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# B-tree

#### 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 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 binary search tree.

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

## Why B-tree?

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.

## B-tree Properties

- For each node
`x`, the keys are stored in increasing order. - In each node, there is a boolean value
`x.leaf`which is true if`x`is a leaf. - If
`n`is the order of the tree, each internal node can contain at most`n – 1`keys along with a pointer to each child. - Each node except root can have at most n children and at least
`n/2`children. - All leaves have the same depth (i.e. height-h of the tree).
- The root has at least 2 children and contains a minimum of 1 key.
- If
`n ≥ 1`, then for any n-key B-tree of height h and minimum degree`t ≥ 2`

,`h ≥ log`

._{t}(n+1)/2

## Operations

### Searching

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.

- Starting from the root node, compare k with the first key of the node.

If`k = the first key of the node`

, return the node and the index. - If
`k.leaf = true`

, return`NULL`(i.e. not found). - If
`k < the first key of the root node`

, search the left child of this key recursively. - If there is more than one key in the current node and
`k > the first key`

, compare k with the next key in the node.

If`k < next key`

, 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. - Repeat steps 1 to 4 until the leaf is reached.

### Searching Example

- Let us search key
`k = 17`

in the tree below of degree 3.

`k`is not found in the root so, compare it with the root key.

- Since
`k > 11`

, go to the right child of the root node.

- Compare k with 16. Since
`k > 16`

, compare k with the next key 18.

- Since
`k < 18`

, k lies between 16 and 18. Search in the right child of 16 or the left child of 18.

- k is found.

## Algorithm for Searching an Element

```
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)
```

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

- Insertion on B-tree
- Deletion on B-tree

## Python Examples

```
/* Searching a key on a B-tree in Python */
/* Create node */
class BTreeNode:
def __init__(self, leaf=False):
self.leaf = leaf
self.keys = []
self.child = []
class BTree:
def __init__(self, t):
self.root = BTreeNode(True)
self.t = t
/* 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 */
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)
/* Insert the key */
def insert_key(self, k):
root = self.root
if len(root.keys) == (2 * self.t) - 1:
temp = BTreeNode()
self.root = temp
temp.child.insert_key(0, root)
self.split(temp, 0)
self.insert_non_full(temp, k)
else:
self.insert_non_full(root, k)
/* Insert non full condition */
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:
while i >= 0 and k[0] < x.keys[i][0]:
i -= 1
i += 1
if len(x.child[i].keys) == (2 * self.t) - 1:
self.split(x, i)
if k[0] > x.keys[i][0]:
i += 1
self.insert_non_full(x.child[i], k)
/* Split */
def split(self, x, i):
t = self.t
y = x.child[i]
z = BTreeNode(y.leaf)
x.child.insert_key(i + 1, z)
x.keys.insert_key(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]
def main():
B = BTree(3)
for i in range(10):
B.insert_key((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()
```

## Searching Complexity on B Tree

Worst case Time complexity: `Θ(log n)`

Average case Time complexity: `Θ(log n)`

Best case Time complexity: `Θ(log n)`

Average case Space complexity: `Θ(n)`

Worst case Space complexity: `Θ(n)`

## B Tree Applications

- databases and file systems
- to store blocks of data (secondary storage media)
- multilevel indexing

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