Annals of Applied Probability

Cutting down trees with a Markov chainsaw

Louigi Addario-Berry, Nicolas Broutin, and Cecilia Holmgren

Full-text: Open access


We provide simplified proofs for the asymptotic distribution of the number of cuts required to cut down a Galton–Watson tree with critical, finite-variance offspring distribution, conditioned to have total progeny $n$. Our proof is based on a coupling which yields a precise, nonasymptotic distributional result for the case of uniformly random rooted labeled trees (or, equivalently, Poisson Galton–Watson trees conditioned on their size). Our approach also provides a new, random reversible transformation between Brownian excursion and Brownian bridge.

Article information

Ann. Appl. Probab., Volume 24, Number 6 (2014), 2297-2339.

First available in Project Euclid: 26 August 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60C05: Combinatorial probability 60F17: Functional limit theorems; invariance principles 05C05: Trees
Secondary: 11Y16: Algorithms; complexity [See also 68Q25]

Cutting down Galton–Watson tree real tree continuum random tree Gromov–Hausdorff convergence


Addario-Berry, Louigi; Broutin, Nicolas; Holmgren, Cecilia. Cutting down trees with a Markov chainsaw. Ann. Appl. Probab. 24 (2014), no. 6, 2297--2339. doi:10.1214/13-AAP978.

Export citation


  • [1] Abraham, R. and Delmas, J.-F. (2013). The forest associated with the record process on a Lévy tree. Stochastic Process. Appl. 123 3497–3517.
  • [2] Abraham, R. and Delmas, J.-F. (2013). Record process on the continuum random tree. ALEA Lat. Am. J. Probab. Math. Stat. 10 225–251.
  • [3] Addario-Berry, L., Broutin, N. and Holmgren, C. (2010). Cutting down trees with a Markov chainsaw (with online slides). YEP VII Seminar (March 2010).
  • [4] Aldous, D. (1991). The continuum random tree. II. An overview. In Stochastic (Durham, 1990) (M. T. Barlow and N. H. Bingham, eds.) 23–70. Cambridge Univ. Press, Cambridge.
  • [5] Aldous, D. (1991). Asymptotic fringe distributions for general families of random trees. Ann. Appl. Probab. 1 228–266.
  • [6] Aldous, D. (1991). The continuum random tree. I. Ann. Probab. 19 1–28.
  • [7] Aldous, D. (1993). The continuum random tree. III. Ann. Probab. 21 248–289.
  • [8] Aldous, D. and Pitman, J. (1998). The standard additive coalescent. Ann. Probab. 26 1703–1726.
  • [9] Aldous, D. and Steele, J. M. (2003). The objective method: Probabilistic combinatorial optimization and local weak convergence. In Discrete and Combinatorial Probability (H. Kesten, ed.) 1–72. Springer, Berlin.
  • [10] Aldous, D. J. (1990). The random walk construction of uniform spanning trees and uniform labelled trees. SIAM J. Discrete Math. 3 450–465.
  • [11] Aldous, D. J. and Pitman, J. (1994). Brownian bridge asymptotics for random mappings. Random Structures Algorithms 5 487–512.
  • [12] Bertoin, J. (2006). Random Fragmentation and Coagulation Processes. Cambridge Univ. Press, Cambridge.
  • [13] Bertoin, J. (2012). Fires on trees. In Annales de l’Institut Henri Poincaré, Probabilités et Statistiques 48 909–921.
  • [14] Bertoin, J. and Miermont, G. (2013). The cut-tree of large Galton–Watson trees and the Brownian CRT. Ann. Appl. Probab. 23 1469–1493.
  • [15] Bertoin, J. and Pitman, J. (1994). Path transformations connecting Brownian bridge, excursion and meander. Bull. Sci. Math. 118 147–166.
  • [16] Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
  • [17] Broder, A. (1989). Generating random spanning trees. In 30th Annual Symposium on Foundations of Computer Science 442–447. IEEE, New York.
  • [18] Burago, D., Burago, Y. and Ivanov, S. (2001). A Course in Metric Geometry. Graduate Studies in Mathematics 33. Amer. Math. Soc., Providence, RI.
  • [19] Chassaing, P. and Marchand, R. (2009). Personal communication.
  • [20] Daley, D. J. and Vere-Jones, D. (2007). An Introduction to the Theory of Point Processes. Vol. II: General Theory and Structure. Springer, New York.
  • [21] Devroye, L. and Janson, S. (2011). Distances between pairs of vertices and vertical profile in conditioned Galton–Watson trees. Random Structures Algorithms 38 381–395.
  • [22] Drmota, M., Iksanov, A., Moehle, M. and Roesler, U. (2009). A limiting distribution for the number of cuts needed to isolate the root of a random recursive tree. Random Structures Algorithms 34 319–336.
  • [23] Fill, J. A., Kapur, N. and Panholzer, A. (2006). Destruction of very simple trees. Algorithmica 46 345–366.
  • [24] Haas, B. and Miermont, G. (2012). Scaling limits of Markov branching trees with applications to Galton–Watson and random unordered trees. Ann. Probab. 40 2589–2666.
  • [25] Holmgren, C. (2008). Random records and cuttings in split trees: Extended abstract. In Fifth Colloquium on Mathematics and Computer Science. 269–281. Assoc. Discrete Math. Theor. Comput. Sci., Nancy.
  • [26] Iksanov, A. and Möhle, M. (2007). A probabilistic proof of a weak limit law for the number of cuts needed to isolate the root of a random recursive tree. Electron. Commun. Probab. 12 28–35.
  • [27] Janson, S. (2004). Random records and cuttings in complete binary trees. In Mathematics and Computer Science. III: Algorithms, Trees, Combinatorics and Probability (Vienna) (M. Drmota, P. Flajolet, D. Gardy and B. Gittenberger, eds.) 241–253. Birkhäuser, Basel.
  • [28] Janson, S. (2006). Random cutting and records in deterministic and random trees. Random Structures Algorithms 29 139–179.
  • [29] Kennedy, D. P. (1976). The distribution of the maximum Brownian excursion. J. Appl. Probab. 13 371–376.
  • [30] Kesten, H. (1986). Subdiffusive behavior of random walk on a random cluster. Ann. Inst. Henri Poincaré Probab. Stat. 22 425–487.
  • [31] Kolchin, V. F. (1986). Random Mappings. Optimization Software Inc. Publications Division, New York.
  • [32] Kuba, M. and Panholzer, A. (2008). Isolating a leaf in rooted trees via random cuttings. Ann. Comb. 12 81–99.
  • [33] Kuba, M. and Panholzer, A. (2008). Isolating nodes in recursive trees. Aequationes Math. 76 258–280.
  • [34] Le Gall, J.-F. (2005). Random trees and applications. Probab. Surv. 2 245–311.
  • [35] Le Gall, J.-F. (2006). Random real trees. Ann. Fac. Sci. Toulouse Math. (6) 15 35–62.
  • [36] Lyons, R. and Peres, Y. (2012). Probability on trees and networks. Unpublished manuscript.
  • [37] McDiarmid, C. (1998). Concentration. In Probabilistic Methods for Algorithmic Discrete Mathematics (M. Habib, C. McDiarmid, J. Ramirez-Alfonsin and B. Reed, eds.) 195–248. Springer, Berlin.
  • [38] Meir, A. and Moon, J. W. (1970). The distance between points in random trees. J. Combin. Theory 8 99–103.
  • [39] Meir, A. and Moon, J. W. (1970). Cutting down random trees. J. Aust. Math. Soc. 11 313–324.
  • [40] Meir, A. and Moon, J. W. (1974). Cutting down recursive trees. Math. Biosci. 21 173–181.
  • [41] Meir, A. and Moon, J. W. (1978). On the altitude of nodes in random trees. Canad. J. Math. 30 997–1015.
  • [42] Miermont, G. (2009). Tessellations of random maps of arbitrary genus. Ann. Sci. Éc. Norm. Supér. (4) 42 725–781.
  • [43] Panholzer, A. (2003). Noncrossing trees revisited: Cutting down and spanning subtrees. In Discrete Random Walks (Paris, 2003) 265–276 (electronic). Assoc. Discrete Math. Theor. Comput. Sci., Nancy.
  • [44] Panholzer, A. (2006). Cutting down very simple trees. Quaest. Math. 29 211–227.
  • [45] Pitman, J. (2006). Combinatorial Stochastic Processes. Lecture Notes in Math. 1875. Springer, Berlin.
  • [46] Riordan, J. (1968). Forests of labeled trees. J. Combin. Theory 5 90–103.
  • [47] Stroock, D. W. (2011). Probability Theory. An Analytic View, 2nd ed. Cambridge Univ. Press, Cambridge.
  • [48] Villani, C. (2009). Optimal Transport: Old and New. Springer, Berlin.
  • [49] Williams, D. (1991). Probability with Martingales. Cambridge Univ. Press, Cambridge.