The Annals of Applied Probability

Search trees: Metric aspects and strong limit theorems

Rudolf Grübel

Full-text: Open access


We consider random binary trees that appear as the output of certain standard algorithms for sorting and searching if the input is random. We introduce the subtree size metric on search trees and show that the resulting metric spaces converge with probability 1. This is then used to obtain almost sure convergence for various tree functionals, together with representations of the respective limit random variables as functions of the limit tree.

Article information

Ann. Appl. Probab., Volume 24, Number 3 (2014), 1269-1297.

First available in Project Euclid: 23 April 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60B99: None of the above, but in this section
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 68Q25: Analysis of algorithms and problem complexity [See also 68W40] 05C05: Trees

Doob–Martin compactification metric trees path length silhouette subtree size metric vector-valued martingales Wiener index


Grübel, Rudolf. Search trees: Metric aspects and strong limit theorems. Ann. Appl. Probab. 24 (2014), no. 3, 1269--1297. doi:10.1214/13-AAP948.

Export citation


  • [1] Biggins, J. D. (1977). Chernoff’s theorem in the branching random walk. J. Appl. Probab. 14 630–636.
  • [2] Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
  • [3] Blackwell, D. and Kendall, D. (1964). The Martin boundary of Pólya’s urn scheme, and an application to stochastic population growth. J. Appl. Probab. 1 284–296.
  • [4] Burago, D., Burago, Y. and Ivanov, S. (2001). A Course in Metric Geometry. Graduate Studies in Mathematics 33. Amer. Math. Soc., Providence, RI.
  • [5] Chauvin, B., Drmota, M. and Jabbour-Hattab, J. (2001). The profile of binary search trees. Ann. Appl. Probab. 11 1042–1062.
  • [6] Chauvin, B., Klein, T., Marckert, J. F. and Rouault, A. (2005). Martingales and profile of binary search trees. Electron. J. Probab. 10 420–435 (electronic).
  • [7] Chauvin, B. and Rouault, A. (2004). Connecting Yule processes, bisection and binary search tree via martingales. J. Iran. Stat. Soc. (JIRSS) 3 89–116.
  • [8] Dennert, F. (2009). Zufällige binäre Bäume: Algorithmen, Asymptotik und Statistik. Ph.D. thesis, Leibniz Univ. Hannover.
  • [9] Dennert, F. and Grübel, R. (2010). On the subtree size profile of binary search trees. Combin. Probab. Comput. 19 561–578.
  • [10] Devroye, L. (1986). A note on the height of binary search trees. J. Assoc. Comput. Mach. 33 489–498.
  • [11] Devroye, L. (1998). Branching processes and their applications in the analysis of tree structures and tree algorithms. In Probabilistic Methods for Algorithmic Discrete Mathematics. Algorithms Combin. 16 249–314. Springer, Berlin.
  • [12] Doob, J. L. (1959). Discrete potential theory and boundaries. J. Math. Mech. 8 433–458; erratum 993.
  • [13] Drmota, M. (2009). Random Trees: An Interplay Between Combinatorics and Probability. Springer, Vienna.
  • [14] Drmota, M., Janson, S. and Neininger, R. (2008). A functional limit theorem for the profile of search trees. Ann. Appl. Probab. 18 288–333.
  • [15] Evans, S. N. (2008). Probability and Real Trees. Lecture Notes in Math. 1920. Springer, Berlin.
  • [16] Evans, S. N., Grübel, R. and Wakolbinger, A. (2012). Trickle-down processes and their boundaries. Electron. J. Probab. 17 1–58.
  • [17] Fuchs, M. (2008). Subtree sizes in recursive trees and binary search trees: Berry–Esseen bounds and Poisson approximations. Combin. Probab. Comput. 17 661–680.
  • [18] Fuchs, M., Hwang, H.-K. and Neininger, R. (2006). Profiles of random trees: Limit theorems for random recursive trees and binary search trees. Algorithmica 46 367–407.
  • [19] Grübel, R. (2009). On the silhouette of binary search trees. Ann. Appl. Probab. 19 1781–1802.
  • [20] Jabbour-Hattab, J. (2001). Martingales and large deviations for binary search trees. Random Structures Algorithms 19 112–127.
  • [21] Kallenberg, O. (1997). Foundations of Modern Probability. Springer, New York.
  • [22] Kaĭmanovich, V. A. and Vershik, A. M. (1983). Random walks on discrete groups: Boundary and entropy. Ann. Probab. 11 457–490.
  • [23] Knuth, D. E. (1973). The Art of Computer Programming. Vol. 3: Sorting and Searching. Addison-Wesley, Reading, MA.
  • [24] Mahmoud, H. M. (1992). Evolution of Random Search Trees. Wiley, New York.
  • [25] Neininger, R. (2002). The Wiener index of random trees. Combin. Probab. Comput. 11 587–597.
  • [26] Neveu, J. (1975). Discrete-Parameter Martingales, Revised ed. North-Holland, Amsterdam.
  • [27] Régnier, M. (1989). A limiting distribution for quicksort. RAIRO Inform. Théor. Appl. 23 335–343.
  • [28] Rösler, U. (1991). A limit theorem for “Quicksort.” RAIRO Inform. Théor. Appl. 25 85–100.
  • [29] Woess, W. (2000). Random Walks on Infinite Graphs and Groups. Cambridge Tracts in Mathematics 138. Cambridge Univ. Press, Cambridge.
  • [30] Woess, W. (2009). Denumerable Markov Chains: Generating Functions, Boundary Theory, Random Walks on Trees. EMS, Zürich.