A limiting distribution for the number of cuts needed to isolate the root of a random recursive tree

  • Authors:

  • Michael Drmota;Alex Iksanov;Martin Moehle;Uwe Roesler

  • Affiliations:

  • Institute of Discrete Mathematics and Geometry, Technical University of Vienna, Wiedner Hauptstrasse 8–10-104, A-1040 Vienna, Austria;Faculty of Cybernetics, National T. Shevchenko University of Kiev, Kiev 01033, Ukraine;Mathematical Institute, University of Duesseldorf, Universitaetsstr. 1, 40225 Duesseldorf, Germany;Mathematisches Seminar, Christian-Albrechts-Universitaet zu Kiel, Ludewig-Meyn-Str. 4, D-24098 Kiel

  • Venue:
  • Random Structures & Algorithms
  • Year:
  • 2009

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Abstract

Let Xn be the number of cuts needed to isolate the root in a random recursive tree with n vertices. We provide a weak convergence result for Xn. The basic observation for its proof is that the probability distributions of $\{X_n: n=2,3,\ldots\}$ are recursively defined by $X_n {{\rm d}\atop{=}} X_{n-D_n} + 1, \; n=2,3,\ldots, \; X_1=0$, where Dn is a discrete random variable with ℙ${\{ D_n = k \} = {1 \over {k(k+1)}} {n \over {n-1}}, \; k=1,2,\ldots, n-1}$, which is independent of $(X_2,\ldots, X_n)$. This distributional recursion was not studied previously in the sense of weak convergence. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2009