (2003). A limit theorem for the contour process of conditioned Galton–Watson trees. Ann. Probab. 31 996–1027.
  • [17] Duquesne, T. and Le Gall, J.-F. (2002). Random trees, Lévy processes and spatial branching processes. Astérisque 281 vi+147.
  • [18] Evans, S. (2000). Snakes and spiders: Brownian motion on real-trees. Probab. Theory Related Fields 117 361–386.
  • [19] Evans, S. N., Pitman, J. and Winter, A. (2006). Rayleigh processes, real trees, and root growth with re-grafting. Probab. Theory Related Fields 134 81–126.
  • [20] Evans, S. N. and Winter, A. (2006). Subtree prune and regraft: A reversible real tree-valued Markov process. Ann. Probab. 34 918–961.
  • [21] Ford, D. J. (2005). Probabilities on cladograms: Introduction to the alpha model. Preprint. arXiv:math.PR/0511246.
  • [22] Ford, D. J. (2006). Phylogenetic trees and the alpha model. Ph.D. thesis in preparation.
  • [23] Geiger, J. and Kauffmann, L. (2004). The shape of large Galton–Watson trees with possibly infinite variance. Random Structures Algorithms 25 311–335.
  • [24] Gnedin, A. and Pitman, J. (2004/06). Regenerative partition structures. Electron. J. Combin. 11 Research Paper 12, 21 pp. (electronic).
  • [25] Gnedin, A. and Pitman, J. (2005). Regenerative composition structures. Ann. Probab. 33 445–479.
  • [26] Gnedin, A., Pitman, J. and Yor, M. (2006). Asymptotic laws for compositions derived from transformed subordinators. Ann. Probab. 34 468–492.
  • [27] Greven, A., Pfaffelhuber, P. and Winter, A. (2006). Convergence in distribution of random metric measure spaces (Λ-coalescent measure trees). Preprint. arXiv:math.PR/0609801.
  • [28] Haas, B. (2003). Loss of mass in deterministic and random fragmentations. Stochastic Process. Appl. 106 245–277.
  • [29] Haas, B. and Miermont, G. (2004). The genealogy of self-similar fragmentations with negative index as a continuum random tree. Electron. J. Probab. 9 57–97 (electronic).
  • [30] James, L. F., Lijoy, A. and Pruenster, I. (2008). Distributions of linear functionals of the two parameter Poisson–Dirichlet random measures. Ann. Appl. Probab. 18 521–551.
  • [31] Kingman, J. F. C. (1978). The representation of partition structures. J. London Math. Soc. (2) 18 374–380.
  • [32] Marchal, P. (2008). A note on the fragmentation of a stable tree. Manuscript in preparation.
  • [33] Miermont, G. (2003). Self-similar fragmentations derived from the stable tree. I. Splitting at heights. Probab. Theory Related Fields 127 423–454.
  • [34] Miermont, G. (2005). Self-similar fragmentations derived from the stable tree. II. Splitting at nodes. Probab. Theory Related Fields 131 341–375.
  • [35] Perman, M., Pitman, J. and Yor, M. (1992). Size-biased sampling of Poisson point processes and excursions. Probab. Theory Related Fields 92 21–39.
  • [36] Pitman, J. (2006). Combinatorial Stochastic Processes. Springer, Berlin.