A bijection for the evolution of $B$-trees

TL;DR

This paper establishes a bijection between sequences of $B$-trees generated by key insertions and a class of increasing trees, enabling analysis of their structural properties and key permutations.

Contribution

It introduces a novel bijection linking $B$-tree evolution sequences to increasing trees, facilitating new analytical approaches.

Findings

Bijection between $B$-tree histories and increasing trees

Characterization of key permutations in $B$-trees

Method for analyzing $B$-tree statistics

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.