Open Access
February 2005 Mixed Poisson approximation of node depth distributions in random binary search trees
Rudolf Grübel, Nikolče Stefanoski
Ann. Appl. Probab. 15(1A): 279-297 (February 2005). DOI: 10.1214/105051604000000611


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.


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Rudolf Grübel. Nikolče Stefanoski. "Mixed Poisson approximation of node depth distributions in random binary search trees." Ann. Appl. Probab. 15 (1A) 279 - 297, February 2005.


Published: February 2005
First available in Project Euclid: 28 January 2005

zbMATH: 1066.68031
MathSciNet: MR2115044
Digital Object Identifier: 10.1214/105051604000000611

Primary: 68Q25
Secondary: 60F05 , 68P10

Keywords: asymptotic normality , Hoare’s selection algorithm , mixed Poisson distributions , Poisson approximation , Random permutations , randomized algorithms

Rights: Copyright © 2005 Institute of Mathematical Statistics

Vol.15 • No. 1A • February 2005
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