The Annals of Probability

Subtree prune and regraft: A reversible real tree-valued Markov process

Steven N. Evans and Anita Winter

Full-text: Open access


We use Dirichlet form methods to construct and analyze a reversible Markov process, the stationary distribution of which is the Brownian continuum random tree. This process is inspired by the subtree prune and regraft (SPR) Markov chains that appear in phylogenetic analysis.

A key technical ingredient in this work is the use of a novel Gromov–Hausdorff type distance to metrize the space whose elements are compact real trees equipped with a probability measure. Also, the investigation of the Dirichlet form hinges on a new path decomposition of the Brownian excursion.

Article information

Ann. Probab. Volume 34, Number 3 (2006), 918-961.

First available in Project Euclid: 27 June 2006

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J25: Continuous-time Markov processes on general state spaces 60J75: Jump processes
Secondary: 92B10: Taxonomy, cladistics, statistics

Dirichlet form continuum random tree Brownian excursion phylogenetic tree Markov chain Monte Carlo simulated annealing path decomposition excursion theory Gromov–Hausdorff metric Prohorov metric


Evans, Steven N.; Winter, Anita. Subtree prune and regraft: A reversible real tree-valued Markov process. Ann. Probab. 34 (2006), no. 3, 918--961. doi:10.1214/009117906000000034.

Export citation


  • Abraham, R. and Serlet, L. (2002). Poisson snake and fragmentation. Electron. J. Probab. 7 15--17 (electronic).
  • Aldous, D. (1991). The continuum random tree. I. Ann. Probab. 19 1--28.
  • Aldous, D. (1991). The continuum random tree. II. An overview. In Stochastic Analysis (M. T. Barlow and N. H. Bingham, eds.) 23--70. Cambridge Univ. Press.
  • Aldous, D. (1993). The continuum random tree III. Ann. Probab. 21 248--289.
  • Aldous, D. J. (2000). Mixing time for a Markov chain on cladograms. Combin. Probab. Comput. 9 191--204.
  • Allen, B. L. and Steel, M. (2001). Subtree transfer operations and their induced metrics on evolutionary trees. Ann. Comb. 5 1--15.
  • Bastert, O., Rockmore, D., Stadler, P. F. and Tinhofer, G. (2002). Landscapes on spaces of trees. Appl. Math. Comput. 131 439--459.
  • Bertoin, J. (1996). Lévy Processes. Cambridge Univ. Press.
  • Billera, L. J., Holmes, S. P. and Vogtmann, K. (2001). Geometry of the space of phylogenetic trees. Adv. in Appl. Math. 27 733--767.
  • Bridson, M. R. and Haefliger, A. (1999). Metric Spaces of Non-Positive Curvature. Springer, Berlin.
  • Burago, D., Burago, Y. and Ivanov, S. (2001). A Course in Metric Geometry. Amer. Math. Soc., Providence, RI.
  • Chiswell, I. (2001). Introduction to $\Lambda$-trees. World Scientific Publishing, River Edge, NJ.
  • Daley, D. J. and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes. Springer, New York.
  • Diaconis, P. and Holmes, S. P. (2002). Random walks on trees and matchings. Electron. J. Probab. 7 6--17 (electronic).
  • Dress, A. W. M. (1984). Trees, tight extensions of metric spaces, and the cohomological dimension of certain groups: A note on combinatorical properties of metric spaces. Adv. in Math. 53 321--402.
  • Dress, A. W. M., Moulton, V. and Terhalle, W. F. (1996). $T$-theory. European J. Combin. 17 161--175.
  • Dress, A. W. M. and Terhalle, W. F. (1996). The real tree. Adv. in Math. 120 283--301.
  • Duquesne, T. and Le Gall, J.-F. (2002). Random Trees, Lévy Processes and Spatial Branching Processes. Astérisque 281. Soc. Math. France, Paris.
  • Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes. Wiley, New York.
  • Evans, S. N., Pitman, J. and Winter, A. (2006). Rayleigh processes, real trees, and root growth with re-grafting. Probab. Theory Related Fields. To appear.
  • Felsenstein, J. (2003). Inferring Phylogenies. Sinauer Associates, Sunderland, MA.
  • Fukushima, M., Oshima, Y. and Takeda, M. (1994). Dirichlet Forms and Symmetric Markov Processes. de Gruyter, Berlin.
  • Gromov, M. (1999). Metric Structures for Riemannian and Non-Riemannian Spaces. Birkhäuser, Boston.
  • Kelley, J. L. (1975). General Topology. Springer, New York.
  • Knight, F. B. (1981). Essentials of Brownian Motion and Diffusion. Amer. Math. Soc., Providence, RI.
  • Le Gall, J.-F. (1999). Spatial Branching Processes, Random Snakes and Partial Differential Equations. Birkhäuser, Basel.
  • Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd ed. Springer, Berlin.
  • Rogers, L. C. G. and Williams, D. (2000). Diffusions, Markov Processes, and Martingales 2. Cambridge Univ. Press.
  • Schweinsberg, J. (2002). An $O(n\sp2)$ bound for the relaxation time of a Markov chain on cladograms. Random Structures Algorithms 20 59--70.
  • Semple, C. and Steel, M. (2003). Phylogenetics. Oxford Univ. Press.
  • Sturm, K.-T. (2004). On the geometry of metric measure spaces. Technical Report 203, SFB 611, Univ. Bonn.
  • Swofford, D. L. and Olsen, G. J. (1990). Phylogeny reconstruction. In Molecular Systematics (D. M. Hillis and G. Moritz, eds.) 411--501. Sinauer Associates, Sunderland, MA.
  • Terhalle, W. F. (1997). $\mathbfR$-trees and symmetric differences of sets. European J. Combin. 18 825--833.
  • Zambotti, L. (2001). A reflected stochastic heat equation as symmetric dynamics with respect to the 3-d Bessel bridge. J. Funct. Anal. 180 195--209.
  • Zambotti, L. (2002). Integration by parts on Bessel bridges and related stochastic partial differential equations. C. R. Math. Acad. Sci. Paris 334 209--212.
  • Zambotti, L. (2003). Integration by parts on $\delta$-Bessel bridges, $\delta>3$ and related SPDEs. Ann. Probab. 31 323--348.