Rank-Balanced Trees

  • Bernhard Haeupler
  • Siddhartha Sen
  • Robert E. Tarjan
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5664)


Since the invention of AVL trees in 1962, a wide variety of ways to balance binary search trees have been proposed. Notable are red-black trees, in which bottom-up rebalancing after an insertion or deletion takes O(1) amortized time and O(1) rotations worst-case. But the design space of balanced trees has not been fully explored. We introduce the rank-balanced tree, a relaxation of AVL trees. Rank-balanced trees can be rebalanced bottom-up after an insertion or deletion in O(1) amortized time and at most two rotations worst-case, in contrast to red-black trees, which need up to three rotations per deletion. Rebalancing can also be done top-down with fixed lookahead in O(1) amortized time. Using a novel analysis that relies on an exponential potential function, we show that both bottom-up and top-down rebalancing modify nodes exponentially infrequently in their heights.


Binary Tree Current Node Search Step Search Path Rank Difference 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Bernhard Haeupler
    • 2
  • Siddhartha Sen
    • 1
  • Robert E. Tarjan
    • 1
    • 3
  1. 1.Princeton UniversityUSA
  2. 2.Massachusetts Institute of TechnologyUSA
  3. 3.HP LaboratoriesPalo AltoUSA

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