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 the use of B-Trees, we must think of the 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 the height of the tree. B-tree is a fat tree. The 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 leaves 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)** 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 the range from k1 and k2.

**7)** B-Tree grows and shrinks from the 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(Logn).

Following is an example B-Tree of minimum degree 3. Note that in practical B-Trees, the value of minimum degree is much more than 3.

**Search**

Search is similar to the search in Binary Search Tree. Let the key to be searched be k. We start from the root and recursively traverse down. For every visited non-leaf node, if the node has the 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.

`// C++ implemntation of search() and traverse() methods ` `#include<iostream> ` `using` `namespace` `std; ` ` ` `// A BTree node ` `class` `BTreeNode ` `{ ` ` ` `int` `*keys; ` `// An array of keys ` ` ` `int` `t; ` `// Minimum degree (defines the range for number of keys) ` ` ` `BTreeNode **C; ` `// An array of child pointers ` ` ` `int` `n; ` `// Current number of keys ` ` ` `bool` `leaf; ` `// Is true when node is leaf. Otherwise false ` `public` `: ` ` ` `BTreeNode(` `int` `_t, ` `bool` `_leaf); ` `// Constructor ` ` ` ` ` `// A function to traverse all nodes in a subtree rooted with this node ` ` ` `void` `traverse(); ` ` ` ` ` `// A function to search a key in the subtree rooted with this node. ` ` ` `BTreeNode *search(` `int` `k); ` `// returns NULL if k is not present. ` ` ` `// Make the BTree friend of this so that we can access private members of this ` `// class in BTree functions ` `friend` `class` `BTree; ` `}; ` ` ` `// A BTree ` `class` `BTree ` `{ ` ` ` `BTreeNode *root; ` `// Pointer to root node ` ` ` `int` `t; ` `// Minimum degree ` `public` `: ` ` ` `// Constructor (Initializes tree as empty) ` ` ` `BTree(` `int` `_t) ` ` ` `{ root = NULL; t = _t; } ` ` ` ` ` `// function to traverse the tree ` ` ` `void` `traverse() ` ` ` `{ ` `if` `(root != NULL) root->traverse(); } ` ` ` ` ` `// function to search a key in this tree ` ` ` `BTreeNode* search(` `int` `k) ` ` ` `{ ` `return` `(root == NULL)? NULL : root->search(k); } ` `}; ` ` ` `// Constructor for BTreeNode class ` `BTreeNode::BTreeNode(` `int` `_t, ` `bool` `_leaf) ` `{ ` ` ` `// Copy the given minimum degree and leaf property ` ` ` `t = _t; ` ` ` `leaf = _leaf; ` ` ` ` ` `// Allocate memory for maximum number of possible keys ` ` ` `// and child pointers ` ` ` `keys = ` `new` `int` `[2*t-1]; ` ` ` `C = ` `new` `BTreeNode *[2*t]; ` ` ` ` ` `// Initialize the number of keys as 0 ` ` ` `n = 0; ` `} ` ` ` `// Function to traverse all nodes in a subtree rooted with this node ` `void` `BTreeNode::traverse() ` `{ ` ` ` `// There are n keys and n+1 children, travers through n keys ` ` ` `// and first n children ` ` ` `int` `i; ` ` ` `for` `(i = 0; i < n; i++) ` ` ` `{ ` ` ` `// If this is not leaf, then before printing key[i], ` ` ` `// traverse the subtree rooted with child C[i]. ` ` ` `if` `(leaf == ` `false` `) ` ` ` `C[i]->traverse(); ` ` ` `cout << ` `" "` `<< keys[i]; ` ` ` `} ` ` ` ` ` `// Print the subtree rooted with last child ` ` ` `if` `(leaf == ` `false` `) ` ` ` `C[i]->traverse(); ` `} ` ` ` `// Function to search key k in subtree rooted with this node ` `BTreeNode *BTreeNode::search(` `int` `k) ` `{ ` ` ` `// Find the first key greater than or equal to k ` ` ` `int` `i = 0; ` ` ` `while` `(i < n && k > keys[i]) ` ` ` `i++; ` ` ` ` ` `// If the found key is equal to k, return this node ` ` ` `if` `(keys[i] == k) ` ` ` `return` `this` `; ` ` ` ` ` `// If the key is not found here and this is a leaf node ` ` ` `if` `(leaf == ` `true` `) ` ` ` `return` `NULL; ` ` ` ` ` `// Go to the appropriate child ` ` ` `return` `C[i]->search(k); ` `} ` |

The above code doesn’t contain driver program. We will be covering the complete program in our next post on B-Tree Insertion.

There are two conventions to define a B-Tree, one is to define by minimum degree (followed in Cormen book), second is define by order. We have followed the minimum degree convention and will be following same in coming posts on B-Tree. The variable names used in the above program are also kept same as Cormen book for better readability.

**Insertion and Deletion**

B-Tree Insertion

B-Tree Deletion

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