B Tree is a specialized m-way tree that can be widely used for disk access. A B-Tree of order m can have at most m-1 keys and m children. One of the main reason of using B tree is its capability to store large number of keys in a single node and large key values by keeping the height of the tree relatively small.
A B tree of order m contains all the properties of an M way tree. In addition, it contains the following properties.
It is not necessary that, all the nodes contain the same number of children but, each node must have m/2 number of nodes.
A B tree of order 4 is shown in the following image.
While performing some operations on B Tree, any property of B Tree may violate such as number of minimum children a node can have. To maintain the properties of B Tree, the tree may split or join.
Searching in B Trees is similar to that in Binary search tree. For example, if we search for an item 49 in the following B Tree. The process will something like following :
Searching in a B tree depends upon the height of the tree. The search algorithm takes O(log n) time to search any element in a B tree.
Insertions are done at the leaf node level. The following algorithm needs to be followed in order to insert an item into B Tree.
Insert the node 8 into the B Tree of order 5 shown in the following image.
8 will be inserted to the right of 5, therefore insert 8.
The node, now contain 5 keys which is greater than (5 -1 = 4 ) keys. Therefore split the node from the median i.e. 8 and push it up to its parent node shown as follows.
Deletion is also performed at the leaf nodes. The node which is to be deleted can either be a leaf node or an internal node. Following algorithm needs to be followed in order to delete a node from a B tree.
If the the node which is to be deleted is an internal node, then replace the node with its in-order successor or predecessor. Since, successor or predecessor will always be on the leaf node hence, the process will be similar as the node is being deleted from the leaf node.
Delete the node 53 from the B Tree of order 5 shown in the following figure.
53 is present in the right child of element 49. Delete it.
Now, 57 is the only element which is left in the node, the minimum number of elements that must be present in a B tree of order 5, is 2. it is less than that, the elements in its left and right sub-tree are also not sufficient therefore, merge it with the left sibling and intervening element of parent i.e. 49.
The final B tree is shown as follows.
Application of B tree
B tree is used to index the data and provides fast access to the actual data stored on the disks since, the access to value stored in a large database that is stored on a disk is a very time consuming process.
Searching an un-indexed and unsorted database containing n key values needs O(n) running time in worst case. However, if we use B Tree to index this database, it will be searched in O(log n) time in worst case.