# B - Trees Datastructure

In a binary search tree, AVL Tree, Red-Black tree etc., every node can have only one value (key) and maximum of two children but there is another type of search tree called B-Tree in which a node can store more than one value (key) and it can have more than two children. B-Tree was developed in the year of 1972 by **Bayer and McCreight** with the name * Height Balanced m-way Search Tree*. Later it was named as B-Tree.

B-Tree can be defined as follows...

B-Tree is a self-balanced search tree with multiple keys in every node and more than two children for every node.

Here, number of keys in a node and number of children for a node is depend on the order of the B-Tree. Every B-Tree has order.**B-Tree of Order m** has the following properties...

**Property #1**- All the**leaf nodes**must be**at same level**.**Property #2**- All nodes except root must have at least**[m/2]-1**keys and maximum of**m-1**keys.**Property #3**- All non leaf nodes except root (i.e. all internal nodes) must have at least**m/2**children.**Property #4**- If the root node is a non leaf node, then it must have**at least 2**children.**Property #5**- A non leaf node with**n-1**keys must have**n**number of children.**Property #6**- All the**key values within a node**must be in**Ascending Order**.

For example, B-Tree of Order 4 contains maximum 3 key values in a node and maximum 4 children for a node.

##### Example

# Operations on a B-Tree

The following operations are performed on a B-Tree...

- Search
- Insertion
- Deletion

# Search Operation in B-Tree

In a B-Ttree, the search operation is similar to that of Binary Search Tree. In a Binary search tree, the search process starts from the root node and every time we make a 2-way decision (we go to either left subtree or right subtree). In B-Tree also search process starts from the root node but every time we make n-way decision where n is the total number of children that node has. In a B-Ttree, the search operation is performed with **O(log n)** time complexity. The search operation is performed as follows...

- Step 1 - Read the search element from the user
- Step 2 - Compare, the search element with first key value of root node in the tree.
- Step 3 - If both are matching, then display "Given node found!!!" and terminate the function
- Step 4 - If both are not matching, then check whether search element is smaller or larger than that key value.
- Step 5 - If search element is smaller, then continue the search process in left subtree.
- Step 6 - If search element is larger, then compare with next key value in the same node and repeate step 3, 4, 5 and 6 until we found exact match or comparision completed with last key value in a leaf node.
- Step 7 - If we completed with last key value in a leaf node, then display "Element is not found" and terminate the function.

# Insertion Operation in B-Tree

In a B-Tree, the new element must be added only at leaf node. That means, always the new keyValue is attached to leaf node only. The insertion operation is performed as follows...

- Step 1 - Check whether tree is Empty.
- Step 2 - If tree is
**Empty**, then create a new node with new key value and insert into the tree as a root node. - Step 3 - If tree is
**Not Empty**, then find a leaf node to which the new key value cab be added using Binary Search Tree logic. - Step 4 - If that leaf node has an empty position, then add the new key value to that leaf node by maintaining ascending order of key value within the node.
- Step 5 - If that leaf node is already full, then
**split**that leaf node by sending middle value to its parent node. Repeat tha same until sending value is fixed into a node. - Step 6 - If the spilting is occuring to the root node, then the middle value becomes new root node for the tree and the height of the tree is increased by one.

##### Example

Construct a **B-Tree of Order 3** by inserting numbers from 1 to 10.