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<!DOCTYPE html>

<head>

window.dataLayer = window.dataLayer || [];

function gtag() { dataLayer.push(arguments); }

gtag('js', new Date());

gtag('config', 'G-HCERBDV76D');

</script>

</head>

<body>

<nav>

Python Tutorial

</a>

</div>

</div>

</div>

<li>

Home

</a>

</li>

<li>

Projects

</a>

</li>

<li>

Free Course

</a>

</li>

<li>

DSA

</a>

</li>

</ul>

</div>

</nav>

</div>

<h3>Page Index</h3>

<ul>

<li><a href="/dsa.html#data-structure-1" title="Data Structures (I)">Data Structures

(I)</a></li>

<li><a href="/dsa.html#data-structure-2" title="Data Structures (II)">Data Structures

(II)</a></li>

<li><a href="/dsa.html#tree-1" title="Tree based DSA (I)">Tree based DSA (I)</a></li>

<li><a href="/dsa.html#tree-2" title="Tree based DSA (II)">Tree based DSA (II)</a></li>

<li><a href="/dsa.html#graph" title="Graph Data Structures and Algorithm">Graph based

DSA</a></li>

<li><a href="/dsa.html#sorting-searching" title="Sorting and Searching">Sorting and

Searching</a></li>

<li><a href="/dsa.html#greedy-algorithm" title="Greedy Algorithms">Greedy Algorithms</a>

</li>

<li><a href="/dsa.html#dynamic-programming" title="Dynamic Programming">Dynamic

Programming</a></li>

<li><a href="/dsa.html#other-algorithms" title="Other Algorithms">Other Algorithms</a>

</li>

</ul>

</div>

In this tutorial, you will learn what a B-tree is. Also, you will find working examples of search operation on a B-tree in C, C++, Java and Python.

B-tree is a special type of self-balancing search tree in which each node can contain more than one key and can have more than two children. It is a generalized form of the <a href="/dsa/binary-search-tree.html">binary search tree</a>.

It is also known as a height-balanced m-way tree.

<figure><img alt="B-tree example" src="//cdn.programiz.com/sites/tutorial2program/files/b-tree.png" title="B-tree" width="588" height="216"><figcaption>B-tree</figcaption></figure><hr><h2 id="why">Why do you need a B-tree data strcuture?</h2>

The need for B-tree arose with the rise in the need for lesser time in accessing the physical storage media like a hard disk. The secondary storage devices are slower with a larger capacity. There was a need for such types of data structures that minimize the disk accesses.

Other data structures such as a binary search tree, avl tree, red-black tree, etc can store only one key in one node. If you have to store a large number of keys, then the height of such trees becomes very large and the access time increases.

However, B-tree can store many keys in a single node and can have multiple child nodes. This decreases the height significantly allowing faster disk accesses.

<hr><h2 id="properties">B-tree Properties</h2>

<ol><li>For each node <var>x</var>, the keys are stored in increasing order.</li>

<li>In each node, there is a boolean value <var>x.leaf</var> which is true if <var>x</var> is a leaf.</li>

<li>If <var>n</var> is the order of the tree, each internal node can contain at most <var>n - 1</var> keys along with a pointer to each child.</li>

<li>Each node except root can have at most n children and at least <var>n/2</var> children.</li>

<li>All leaves have the same depth (i.e. height-h of the tree).</li>

<li>The root has at least 2 children and contains a minimum of 1 key.</li>

<li>If <var>n ≥ 1</var>, then for any n-key B-tree of height h and minimum degree <code>t ≥ 2</code>, <code>h ≥ logt (n+1)/2</code>.</li>

</ol><hr><h2 id="operations">Operations on a B-tree</h2>

<h3>Searching an element in a B-tree</h3>

Searching for an element in a B-tree is the generalized form of searching an element in a Binary Search Tree. The following steps are followed.

<ol><li>Starting from the root node, compare k with the first key of the node.

If <code>k = the first key of the node</code>, return the node and the index.</li>

<li>If <code>k.leaf = true</code>, return <var>NULL</var> (i.e. not found).</li>

<li>If <code>k < the first key of the root node</code>, search the left child of this key recursively.</li>

<li>If there is more than one key in the current node and <code>k > the first key</code>, compare k with the next key in the node.

If <code>k < next key</code>, search the left child of this key (ie. k lies in between the first and the second keys).

Else, search the right child of the key.</li>

<li>Repeat steps 1 to 4 until the leaf is reached.</li>

</ol><h3>Searching Example</h3>

<ol><li>Let us search key <code>k = 17</code> in the tree below of degree 3.

<li><var>k</var> is not found in the root so, compare it with the root key.

<figure><img alt="Not found on the root node" src="//cdn.programiz.com/sites/tutorial2program/files/search-2.png" title="Not found on the root node" width="368" height="216"><figcaption>k is not found on the root node</figcaption></figure></li>

<li>Since <code>k > 11</code>, go to the right child of the root node.

<figure><img alt="Go to the right subtree" src="//cdn.programiz.com/sites/tutorial2program/files/search-3.png" title="Go to the right subtree" width="368" height="216"><figcaption>Go to the right subtree</figcaption></figure></li>

<li>Compare k with 16. Since <code>k > 16</code>, compare k with the next key 18.

<figure><img alt="Compare with the keys from left to right" src="//cdn.programiz.com/sites/tutorial2program/files/search-4.png" title="Compare with the keys from left to right" width="368" height="216"><figcaption>Compare with the keys from left to right</figcaption></figure></li>

<li>Since <code>k < 18</code>, k lies between 16 and 18. Search in the right child of 16 or the left child of 18.

<figure><img alt="k lies in between 16 and 18" src="//cdn.programiz.com/sites/tutorial2program/files/search-5.png" title="k lies in between 16 and 18" width="368" height="216"><figcaption>k lies in between 16 and 18</figcaption></figure></li>

<li>k is found.

<figure><img alt="k is found" src="//cdn.programiz.com/sites/tutorial2program/files/search-6.png" title="k is found" width="368" height="216"><figcaption>k is found</figcaption></figure></li>

</ol><hr><h2 id="algorithm">Algorithm for Searching an Element</h2>

<div class="pre-code-wrapper"><div title="Click to copy" class="copy-code-button"></div><pre class="unspecified-exec" style="max-height: 600px;"><code class="dsa hljs kotlin">BtreeSearch(x, k)

i = 1

while i ≤ n[x] and k ≥ keyi[x] // n[x] means number of keys in x node

do i = i + 1

if i n[x] and k = keyi[x]

then return (x, i)

if leaf [x]

then return NIL

else

return BtreeSearch(ci[x], k)

</code></pre></div>

To learn more about different B-tree operations, please visit

<ul><li><a href="/dsa/insertion-into-a-b-tree.html">Insertion on B-tree</a></li>

<li><a href="/dsa/deletion-from-a-b-tree.html">Deletion on B-tree</a></li>

</ul><hr><h2 id="code">B-tree operations code in Python, Java, and C/C++</h2>

<div id="py1" onclick="changepy()" class="tabbed-editor__node tabbed-editor__node--active"><a href="#python-code">Python</a></div>

</div>

<div class="pre-code-wrapper"><div title="Click to copy" class="copy-code-button"></div><pre class="exec" style="max-height: 600px;"><code class="python hljs"># Searching a key on a B-tree in Python

# Create a node

class BTreeNode:

def __init__(self, leaf=False):

self.leaf = leaf

self.keys = []

self.child = []

# Tree

class BTree:

def __init__(self, t):

self.root = BTreeNode(True)

self.t = t

# Insert node

def insert(self, k):

root = self.root

if len(root.keys) == (2 * self.t) - 1:

temp = BTreeNode()

self.root = temp

temp.child.insert(0, root)

self.split_child(temp, 0)

self.insert_non_full(temp, k)

else:

self.insert_non_full(root, k)

# Insert nonfull

def insert_non_full(self, x, k):

i = len(x.keys) - 1

if x.leaf:

x.keys.append((None, None))

while i >= 0 and k[0] < x.keys[i][0]:

x.keys[i + 1] = x.keys[i]

i -= 1

x.keys[i + 1] = k

else:

i -= 1

i += 1

if len(x.child[i].keys) == (2 * self.t) - 1:

self.split_child(x, i)

if k[0] > x.keys[i][0]:

i += 1

self.insert_non_full(x.child[i], k)

# Split the child

def split_child(self, x, i):

t = self.t

y = x.child[i]

z = BTreeNode(y.leaf)

x.child.insert(i + 1, z)

x.keys.insert(i, y.keys[t - 1])

z.keys = y.keys[t: (2 * t) - 1]

y.keys = y.keys[0: t - 1]

if not y.leaf:

z.child = y.child[t: 2 * t]

y.child = y.child[0: t - 1]

# Print the tree

def print_tree(self, x, l=0):

print("Level ", l, " ", len(x.keys), end=":")

for i in x.keys:

print(i, end=" ")

print()

l += 1

if len(x.child) > 0:

for i in x.child:

self.print_tree(i, l)

# Search key in the tree

def search_key(self, k, x=None):

if x is not None:

i = 0

while i < len(x.keys) and k > x.keys[i][0]:

i += 1

if i < len(x.keys) and k == x.keys[i][0]:

return (x, i)

elif x.leaf:

return None

else:

return self.search_key(k, x.child[i])

else:

return self.search_key(k, self.root)

def main():

B = BTree(3)

for i in range(10):

B.insert((i, 2 * i))

B.print_tree(B.root)

if B.search_key(8) is not None:

print("\nFound")

else:

print("\nNot Found")

if __name__ == '__main__':

main()</code></pre></div>

</div>

// Searching a key on a B-tree in Java

public class BTree {

private int T;

// Node creation

public class Node {

int n;

int key[] = new int[2 * T - 1];

Node child[] = new Node[2 * T];

boolean leaf = true;

public int Find(int k) {

for (int i = 0; i < this.n; i++) {

if (this.key[i] == k) {

return i;

}

return -1;

};

}

public BTree(int t) {

T = t;

root = new Node();

root.n = 0;

root.leaf = true;

}

private Node root;

// Search key

private Node Search(Node x, int key) {

int i = 0;

if (x == null)

return x;

for (i = 0; i < x.n; i++) {

if (key < x.key[i]) {

break;

}

if (key == x.key[i]) {

return x;

}

if (x.leaf) {

return null;

} else {

return Search(x.child[i], key);

}

// Splitting the node

private void Split(Node x, int pos, Node y) {

Node z = new Node();

z.leaf = y.leaf;

z.n = T - 1;

for (int j = 0; j < T - 1; j++) {

z.key[j] = y.key[j + T];

}

if (!y.leaf) {

for (int j = 0; j < T; j++) {

z.child[j] = y.child[j + T];

}

y.n = T - 1;

for (int j = x.n; j >= pos + 1; j--) {

x.child[j + 1] = x.child[j];

}

x.child[pos + 1] = z;

for (int j = x.n - 1; j >= pos; j--) {

x.key[j + 1] = x.key[j];

}

x.key[pos] = y.key[T - 1];

x.n = x.n + 1;

}

// Inserting a value

public void Insert(final int key) {

Node r = root;

if (r.n == 2 * T - 1) {

Node s = new Node();

root = s;

s.leaf = false;

s.n = 0;

s.child[0] = r;

Split(s, 0, r);

insertValue(s, key);

} else {

insertValue(r, key);

}

// Insert the node

final private void insertValue(Node x, int k) {

if (x.leaf) {

int i = 0;

for (i = x.n - 1; i >= 0 && k < x.key[i]; i--) {

x.key[i + 1] = x.key[i];

}

x.key[i + 1] = k;

x.n = x.n + 1;

} else {

int i = 0;

for (i = x.n - 1; i >= 0 && k < x.key[i]; i--) {

}

;

i++;

Node tmp = x.child[i];

if (tmp.n == 2 * T - 1) {

Split(x, i, tmp);

if (k > x.key[i]) {

i++;

}

insertValue(x.child[i], k);

}

public void Show() {

Show(root);

}

// Display

private void Show(Node x) {

assert (x == null);

for (int i = 0; i < x.n; i++) {

System.out.print(x.key[i] + " ");

}

if (!x.leaf) {

for (int i = 0; i < x.n + 1; i++) {

Show(x.child[i]);

}

// Check if present

public boolean Contain(int k) {

if (this.Search(root, k) != null) {

return true;

} else {

return false;

}

public static void main(String[] args) {

BTree b = new BTree(3);

b.Insert(8);

b.Insert(9);

b.Insert(10);

b.Insert(11);

b.Insert(15);

b.Insert(20);

b.Insert(17);

b.Show();

if (b.Contain(12)) {

System.out.println("\nfound");

} else {

System.out.println("\nnot found");

}

;

}

}</code></pre></div>

</div>

<div class="pre-code-wrapper"><div title="Click to copy" class="copy-code-button"></div><pre class="exec" style="max-height: 600px;"><code class="c hljs cpp">// Searching a key on a B-tree in C

#include <stdio.h>

#include <stdlib.h>

#define MAX 3

#define MIN 2

struct BTreeNode {

int val[MAX + 1], count;

struct BTreeNode *link[MAX + 1];

};

struct BTreeNode *root;

// Create a node

struct BTreeNode *createNode(int val, struct BTreeNode *child) {

struct BTreeNode *newNode;

newNode = (struct BTreeNode *)malloc(sizeof(struct BTreeNode));

newNode->val[1] = val;

newNode->count = 1;

newNode->link[0] = root;

newNode->link[1] = child;

return newNode;

}

// Insert node

void insertNode(int val, int pos, struct BTreeNode *node,

struct BTreeNode *child) {

int j = node->count;

while (j > pos) {

node->val[j + 1] = node->val[j];

node->link[j + 1] = node->link[j];

j--;

}

node->val[j + 1] = val;

node->link[j + 1] = child;

node->count++;

}

// Split node

void splitNode(int val, int *pval, int pos, struct BTreeNode *node,

struct BTreeNode *child, struct BTreeNode **newNode) {

int median, j;

if (pos > MIN)

median = MIN + 1;

else

median = MIN;

*newNode = (struct BTreeNode *)malloc(sizeof(struct BTreeNode));

j = median + 1;

while (j <= MAX) {

(*newNode)->val[j - median] = node->val[j];

(*newNode)->link[j - median] = node->link[j];

j++;

}

node->count = median;

(*newNode)->count = MAX - median;

if (pos <= MIN) {

insertNode(val, pos, node, child);

} else {

insertNode(val, pos - median, *newNode, child);

}

*pval = node->val[node->count];

(*newNode)->link[0] = node->link[node->count];

node->count--;

}

// Set the value

int setValue(int val, int *pval,

struct BTreeNode *node, struct BTreeNode **child) {

int pos;

if (!node) {

*pval = val;

*child = NULL;

return 1;

}

if (val < node->val[1]) {

pos = 0;

} else {

for (pos = node->count;

(val < node->val[pos] && pos > 1); pos--)

;

if (val == node->val[pos]) {

printf("Duplicates are not permitted\n");

return 0;

}

if (setValue(val, pval, node->link[pos], child)) {

if (node->count < MAX) {

insertNode(*pval, pos, node, *child);

} else {

splitNode(*pval, pval, pos, node, *child, child);

return 1;

}

return 0;

}

// Insert the value

void insert(int val) {

int flag, i;

struct BTreeNode *child;

flag = setValue(val, &i, root, &child);

if (flag)

root = createNode(i, child);

}

// Search node

void search(int val, int *pos, struct BTreeNode *myNode) {

if (!myNode) {

return;

}

if (val < myNode->val[1]) {

*pos = 0;

} else {

for (*pos = myNode->count;

(val < myNode->val[*pos] && *pos > 1); (*pos)--)

;

if (val == myNode->val[*pos]) {

printf("%d is found", val);

return;

}

search(val, pos, myNode->link[*pos]);

return;

}

// Traverse then nodes

void traversal(struct BTreeNode *myNode) {

int i;

if (myNode) {

for (i = 0; i < myNode->count; i++) {

traversal(myNode->link[i]);

printf("%d ", myNode->val[i + 1]);

}

traversal(myNode->link[i]);

}

int main() {

int val, ch;

insert(8);

insert(9);

insert(10);

insert(11);

insert(15);

insert(16);

insert(17);

insert(18);

insert(20);

insert(23);

traversal(root);

printf("\n");

search(11, &ch, root);

}</code></pre></div>

</div>

<div class="pre-code-wrapper"><div title="Click to copy" class="copy-code-button"></div><pre class="exec" style="max-height: 600px;"><code class="cpp hljs">// Searching a key on a B-tree in C++

#include <iostream>

using namespace std;

class TreeNode {

int *keys;

int t;

TreeNode **C;

int n;

bool leaf;

public:

TreeNode(int temp, bool bool_leaf);

void insertNonFull(int k);

void splitChild(int i, TreeNode *y);

void traverse();

TreeNode *search(int k);

friend class BTree;

};

class BTree {

TreeNode *root;

int t;

public:

BTree(int temp) {

root = NULL;

t = temp;

}

void traverse() {

if (root != NULL)

root->traverse();

}

TreeNode *search(int k) {

return (root == NULL) ? NULL : root->search(k);

}

void insert(int k);

};

TreeNode::TreeNode(int t1, bool leaf1) {

t = t1;

leaf = leaf1;

keys = new int[2 * t - 1];

C = new TreeNode *[2 * t];

n = 0;

}

void TreeNode::traverse() {

int i;

for (i = 0; i < n; i++) {

if (leaf == false)

C[i]->traverse();

cout << " " << keys[i];

}

if (leaf == false)

C[i]->traverse();

}

TreeNode *TreeNode::search(int k) {

int i = 0;

while (i < n && k > keys[i])

i++;

if (keys[i] == k)

return this;

if (leaf == true)

return NULL;

return C[i]->search(k);

}

void BTree::insert(int k) {

if (root == NULL) {

root = new TreeNode(t, true);

root->keys[0] = k;

root->n = 1;

} else {

if (root->n == 2 * t - 1) {

TreeNode *s = new TreeNode(t, false);

s->C[0] = root;

s->splitChild(0, root);

int i = 0;

if (s->keys[0] < k)

i++;

s->C[i]->insertNonFull(k);

root = s;

} else

root->insertNonFull(k);

}

void TreeNode::insertNonFull(int k) {

int i = n - 1;

if (leaf == true) {

while (i >= 0 && keys[i] > k) {

keys[i + 1] = keys[i];

i--;

}

keys[i + 1] = k;

n = n + 1;

} else {

while (i >= 0 && keys[i] > k)

i--;

if (C[i + 1]->n == 2 * t - 1) {

splitChild(i + 1, C[i + 1]);

if (keys[i + 1] < k)

i++;

}

C[i + 1]->insertNonFull(k);

}

void TreeNode::splitChild(int i, TreeNode *y) {

TreeNode *z = new TreeNode(y->t, y->leaf);

z->n = t - 1;

for (int j = 0; j < t - 1; j++)

z->keys[j] = y->keys[j + t];

if (y->leaf == false) {

for (int j = 0; j < t; j++)

z->C[j] = y->C[j + t];

}

y->n = t - 1;

for (int j = n; j >= i + 1; j--)

C[j + 1] = C[j];

C[i + 1] = z;

for (int j = n - 1; j >= i; j--)

keys[j + 1] = keys[j];

keys[i] = y->keys[t - 1];

n = n + 1;

}

int main() {

BTree t(3);

t.insert(8);

t.insert(9);

t.insert(10);

t.insert(11);

t.insert(15);

t.insert(16);

t.insert(17);

t.insert(18);

t.insert(20);

t.insert(23);

cout << "The B-tree is: ";

t.traverse();

int k = 10;

(t.search(k) != NULL) ? cout << endl

<< k << " is found"

: cout << endl

<< k << " is not Found";

k = 2;

(t.search(k) != NULL) ? cout << endl

<< k << " is found"

: cout << endl

<< k << " is not Found\n";

}</code></pre></div>

</div>

<hr><h2 id="complexity">Searching Complexity on B Tree</h2>

<div class="clearfix"></div>Worst case Time complexity: <code>Θ(log n)</code>

Average case Time complexity: <code>Θ(log n)</code>

Best case Time complexity: <code>Θ(log n)</code>

Average case Space complexity: <code>Θ(n)</code>

Worst case Space complexity: <code>Θ(n)</code>

<hr><h2 id="applications">B Tree Applications</h2>

<ul><li>databases and file systems</li>

<li>to store blocks of data (secondary storage media)</li>

<li>multilevel indexing</li>

</ul></div>

</div>

<div class="tutorial-toc"><div class="tutorial-toc__inner"><h3 class="tutorial-toc__title">Table of Contents

<li><a href="#properties">B-tree Properties</a></li>

<li><a href="#operations">Operations</a></li>

<li><a href="#algorithm">Algorithm for Searching an Element</a></li>

<li><a href="#code">Python, Java and C/C++ Examples</a></li>

<li><a href="#complexity">Searching Complexity on B Tree</a></li>

<li><a href="#applications">B+ Tree Applications</a></li>

</ul></div></div></div> </div>

<div>

<ul>