In a B-tree of order 5 , the following elements are inserted as follows

3,14,7,1,8,5,11,17,13,6,23,12,20,26,4,16,18,24,25,19

How many elements are present in the root after inserting the element 23?

*
4
3
1
2

To determine how many elements are present in the root of a B-tree of order 5 after inserting the specified elements, we can follow these steps:

1. **B-tree properties**: In a B-tree of order \( m \), each node can contain a maximum of \( m-1 \) keys and must have at least \( \lceil m/2 \rceil - 1 \) keys (except for the root).

2. **Insert elements**: We'll insert the elements one by one and observe when splits occur, which affects how many keys end up in the root.

### Insertions and Splits:

1. **Insert 3**: [3]
2. **Insert 14**: [3, 14]
3. **Insert 7**: [3, 7, 14]
4. **Insert 1**: [1, 3, 7, 14]
5. **Insert 8**: [1, 3, 7, 8, 14] (now full)
6. **Insert 5**: This causes a split:
   - **New Root**: [7]
   - **Left Child**: [1, 3, 5, 6]
   - **Right Child**: [8, 14]

7. **Insert 11**: [8, 11, 14] (in right child)
8. **Insert 17**: [8, 11, 14, 17]
9. **Insert 13**: [8, 11, 13, 14, 17] (now full)
10. **Insert 6**: Causes another split:
   - **Root**: [11]
   - **Left Child**: [7]
   - **Right Child**: [13, 14]

11. **Insert 23**: Added to the right child [13, 14, 17, 23] (now full)
12. **Insert 12**: This causes another split:
   - **Root**: [11, 14]
   - **Left Child**: [7]
   - **Middle Child**: [12]
   - **Right Child**: [17, 23]

### Final Count in Root After Inserting 23

After all these insertions and the associated splits, the root contains **two elements**: 11 and 14.

### Conclusion

Therefore, the number of elements present in the root after inserting the element 23 is **2**.

If you have any questions or need further details, feel free to ask! Here are some follow-up questions you might consider:

1. What happens if we insert more elements into this B-tree?
2. Can you explain how B-trees balance themselves during insertions?
3. How would the structure change if the order of the B-tree was increased to 6?
4. What are the advantages of using B-trees over other data structures like binary search trees?
5. How does deletion in a B-tree affect the number of elements in the root? 

**Tip**: When working with B-trees, always keep track of the maximum and minimum number of keys per node to understand when splits or merges will occur.

In a B-tree of order 5, how many elements are present in the root after inserting the element 23?

Explore the process of inserting elements into a B-tree of order 5 and learn how many elements are present in the root after inserting 23. This detailed solution highlights B-tree properties and insertion methods, ideal for Grades 10-12 students.

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