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May 2004 Spanning tree size in random binary search trees
Alois Panholzer, Helmut Prodinger
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Ann. Appl. Probab. 14(2): 718-733 (May 2004). DOI: 10.1214/105051604000000071


This paper deals with the size of the spanning tree of p randomly chosen nodes in a binary search tree. It is shown via generating functions methods, that for fixed p, the (normalized) spanning tree size converges in law to the Normal distribution. The special case p=2 reproves the recent result (obtained by the contraction method by Mahmoud and Neininger [Ann. Appl. Probab. 13 (2003) 253–276]), that the distribution of distances in random binary search trees has a Gaussian limit law. In the proof we use the fact that the spanning tree size is closely related to the number of passes in Multiple Quickselect. This parameter, in particular, its first two moments, was studied earlier by Panholzer and Prodinger [Random Structures Algorithms 13 (1998) 189–209]. Here we show also that this normalized parameter has for fixed p-order statistics a Gaussian limit law. For p=1 this gives the well-known result that the depth of a randomly selected node in a random binary search tree converges in law to the Normal distribution.


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Alois Panholzer. Helmut Prodinger. "Spanning tree size in random binary search trees." Ann. Appl. Probab. 14 (2) 718 - 733, May 2004.


Published: May 2004
First available in Project Euclid: 23 April 2004

zbMATH: 1126.68031
MathSciNet: MR2052899
Digital Object Identifier: 10.1214/105051604000000071

Primary: 05C05 , 60C05
Secondary: 60F05 , 68P05

Keywords: Binary search trees , Limiting distribution , Multiple Quickselect , quasi power theorem , spanning tree size

Rights: Copyright © 2004 Institute of Mathematical Statistics

Vol.14 • No. 2 • May 2004
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