Cutting Edges at Random in Large Recursive Trees

  • Erich Baur
  • Jean BertoinEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 100)


We comment on old and new results related to the destruction of a random recursive tree (RRT), in which its edges are cut one after the other in a uniform random order. In particular, we study the number of steps needed to isolate or disconnect certain distinguished vertices when the size of the tree tends to infinity. New probabilistic explanations are given in terms of the so-called cut-tree and the tree of component sizes, which both encode different aspects of the destruction process. Finally, we establish the connection to Bernoulli bond percolation on large RRT’s and present recent results on the cluster sizes in the supercritical regime.


Random recursive tree Destruction of graphs Isolation of nodes Disconnection Supercritical percolation Cluster sizes Fluctuations 


  1. 1.
    L. Addario-Berry, N. Broutin, C. Holmgren, Cutting down trees with a Markov chainsaw. Ann. Appl. Probab. 24(6), 2297–2339 (2014)Google Scholar
  2. 2.
    D. Aldous, The Continuum Random Tree III. Ann. Probab. 21, 248–289 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    A.-L. Barabási, R. Albert, Emergence of scaling in random networks. Science 286(5439), 509–512 (1999)CrossRefMathSciNetGoogle Scholar
  4. 4.
    D. Barraez, S. Boucheron, W. de la Fernandez, Vega, On the fluctuations of the giant component. Comb. Probab. Comput. 9, 287–304 (2000)CrossRefzbMATHGoogle Scholar
  5. 5.
    E. Baur, Percolation on random recursive trees. Preprint (2014). arXiv:1407.2508
  6. 6.
    J. Bertoin, Fires on trees. Ann. Inst. Henri Poincaré Probab. Stat. 48, 909–921 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    J. Bertoin, Sizes of the largest clusters for supercritical percolation on random recursive trees. Random Struct. Algorithms 44–1, 1098–2418 (2014)MathSciNetGoogle Scholar
  8. 8.
    J. Bertoin, On the non-Gaussian fluctuations of the giant cluster for percolation on random recursive trees. Electron. J. Probab. 19(24), 1–15 (2014)MathSciNetGoogle Scholar
  9. 9.
    J. Bertoin, The cut-tree of large recursive trees. To appear in Ann. Instit. Henri Poincaré Probab. Stat.Google Scholar
  10. 10.
    J. Bertoin, G. Uribe Bravo, Supercritical percolation on large scale-free random trees. To appear in Ann. Appl. Probab.Google Scholar
  11. 11.
    J. Bertoin, G. Miermont, The cut-tree of large Galton-Watson trees and the Brownian CRT. Ann. Appl. Probab. 23, 1469–1493 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    E. Bolthausen, A.-S. Sznitman, On Ruelle’s probability cascades and an abstract cavity method. Comm. Math. Phys. 197–2, 247–276 (1998)CrossRefMathSciNetGoogle Scholar
  13. 13.
    B. Chauvin, T. Klein, J.-F. Marckert, A. Rouault, Martingales and profile of binary search trees. Electron. J. Probab. 10, 420–435 (2005)CrossRefMathSciNetGoogle Scholar
  14. 14.
    D. Dieuleveut,The vertex-cut-tree of Galton-Watson trees converging to a stable tree. To appear in Ann. Appl. Probab.Google Scholar
  15. 15.
    M. Drmota, Random Trees (Springer, New York, Vienna, 2009)CrossRefzbMATHGoogle Scholar
  16. 16.
    M. Drmota, A. Iksanov, M. Möhle, U. Rösler, A limiting distribution for the number of cuts needed to isolate the root of a random recursive tree. Random Struct. Algorithms 34–3, 319–336 (2009)CrossRefGoogle Scholar
  17. 17.
    K.B. Erickson, Strong renewal theorems with infinite mean. Trans. Am. Math. Soc. 151, 263–291 (1970)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    C. Goldschmidt, J.B. Martin, Random recursive trees and the Bolthausen-Sznitman coalescent. Electron. J. Probab. 10, 718–745 (2005)CrossRefMathSciNetGoogle Scholar
  19. 19.
    B. Haas, G. Miermont, Scaling limits of Markov branching trees with applications to Galton-Watson and random unordered trees. Ann. Probab. 40–6, 2589–2666 (2012)CrossRefMathSciNetGoogle Scholar
  20. 20.
    C. Holmgren, Random records and cuttings in binary search trees. Comb. Probab. Comput. 19–3, 391–424 (2010)CrossRefMathSciNetGoogle Scholar
  21. 21.
    C. Holmgren, A weakly 1-stable distribution for the number of random records and cuttings in split trees. Adv. Appl. Probab. 43–1, 151–177 (2011)CrossRefMathSciNetGoogle Scholar
  22. 22.
    A. Iksanov, M. Möhle, A probabilistic proof of a weak limit law for the number of cuts needed to isolate the root of a random recursive tree. Electron. Comm. Probab. 12, 28–35 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    S. Janson, Random records and cuttings in complete binary trees, in Mathematics and Computer Science III, Algorithms, Trees, Combinatorics and Probabilities (Vienna 2004), ed. by M. Drmota, P. Flajolet, D. Gardy, B. Gittenberger (Basel, Birkhäuser, 2004), pp. 241–253Google Scholar
  24. 24.
    S. Janson, Random cutting and records in deterministic and random trees. Random Struct. Algorithms 29–2, 139–179 (2006)CrossRefMathSciNetGoogle Scholar
  25. 25.
    O. Kallenberg, Foundations of Modern Probability, Probability and its Applications (Springer, New York, 2002)CrossRefGoogle Scholar
  26. 26.
    M. Kuba, A. Panholzer, On the degree distribution of the nodes in increasing trees. J. Combin. Theory, Series A 114–4, 597–618 (2007)Google Scholar
  27. 27.
    M. Kuba, A. Panholzer, Isolating nodes in recursive trees. Aequ. Math. 76, 258–280 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    M. Kuba, A. Panholzer, Multiple isolation of nodes in recursive trees. OJAC 9, 26 (2014)Google Scholar
  29. 29.
    H. Mahmoud, Evolution of Random Search Trees (Wiley, New York, 1992)zbMATHGoogle Scholar
  30. 30.
    H. Mahmoud, R.T. Smythe, A survey of recursive trees. Theor. Probab. Math. Stat. 51, 1–29 (1994)zbMATHMathSciNetGoogle Scholar
  31. 31.
    A. Meir, J.W. Moon, Cutting down random trees. J. Austral. Math. Soc. 11, 313–324 (1970)CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    A. Meir, J.W. Moon, Cutting down recursive trees. Math. Biosci. 21, 173–181 (1974)CrossRefzbMATHGoogle Scholar
  33. 33.
    A. Panholzer, Cutting down very simple trees. Quaest. Math. 29–2, 211–227 (2006)CrossRefMathSciNetGoogle Scholar
  34. 34.
    J. Pitman, Combinatorial Stochastic Processes. École d’été de Probabilités de St. Flour, Lecture Notes in Mathematics 1875 (Springer, Berlin, 2006)Google Scholar
  35. 35.
    B. Pittel, On tree census and the giant component in sparse random graphs. Random Struct. Algorithms 1, 311–342 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  36. 36.
    J. Schweinsberg, Dynamics of the evolving Bolthausen-Sznitman coalescent. Electron. J. Probab. 17, 1–50 (2012)CrossRefMathSciNetGoogle Scholar
  37. 37.
    V.E. Stepanov, On the probability of connectedness of a random graph \(\cal {G}\) \(_m\) (t). Theory Probab. Appl. 15–1, 55–67 (1970)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.ENS LyonLyonFrance
  2. 2.Universität ZürichZürichSwitzerland

Personalised recommendations