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On the Number of Distinct Fringe Subtrees in Binary Search Trees

Author Stephan Wagner

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Author Details

Stephan Wagner
  • Institute of Discrete Mathematics, TU Graz, Austria
  • Department of Mathematics, Uppsala University, Sweden

Funding

Supported by the Swedish research council (VR), grant 2022-04030.

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Stephan Wagner. On the Number of Distinct Fringe Subtrees in Binary Search Trees. In 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 302, pp. 13:1-13:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.AofA.2024.13

Abstract

A fringe subtree of a rooted tree is a subtree that consists of a vertex and all its descendants. The number of distinct fringe subtrees in random trees has been studied by several authors, notably because of its connection to tree compaction algorithms. Here, we obtain a very precise result for binary search trees: it is shown that the number of distinct fringe subtrees in a binary search tree with n leaves is asymptotically equal to (c₁n)/(log n) for a constant c₁ ≈ 2.4071298335, both in expectation and with high probability. This was previously shown to be a lower bound, our main contribution is to prove a matching upper bound. The method is quite general and can also be applied to similar problems for other tree models.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Enumeration
  • Theory of computation → Randomness, geometry and discrete structures
  • Theory of computation → Data compression
Keywords
  • Fringe subtrees
  • binary search trees
  • tree compression
  • minimal DAG
  • asymptotics

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