B-Tree

A B-Tree is a data structure used for organizing large amounts of data on disk. It is designed to optimize disk I/O operations by minimizing the number of disk accesses required to locate a desired node. B-Trees are balanced and have multiple child nodes.

Building a B-Tree

To build a B-Tree, we start with an empty root node and add values to the tree in ascending order. The maximum number of pointers in a B-Tree is determined by the order of the tree, which is the maximum number of child nodes a parent node can have.

For example, a B-Tree of order 4 can have a maximum of 4 child nodes for each parent node.

Solution

We are given the following set of key values: (2, 3, 5, 7, 11, 17, 19, 23, 29, 31)

We need to construct a B-Tree for this set of values, assuming that the tree is initially empty and values are added in ascending order. We also need to construct B-Trees for the cases where the number of pointers is 4 and determine which element is the root node of the tree.

B-Tree of Order 4

The maximum number of pointers in a B-Tree of order 4 is 4.

To construct a B-Tree of order 4 for the given set of key values, we follow these steps:

1. Insert 2 as the root node of the tree.

2. Insert 3 as a child node of the root.

3. Insert 5 as a child node of the root.

4. Insert 7 as a child node of the root.

5. Insert 11 as a child node of the root.

6. Split the root node into two nodes and promote 5 to the root.

7. Insert 2 and 3 as child nodes of the left node.

8. Insert 7 and 11 as child nodes of the right node.

9. Split the left node into two nodes and promote 3 to the root.

10. Insert 2 as a child node of the left node.

11. Insert 5 as a child node of the right node.

12. Split the right node into two nodes and promote 7 to the root.

13. Insert 11 as a child node of the right node.

14. Insert 17 as a child node of the right node.

15. Split the right node into two nodes and promote 11 to the root.

16. Insert 7 as a child node of the left node.

17. Insert 17 and 19 as child nodes of the right node.

18. Split the right node into two nodes and promote 17 to the root.

19. Insert 11 and 19 as child nodes of the right node.

20. Insert 23 as a child node of the right node.

21. Split the right node into two nodes and promote 19 to the root.

22. Insert 17 and 23 as child nodes of the right node.

23. Insert 29 as a child node of the right node.

24. Split the right node into two nodes and promote 23 to the root.

25. Insert 19 and 29 as child nodes of the