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# B- Trees

**Introduction to B- Trees**

**A B – Tree can be defined as a multi-way search tree of order ***m* such that it follows the conditions stated:

- All the internal nodes (except the root node) have at least
children and at most*m/2**m*children. - The non – leaf root node may have
**at most***m*non empty child and at least two child nodes. - In a B- Tree, the root node is the only node that can exist without no child nodes.
**If a node has***m*children then it must have*(m – 1)***values.**All these values of a particular node are in increasing order.- All the values of a node that appear on the leftmost child of the node are smaller than the first value of that node. Similarly, all the values that appear on the rightmost child of a node are greater than the last value of that node.
- If
and*a*are any two*b**i*and*th***(i+1)th values of the node , where***x < y*, then all the values appearing on the (i+1)th**sub tree**of the that node are greater than*x*and less than*y.* - All the leaf nodes should
**appear on the same level.** - The primary goal to use a B-tree is to
**reduce the number of disk access.**Majority of tree operations such as searching, traversal ,insertion, etc, take*O(n)*time, where*n*is the height of the tree. - A B-Tree of degree
*T*can have –*T-1*minimum keys.*2T-1*maximum keys.

**Operations in a B-Tree**

- Insertion.
- Deletion

**Representation of B- Tree**

The structure of a node in B-Tree can be represented as follows:

- struct btree_node
- {
- int c;
- int val
**[M + 1];** - struct btree_node *child
**[M + 1]** - };

- Here
represents the number of children a particular node can have.**c** - Array
represents the values of the node stored in it.**val** - Array
**child** - The macro
**M**

**Creation of a B-Tree:**

Let us consider a B-Tree of order **5**. It means we can have a maximum number of **4 values **that any node can hold.

For reference we see the given image. we can see that any node can have at most 4 values.

But if in any case we are given the degree of a tree; for instance consider **t = 2, **then,

- Minimum number of keys that a tree can hold can be calculated as
**min_key = t – 1=> 2 – 1 => min_key = 1**

- Maximum number of keys that a tree can hold can be calculated as
**max_key = 2t – 1 => 2(2) – 1 => 4 – 1 => max_key = 3**

*20 , 30 , 35 , 85 , 5 , 10 , 40 , 55 , 60 , 38 , 25 , 7 , 78.*

#### Step 1:

The first three elements * 20 , 30 , 35 *are added to a node.

#### Step 2:

The element * 85* is added to the node. Now if we jump back to the previous section we can see that a B-tree with

*can have at most 3 key values. Thus the node is split from the median.*

**degree****t=2****Note: How to calculate median for splitting the node-**

- Let
*x*be a given key values in a node. *n*be the length of the node .**Meadian =***n[x] / 2.*

Hence here we can see the length * n[x] = 4*.

Thus, Median = ** n[x] / 2 **=>

**4 / 2 = 2.**This means that the node will be split from the second key of the node.

#### Step 3:

As the node is split * 30 *becomes the new root node of the tree.

#### Step 4:

The element ** 5 **is inserted.

#### Step 5:

The element * 10* is inserted.

#### Step 6:

The element* 40* is inserted

#### Step 7:

The element * 55* is added to the node.

#### Step 8:

**To conserve the degree of the tree**, we need to split the node wherewas added. To do so we**55****calculate the median**which comes out to be.**2**- The node is split from the 2nd key.
moves upwards to the root node.**40** - The tree structure is shown as follows.

#### Step 9:

The elements * 38 and 60 *are added to the B-Tree.

#### Step 10:

- The element
is added to the node. The node has to split as each node in this tree can have at most 3 keys.**25** - The node is split from the Median, i.e,
**2***.*

#### Step 11:

The node is split from the 2nd key. Thus, 10 moves upwards to the parent / root node.

#### Step 12:

The elements * 7 *and

*are added to the B-Tree*

**78**#### Step 13:

- As the node is split the element
moves upward towards the parent / root node.**60** - Now, again we can see that the
**root node**is to be split because each node can hold at most 3 keys.

#### Step 14:

- Again, we calculate the median of the node. The median can be calculated as follows,
**Median = n[x] / 2.****Median = 4 / 2.****Median = 2.**

- The node is split from the 2nd key which moves upwards and becomes the new root of the tree. The tree structure that follows is shown as follows,

- The final B-Tree after the concurrent insertion and splitting is shown as follows.
- This B-Tree fulfills all the properties of a B-Tree.
- Each node has either
**at least 1 key or at most 3 keys.** - Any node with number of keys exceeding the given number is split and re-arranged accordingly.

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