The Annals of Applied Probability

Mixed Poisson approximation of node depth distributions in random binary search trees

Rudolf Grübel and Nikolče Stefanoski

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We investigate the distribution of the depth of a node containing a specific key or, equivalently, the number of steps needed to retrieve an item stored in a randomly grown binary search tree. Using a representation in terms of mixed and compounded standard distributions, we derive approximations by Poisson and mixed Poisson distributions; these lead to asymptotic normality results. We are particularly interested in the influence of the key value on the distribution of the node depth. Methodologically our message is that the explicit representation may provide additional insight if compared to the standard approach that is based on the recursive structure of the trees. Further, in order to exhibit the influence of the key on the distributional asymptotics, a suitable choice of distance of probability distributions is important. Our results are also applicable in connection with the number of recursions needed in Hoare’s [Comm. ACM 4 (1961) 321–322] selection algorithm FIND.

Article information

Ann. Appl. Probab. Volume 15, Number 1A (2005), 279-297.

First available in Project Euclid: 28 January 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 68Q25: Analysis of algorithms and problem complexity [See also 68W40]
Secondary: 68P10: Searching and sorting 60F05: Central limit and other weak theorems

Asymptotic normality mixed Poisson distributions Poisson approximation random permutations randomized algorithms Hoare’s selection algorithm


Grübel, Rudolf; Stefanoski, Nikolče. Mixed Poisson approximation of node depth distributions in random binary search trees. Ann. Appl. Probab. 15 (2005), no. 1A, 279--297. doi:10.1214/105051604000000611.

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  • Arnold, B. C., Balakrishnan, N. and Nagaraja, H. N. (1998). Records. Wiley, New York.
  • Arora, S. and Dent, W. (1969). Randomized binary search technique. Comm. ACM 12 77--80.
  • Barbour, A. D., Holst, L. and Janson, S. (1992). Poisson Approximation. Clarendon Press, Oxford.
  • Cormen, Th. H., Leiserson, Ch. E. and Rivest, R. L. (1990). Introduction to Algorithms. MIT Press.
  • Devroye, L. (1988). Applications of the theory of records in the study of random trees. Acta Inform. 26 123--130.
  • Devroye, L. and Neininger, R. (2004). Distances and finger search in random binary search trees. SIAM J. Comput. 33 647--658.
  • Graham, R. L., Knuth, D. E. and Patashnik, O. (1989). Concrete Mathematics, 2nd ed. Addison--Wesley, Reading, MA.
  • Grübel, R. (1998). Hoare's selection algorithm: A Markov chain approach. J. Appl. Probab. 35 36--45.
  • Grübel, R. and Rösler, U. (1996). Asymptotic distribution theory for Hoare's selection algorithm. Adv. in Appl. Probab. 28 252--269.
  • Hoare, C. A. R. (1961). Algorithm 63: Partition, Algorithm 64: Quicksort, Algorithm 65: Find. Comm. ACM 4 321--322.
  • Kirschenhofer, P. and Prodinger, H. (1998). Comparisons in Hoare's Find algorithm. Combin. Probab. Comput. 7 111--120.
  • Knuth, D. E. (1973). Sorting and Searching. The Art of Computer Programming 3. Addison--Wesley, Reading, MA.
  • Louchard, G. (1987). Exact and asymptotic distributions in digital binary search trees. Theor. Inform. Appl. 21 479--496.
  • Mahmoud, H. M. (1992). Evolution of Random Search Trees. Wiley, New York.
  • Mahmoud, H. M. (2000). Sorting: A Distribution Theory. Wiley, New York.
  • Mahmoud, H. M. and Neininger, R. (2003). Distribution of distances in random binary search trees. Ann. Appl. Probab. 13 253--276.
  • Sedgewick, R. and Flajolet, Ph. (1996). An Introduction to the Analysis of Algorithms. Addison--Wesley, Reading, MA.