Binary Search Trees

This research is all concerned with the development of an analysis and simulations of the effect of mixed deletions and insertions in binary search trees. Previously it was believed that the average depth of a node in a tree subjected to updates was decreased towards an optimal O(log n) when using the usual algorithms (See e.g. Knuth Vol. 3 [8] ) This was supported to some extent by a complete analysis for trees of only three nodes by Jonassen and Knuth [7] in 1978. Eppinger [6] performed some simulations showing that this was likely false, and conjectured, based on the experimental evidence, that the depth grew as O(log^3 n), but offered no analysis or explanation as to why this should occur. Essentially, this problem concerning one of the most widely used data structures had remained open for 20 years.

Both my MSc [1] and Ph.D. [2] focused on this problem. This research indicates that in fact the depth grows as O(n^{1/2}). A proof under certain simplifying assumptions was given in the theoretical analysis in Algorithmica [3], while a set of simulations was presented together with a less formal analysis of the more general case, in the Computer Journal [4] , intended for a wider audience. A complete analysis for the most general case is still open.

More recently P. Evans [5] has demonstrated that asymmetry may be even more dangerous in smaller doses! Algorithms with only occasional asymmetric moves tend to develop trees with larger skews, although the effects take much longer.


  1. Joseph Culberson. Updating Binary Trees. MSc Thesis, University of Waterloo Department of Computer Science, 1984. (Available as Waterloo Research Report CS-84-08.) Abstract
  2. Joseph Culberson. The Effect of Asymmetric Deletions on Binary Search Trees. Ph. D. Thesis University of Waterloo Department of Computer Science, May 1986. (Available as Waterloo Research Report CS-86-15.) Abstract
  3. Joseph Culberson J. Ian Munro. Analysis of the standard deletion algorithms in exact fit domain binary search trees. Algorithmica, vol 6, 295-311, 1990. Abstract
  4. Joseph Culberson J. Ian Munro. Explaining the behavior of binary search trees under prolonged updates: A model and simulations. The Computer Journal, vol 32(1), 68-75, February 1989. Abstract
  5. Joseph C. Culberson and Patricia A. Evans Asymmetry in Binary Search Tree Update Algorithms Technical Report TR94-09

    Other References

  6. Jeffery L. Eppinger. An empirical study of insertion and deletion in binary trees. Communications of the ACM vol. 26, September 1983.
  7. Arne T. Jonassen and Donald E. Knuth A trivial algorithm whose analysis isn't. Journal of Computer and System Sciences, 16:301-322, 1978.
  8. D. E. Knuth Sorting and Searching Volume III, The Art of Computer Programming Addison-Wesley Publishing Company, Inc., Reading, Massachusetts, 1973.

Joseph Culberson

May 10, 1994