## B Trees

# Introduction

B-Tree is a self-balancing search tree. In most of the other self-balancing search trees (like AVL and Red Black Trees), it is assumed that everything is in main memory. To understand use of B-Trees, we must think of huge amount of data that cannot fit in main memory. When the number of keys is high, the data is read from disk in the form of blocks. Disk access time is very high compared to main memory access time. The main idea of using B-Trees is to reduce the number of disk accesses. Most of the tree operations (search, insert, delete, max, min, etc) require O(h) disk accesses where h is height of the tree. B-tree is a fat tree. Height of B-Trees is kept low by putting maximum possible keys in a B-Tree node. Generally, a B-Tree node size is kept equal to the disk block size. Since h is low for B-Tree, total disk accesses for most of the operations are reduced significantly compared to balanced Binary Search Trees like AVL Tree, Red Black Tree, etc.

Properties of B-Tree:

1) All leaf nodes are at same level.

2) A B-Tree is defined by the term minimum degree ‘t’. The value of t depends upon disk block size.

3) Every node except root must contain at least t-1 keys. Root may contain minimum 1 key.

4) All nodes (including root) may contain at most 2t – 1 keys.

5) Speaking in terms of order n , each node must cantain at minimum ciel(m/2) -1 keys and maximum m-1 keys.

5) Number of children of a node is equal to the number of keys in it plus 1.

6) All keys of a node are sorted in increasing order. The child between two keys k1 and k2 contains all keys in range from k1 and k2.

7) B-Tree grows and shrinks from root which is unlike Binary Search Tree. Binary Search Trees grow downward and also shrink from downward.

8) Like other balanced Binary Search Trees, time complexity to search, insert and delete is O(Log(n)).

### Search:

Search is similar to search in Binary Search Tree. Let the key to be searched be k. We start from root and recursively traverse down. For every visited non-leaf node, if the node has key, we simply return the node. Otherwise we recur down to the appropriate child (The child which is just before the first greater key) of the node. If we reach a leaf node and don’t find k in the leaf node, we return NULL.

### Traverse:

Traversal is also similar to Inorder traversal of Binary Tree. We start from the leftmost child, recursively print the leftmost child, then repeat the same process for remaining children and keys. In the end, recursively print the rightmost child.

### Insert

First we search and find out to which node the key should belong to and we insert it into it. Afterwards we look for and fix these problems:

If the number of keys is too high(greater than t – 1) then we move the middle key to the nodes parent. We do this recursively up until root. If the number of keys in the root is too high, then we make the middle key to be the new root of the whole tree and connect it to the node it was in before.

### Time Analysis for B-Tree:

Suppose a B-tree has n elements and M is the maximum number of children a node can have. What is the maximum depth the tree could have? What is the minimum depth the tree could have?

- The worst-case depth (maximum depth) of a B-tree is: logM/2 n.
- The best-case depth (minimum depth) of a B-tree is: logM n.