# B-trees Require Fewer Comparisons Than Balanced Binary Search Trees

DevFeed: [B-trees Require Fewer Comparisons Than Balanced Binary Search Trees](<https://devfeed.tech/articles/b-trees-require-fewer-comparisons-than-balanced-binary-search-trees-25085.md>)

Original publisher: [Read original article](<https://databasearchitects.blogspot.com/2024/06/b-trees-require-fewer-comparisons-than.html>)

Author: Viktor Leis (noreply@blogger.com)

Published: 2024-06-06T13:59:00Z

Content type: article

Language: en

Sources: [Database Architects](<https://devfeed.tech/sources/database-architects.md>)

Topics: [Data structures](<https://devfeed.tech/topics/data-structures.md>), [data](<https://devfeed.tech/topics/data.md>)

Tags: [comparisons](<https://devfeed.tech/tags/comparisons.md>), [data-structure](<https://devfeed.tech/tags/data-structure.md>), [structure](<https://devfeed.tech/tags/structure.md>), [theory](<https://devfeed.tech/tags/theory.md>)

## AI overview

The article compares B-trees with balanced binary search trees by analyzing the number of comparisons required for lookup operations. It explains that as the B-tree degree increases, the comparison bound approaches the lower bound, and for degree k>=8, B-trees are guaranteed to use fewer comparisons than AVL trees.

## Source excerpt

Due to better access locality, B-trees are faster than binary search trees in practice -- but are they also better in theory? To answer this question, let's look at the number of comparisons required for a search operation. Assuming we store n elements in a binary search tree, the lower bound for the number of comparisons is log2 n in the worst case. However, this is only achievable for a perfectly balanced tree. Maintaining such a tree's perfect balance during insert/delete operations requires O(n) time in the worst case. Balanced binary search trees, therefore, leave some slack in terms of how balanced they are and have slightly worse bounds. For example, it is well known that an AVL tree guarantees at most 1.44 log2 n comparisons, and a Red-Black tree guarantees 2 log2 n comparisons. In other words, AVL trees require at most 1.44 times the minimum number of comparisons, and Red-Black trees require up to twice the minimum. How many comparisons does a B-tree need? In B-trees with degree k, each node (except the root) has between k and 2k children. For k=2, a B-tree is essentially the same data structure as a Red-Black tree and therefore provides the same guarantee of 2 log2 n comparisons. So how about larger, more realistic values of k? To analyze the general case, we start with a B-tree that has the highest possible height for n elements. The height is maximal when each node has only k children (for simplicity, this analysis ignores the special case of underfull root nodes). This implies that the worst-case height of a B-tree is logk n. During a lookup, one has to perform a binary search that takes log2 k comparisons in each of the logk n nodes. So in total, we have log2 k * logk n = log2 n comparisons. This actually matches the best case, and to construct the worst case, we have to modify the tree somewhat. On one (and only one) arbitrary path from the root to a single leaf node, we increase the number of children from k to 2k. In this situation, the tree height