BTree : Searching and Insertion
Reading time: 15 minutes  Coding time: 9 minutes
A Btree is a tree data structure that keeps data sorted and allows searches, insertions, and deletions in logarithmic amortized time. Unlike selfbalancing binary search trees, it is optimized for systems that read and write large blocks of data. It is most commonly used in database and file systems.
BTree is a selfbalanced search tree with multiple keys in every node and more than two children for every node. Basic properties associated with BTree:
 All the leaf nodes must be at same level.
 All nodes except root must have at least [m/2]1 keys and maximum of m1 keys.
 All non leaf nodes except root (i.e. all internal nodes) must have at least m/2 children.
 If the root node is a non leaf node, then it must have at least 2 children.
 A non leaf node with n1 keys must have n number of children.
 All the key values within a node must be in Ascending Order.
The figure depicts the basic construction of BTree.
Algorithm
 Basic operations associated with BTree:

Searching a node in a BTree
 Perform a binary search on the records in the current node.
 If a record with the search key is found, then return that record.
 If the current node is a leaf node and the key is not found, then report an unsuccessful search.
 Otherwise, follow the proper branch and repeat the process.

Insertion Operation of a node in a BTree depending on two cases:
 x is a leaf node. Then we find where k belongs in the array of keys, shift everything over to the left, and stick k in there.

x is not a leaf node:
 We can't just stick k in because it doesn't have any children; children are really only created when we split a node, so we don't get an unbalanced tree.
 We find a child of x where we can (recursively) insert k.
 We read that child in from disk. If that child is full, we split it and figure out which one k belongs in.
 Then we recursively insert k into this child (which we know is nonfull, because if it were, we would have split it).
The figure illustrates the steps of insertion of an element in a BTree.
Complexity

Worst case search time complexity:
Θ(logn)

Average case search time complexity:
Θ(logn)

Best case search time complexity:
Θ(logn)

Average case Space complexity:
Θ(n)

Worst case Space complexity:
Θ(n)
Implementations
#include<stdio.h>
#include<conio.h>
#include<iostream>
using namespace std;
struct BTreeNode
{
int *data;
BTreeNode **child_ptr;
bool leaf;
int n;
}*root = NULL, *np = NULL, *x = NULL;
BTreeNode * init()
{
int i;
np = new BTreeNode;
np>data = new int[5];
np>child_ptr = new BTreeNode *[6];
np>leaf = true;
np>n = 0;
for (i = 0; i < 6; i++)
{
np>child_ptr[i] = NULL;
}
return np;
}
void searchOrTraverse(BTreeNode *p)
{
cout<<endl;
int i;
for (i = 0; i < p>n; i++)
{
if (p>leaf == false)
{
searchOrTraverse(p>child_ptr[i]);
}
cout << " " << p>data[i];
}
if (p>leaf == false)
{
searchOrTraverse(p>child_ptr[i]);
}
cout<<endl;
}
void sort(int *p, int n)
{
int i, j, temp;
for (i = 0; i < n; i++)
{
for (j = i; j <= n; j++)
{
if (p[i] > p[j])
{
temp = p[i];
p[i] = p[j];
p[j] = temp;
}
}
}
}
int childSplitOp(BTreeNode *x, int i)
{
int j, mid;
BTreeNode *np1, *np3, *y;
np3 = init();
np3>leaf = true;
if (i == 1)
{
mid = x>data[2];
x>data[2] = 0;
x>n;
np1 = init();
np1>leaf = false;
x>leaf = true;
for (j = 3; j < 5; j++)
{
np3>data[j  3] = x>data[j];
np3>child_ptr[j  3] = x>child_ptr[j];
np3>n++;
x>data[j] = 0;
x>n;
}
for (j = 0; j < 6; j++)
{
x>child_ptr[j] = NULL;
}
np1>data[0] = mid;
np1>child_ptr[np1>n] = x;
np1>child_ptr[np1>n + 1] = np3;
np1>n++;
root = np1;
}
else
{
y = x>child_ptr[i];
mid = y>data[2];
y>data[2] = 0;
y>n;
for (j = 3; j < 5; j++)
{
np3>data[j  3] = y>data[j];
np3>n++;
y>data[j] = 0;
y>n;
}
x>child_ptr[i + 1] = y;
x>child_ptr[i + 1] = np3;
}
return mid;
}
void insertionOp(int a)
{
int i, temp;
x = root;
if (x == NULL)
{
root = init();
x = root;
}
else
{
if (x>leaf == true && x>n == 5)
{
temp = childSplitOp(x, 1);
x = root;
for (i = 0; i < (x>n); i++)
{
if ((a > x>data[i]) && (a < x>data[i + 1]))
{
i++;
break;
}
else if (a < x>data[0])
{
break;
}
else
{
continue;
}
}
x = x>child_ptr[i];
}
else
{
while (x>leaf == false)
{
for (i = 0; i < (x>n); i++)
{
if ((a > x>data[i]) && (a < x>data[i + 1]))
{
i++;
break;
}
else if (a < x>data[0])
{
break;
}
else
{
continue;
}
}
if ((x>child_ptr[i])>n == 5)
{
temp = childSplitOp(x, i);
x>data[x>n] = temp;
x>n++;
continue;
}
else
{
x = x>child_ptr[i];
}
}
}
}
x>data[x>n] = a;
sort(x>data, x>n);
x>n++;
}
int main()
{
int i, n, t;
cout<<"Please enter the number elements for insertion operation\n";
cin>>n;
for(i = 0; i < n; i++)
{
cout<<"Please enter the value\n";
cin>>t;
insertionOp(t);
}
cout<<"Traversing the constructed tree\n";
searchOrTraverse(root);
getch();
}
Applications
BTree has many important applications in computer science.

Btrees are preferred when decision points, called nodes, are on hard disk rather than in randomaccess memory (RAM)

Btrees save time by using nodes with many branches (called children), compared with binary trees, in which each node has only two children, thereby speeding up the process.