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A bijection for the evolution of $B$-trees

Fabian Burghart, Stephan Wagner

TL;DR

This work establishes a bijection between histories of inserting keys into $B$-trees of order $2m+1$ and a class of rooted plane trees called $(2m+1)$-historic trees, enabling precise combinatorial accounting of leaves, branchings, and permutations that realize a history. It provides a recursive, algorithmic description for generating all permutations associated with a given history and introduces a digraph-based framework to connect histories to topological labelings of a constructed graph, thereby enabling systematic enumeration. For the case $m=1$, the authors derive explicit generating-function recurrences and asymptotics for the number of histories, and they compute the distributional statistics of leaves (external vertices) under random insertions, including mean, variance, and a central limit theorem; they extend the approach to general $m$ via higher-order differential equations and conjectured asymptotics. The historic-tree approach offers an alternative to urn-based analyses of $B$-trees, provides concrete readouts of permutation counts for histories, and sets the stage for further generalizations, including the behavior when $m$ grows with $n$ and additional statistics of tree evolution.

Abstract

A $B$-tree is a type of search tree where every node (except possibly for the root) contains between $m$ and $2m$ keys for some positive integer $m$, and all leaves have the same distance to the root. We study sequences of $B$-trees that can arise from successively inserting keys, and in particular present a bijection between such sequences (which we call histories) and a special type of increasing trees. We describe the set of permutations for the keys that belong to a given history, and also show how to use this bijection to analyse statistics associated with $B$-trees.

A bijection for the evolution of $B$-trees

TL;DR

This work establishes a bijection between histories of inserting keys into -trees of order and a class of rooted plane trees called -historic trees, enabling precise combinatorial accounting of leaves, branchings, and permutations that realize a history. It provides a recursive, algorithmic description for generating all permutations associated with a given history and introduces a digraph-based framework to connect histories to topological labelings of a constructed graph, thereby enabling systematic enumeration. For the case , the authors derive explicit generating-function recurrences and asymptotics for the number of histories, and they compute the distributional statistics of leaves (external vertices) under random insertions, including mean, variance, and a central limit theorem; they extend the approach to general via higher-order differential equations and conjectured asymptotics. The historic-tree approach offers an alternative to urn-based analyses of -trees, provides concrete readouts of permutation counts for histories, and sets the stage for further generalizations, including the behavior when grows with and additional statistics of tree evolution.

Abstract

A -tree is a type of search tree where every node (except possibly for the root) contains between and keys for some positive integer , and all leaves have the same distance to the root. We study sequences of -trees that can arise from successively inserting keys, and in particular present a bijection between such sequences (which we call histories) and a special type of increasing trees. We describe the set of permutations for the keys that belong to a given history, and also show how to use this bijection to analyse statistics associated with -trees.
Paper Structure (12 sections, 8 theorems, 34 equations, 2 figures, 2 tables)

This paper contains 12 sections, 8 theorems, 34 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction and main results
  2. B-trees and their insertion algorithm
  3. Main results
  4. The bijection
  5. The permutations associated with a history
  6. Preparatory lemmas
  7. The algorithm
  8. An example
  9. Proofs
  10. The number of histories
  11. Statistics of B-trees via historic trees
  12. Conclusion and perspective

Key Result

Theorem 1

Let $n,m\geq 1$. There is a bijection between $\mathscr H_m(n)$ and the set of all trees $H_n$ satisfying the following properties:

Figures (2)

  • Figure 1: A history of $B$-trees of order $2m+1=3$ on the left, with the corresponding historic tree $H_9$ shown on the right. The external vertices of $H_9$ are shown in white, and are connected by dotted lines. The vertices $3,5,8$, and $9$ are the branchings of $H_9$.
  • Figure 2: Steps in the algorithm of Section \ref{['section:perm']}, as performed in Subsection \ref{['subsection:ex']}.

Theorems & Definitions (18)

  • Theorem 1
  • Proposition 2
  • Proposition 3
  • Remark 4
  • proof : Proof of Proposition \ref{['prop:bijection']}
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • Lemma 7
  • ...and 8 more