B and B+ tree
- 1. B AND B+ TREE
- 2. Details Of B Tree
- 3. Definition:- B-Tree is a generalization of a binary search tree designed especially for use on disk. So, B-trees used to represent very large dictionaries that reside on disk. This is the special case of M-Way tree. B-Tree Consists of Root Node, Branch Node and Leaf Node containing indexed field value in ending nodes of the tree. B-trees are a good example of a data structure for external memory. It is commonly used in databases and filesystems. In order to maintain the pre-defined range, internal nodes may be joined or split. Each internal node of a B-tree contains a number of keys. The keys act as separation values which divide its subtrees. B-trees may waste some space (memory) * Since nodes are not entirely full. If the B-tree is balanced, its search / insert / add operations take about log(n) steps.
- 4. Properties of B Tree:- B-Tree of order n Not empty => root has at least 2 children. Remaining internal nodes (if any) have at least ceil (m/2) children. All leaves are at the same depth in the tree. Non-leaf node with k children has k-1 keys. It may contain large number of keys.
- 5. Operation on B+ Tree:- Searching. Insertion. Deletion.
- 6. Searching:- In B-Tree search start from Root node but every time we make 2-way decision. Steps for searching :- Let the key to be searched be k. Search is similar to search in Binary Search Tree. 1. We start from root and recursively traverse down. 2. For every visited non-leaf node, if the node has key, we simply return the node. 3. Otherwise we recur down to the appropriate child (The child which is just before the first greater key) of the node. 4. If we reach a leaf node and don’t find k in the leaf node, we return NULL.
- 7. Insertion:- Steps for insertion :- Let the key to be inserted be k. 1. A new key is always inserted at leaf node. 2. Like BST, we start from root and traverse down till we reach a leaf node. 3. Once we reach a leaf node, we insert the key in that leaf node. Unlike BSTs, we have a predefined range on number of keys that a node can contain. 4. So before inserting a key to node, we make sure that the node has extra space. 5. To insert a new key, we go down from root to leaf. 6. Before traversing down to a node, we first check if the node is full. 7. If the node is full, we split it to create space.
- 8. Insertion Example :- key :- 1,12,8,2,25,6,14,28,17,7,52,16,48,68. Order = 5 Procedure for adding key in b-tree Step1. Step2.
- 10. Deletion:- 1. Deletion from a B-tree is more complicated than insertion, because we can delete a key from any node-not just a leaf. 2. when we delete a key from an internal node, we will have to rearrange the node’s children. 3. Just as we had to ensure that a node didn’t get too big due to insertion 4. we must ensure that a node doesn’t get too small during deletion. 5. a simple approach to deletion might have to back up if a node (other than the root) along the path to where the key is to be deleted has the minimum number of keys. 6. Since most of the keys in a B-tree are in the leaves, deletion operations are most often used to delete keys from leaves. 7. The recursive delete procedure then acts in one downward pass through the tree, without having to back up. 8. When deleting a key in an internal node, however, the procedure makes a downward pass through the tree but may have to return to the node from which the key was deleted to replace the key with its predecessor or successor. 9. No of children must be according to the ceil of m/2.
- 11. Deletion Example:- Given Tree:-
- 12. 6 deleted
- 13. 13 deleted
- 14. 7 deleted
- 15. Analysis Of B Tree For a B-Tree of order M 1. Each internal node has up to M-1 keys to search 2. Each internal node has between M/2 and M children 3. Depth of B-Tree storing N items is O(log M/2 N) Find: Run time is: 1. O(log M) to binary search which branch to take at each node. But M is small compared to N. 2. Total time to find an item is O(depth*log M) = O(log N)
- 16. Details Of B+ Tree
- 17. Definition:- The most commonly implemented form of the B-Tree is the B+-Tree. The order of the B+ tree determines the number of subtrees, which exactly follows the B-Tree definition. A B+ tree is a data structure often used in the implementation of database indexes. Only Leaf nodes store references to data. A leaf node may store more or less data records than an internal node stores keys. It contains index pages and data pages. The data pages always appear as leaf nodes in the tree. The root node and intermediate nodes are always index pages.
- 18. Properties of B+ Tree:- For a B-tree of order m: All data are on the same level. If a B+ tree consists of a single node, then the node is both a root and a leaf. Each internal node other than the root contains between m/2 search keys. The capacity of a leaf has to be 50% or more.
- 19. Operation on B+ Tree:- Searching. Insertion. Deletion.
- 20. Searching:- 1. Searching just like in a binary search tree. 2. Starts at the root, works down to the leaf level. 3. Does a comparison of the search value and the current “separation value”, goes left or right. 4. Since no structure change in a B+ tree during a searching process. 5. So just compare the key value with the data in the tree, then give the result back.
- 21. Insertion:- 1. A search is first performed, using the value to be added. 2. After the search is completed, the location for the new value is known. 3. If the tree is empty, add to the root. 4. Once the root is full, split the data into 2 leaves, using the root to hold keys and pointers. 5. If adding an element will overload a leaf, take the median and split it.
- 22. Insertion Example :- Example #1: insert 28 into the below tree. Result:-
- 23. Example #2: insert 70 into below tree Result:-
- 24. Example #3: insert 95 into below tree Result:-
- 25. Deletion:- 1. Deletion, like insertion, begins with a search. 2. When the item to be deleted is located and removed, the tree must be checked to make sure no rules are violated. 3. The rule to focus on is to ensure that each node has at least ceil (n/2) pointers.
- 26. Deletion Example :- Example #1: delete 70 from the tree Result:-
- 27. Example #2: delete 25 from below tree Result:- Replace 25 with 28 in the index page. On Next Slide…
- 29. Example #3: delete 60 from the below tree Result:-
- 30. Analysis Of B+ Tree For a B+-Tree Searching Operation:- 1. During the search operation, h nodes are read from the disk to the main memory where h is the height of the B+-tree. 2. So, the height of the B+-tree is h = O(logn), and the time complexity of each linear search is O(t), Thus, the total time complexity of the B+-tree search operation is O(t logt n). For a B+-Tree Insertion Operation:- 1. The time complexity of the insertion algorithm is O(t · logtn). 2. where the cofficient t is due to the operations performed for each node in the main memory.
- 31. Analysis Of B+ Tree For a B+-Tree Deletion Operation:- 1. The time complexity of the insertion algorithm is O(t · logtn). 2. Each time we encounter a node having exactly t children (non-leaf node) or keys (leaf node)