The Annals of Probability

Regenerative tree growth: Binary self-similar continuum random trees and Poisson–Dirichlet compositions

Jim Pitman and Matthias Winkel

Full-text: Open access


We use a natural ordered extension of the Chinese Restaurant Process to grow a two-parameter family of binary self-similar continuum fragmentation trees. We provide an explicit embedding of Ford’s sequence of alpha model trees in the continuum tree which we identified in a previous article as a distributional scaling limit of Ford’s trees. In general, the Markov branching trees induced by the two-parameter growth rule are not sampling consistent, so the existence of compact limiting trees cannot be deduced from previous work on the sampling consistent case. We develop here a new approach to establish such limits, based on regenerative interval partitions and the urn-model description of sampling from Dirichlet random distributions.

Article information

Ann. Probab. Volume 37, Number 5 (2009), 1999-2041.

First available in Project Euclid: 21 September 2009

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.)

Regenerative composition Poisson–Dirichlet composition Chinese Restaurant Process Markov branching model self-similar fragmentation continuum random tree ℝ-tree recursive random tree phylogenetic tree


Pitman, Jim; Winkel, Matthias. Regenerative tree growth: Binary self-similar continuum random trees and Poisson–Dirichlet compositions. Ann. Probab. 37 (2009), no. 5, 1999--2041. doi:10.1214/08-AOP445.

Export citation


  • [1] Aldous, D. (1991). The continuum random tree. I. Ann. Probab. 19 1–28.
  • [2] Aldous, D. (1996). Probability distributions on cladograms. In Random Discrete Structures (Minneapolis, MN, 1993). IMA Vol. Math. Appl. 76 1–18. Springer, New York.
  • [3] Bertoin, J. (2002). Self-similar fragmentations. Ann. Inst. H. Poincaré Probab. Statist. 38 319–340.
  • [4] Bertoin, J. and Rouault, A. (2005). Discretization methods for homogeneous fragmentations. J. London Math. Soc. (2) 72 91–109.
  • [5] Blei, D. M., Griffiths, T. L. and Jordan, M. I. (2008). The nested Chinese restaurant process and hierarchical topic models. Preprint. Available at arXiv:0710.0845v2.
  • [6] Dong, R., Goldschmidt, C. and Martin, J. B. (2006). Coagulation-fragmentation duality, Poisson–Dirichlet distributions and random recursive trees. Ann. Appl. Probab. 16 1733–1750.
  • [7] Duquesne, T. and Winkel, M. (2007). Growth of Lévy trees. Probab. Theory Related Fields 139 313–371.
  • [8] 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.
  • [9] Evans, S. N. and Winter, A. (2006). Subtree prune and regraft: A reversible real tree-valued Markov process. Ann. Probab. 34 918–961.
  • [10] Ford, D. J. (2005). Probabilities on cladograms: Introduction to the alpha model. Preprint. Available at arXiv:math/0511246v1.
  • [11] Gnedin, A. and Pitman, J. (2007). Exchangeable Gibbs partitions and Stirling triangles. Preprint. Available at arXiv:math/0412494v1.
  • [12] Gnedin, A. and Pitman, J. (2005). Exchangeable Gibbs partitions and Stirling triangles. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 325 83–102, 244–245.
  • [13] Gnedin, A. and Pitman, J. (2005). Regenerative composition structures. Ann. Probab. 33 445–479.
  • [14] Gnedin, A., Pitman, J. and Yor, M. (2006). Asymptotic laws for compositions derived from transformed subordinators. Ann. Probab. 34 468–492.
  • [15] Greenwood, P. and Pitman, J. (1980). Construction of local time and Poisson point processes from nested arrays. J. London Math. Soc. (2) 22 182–192.
  • [16] Greven, A., Pfaffelhuber, P. and Winter, A. (2009). Convergence in distribution of random metric measure spaces (Λ-coalescent measure trees). Probab. Theory Related Fields 145 285–322.
  • [17] 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).
  • [18] Haas, B., Miermont, G., Pitman, J. and Winkel, M. (2008). Continuum tree asymptotics of discrete fragmentations and applications to phylogenetic models. Ann. Probab. 36 1790–1837.
  • [19] Haas, B., Pitman, J. and Winkel, M. (2007). Spinal partitions and invariance under re-rooting of continuum random trees. Ann. Probab. Preprint. Available at arXiv:0705.3602v1.
  • [20] Lamperti, J. (1972). Semi-stable Markov processes. I. Z. Wahrsch. Verw. Gebiete 22 205–225.
  • [21] Marchal, P. (2003). Constructing a sequence of random walks strongly converging to Brownian motion. In Discrete Random Walks (Paris, 2003). Discrete Math. Theor. Comput. Sci. Proc., AC 181–190 (electronic). Assoc. Discrete Math. Theor. Comput. Sci., Nancy.
  • [22] Mauldin, R. D., Sudderth, W. D. and Williams, S. C. (1992). Pólya trees and random distributions. Ann. Statist. 20 1203–1221.
  • [23] Pitman, J. (1997). Partition structures derived from Brownian motion and stable subordinators. Bernoulli 3 79–96.
  • [24] Pitman, J. (2006). Combinatorial Stochastic Processes. Lecture Notes in Math. 1875. Springer, Berlin.
  • [25] Rémy, J.-L. (1985). Un procédé itératif de dénombrement d’arbres binaires et son application à leur génération aléatoire. RAIRO Inform. Théor. 19 179–195.