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.