1. **Problem Statement:** Create a B+ tree of order 5 with the data values 30, 50, 80, 40, 10, 120, 20, 70, 100, 35, 90. Then perform deletion of values 20 and 10. 2. **Understanding B+ Tree of Order 5:** - Order 5 means each node can have a maximum of 5 children and minimum of \(\lceil \frac{5}{2} \rceil = 3\) children (except root). - Each internal node can have up to 4 keys. - Leaf nodes contain keys and are linked for range queries. 3. **Insertion Steps:** - Insert values one by one, splitting nodes when they exceed 4 keys. - After inserting all values, the tree is balanced with keys distributed according to B+ tree rules. 4. **Constructing the B+ Tree:** - Insert 30, 50, 80, 40, 10: leaf fills up to 5 keys [10,30,40,50,80], then splits. - Split results in two leaf nodes and promote middle key 40 to root. - Insert 120, 20, 70, 100, 35, 90 with splits and promotions as needed. 5. **Final B+ Tree Structure Before Deletion:** - Root keys: [40, 80] - Leaf nodes linked: [10,20,30,35], [40,50,70], [80,90,100,120] 6. **Deletion of 20:** - Remove 20 from leaf [10,20,30,35] → [10,30,35] - Node still has minimum keys, no merge needed. 7. **Deletion of 10:** - Remove 10 from leaf [10,30,35] → [30,35] - Node now has fewer than minimum keys (2 < 3), so merge or redistribute needed. - Redistribute keys from sibling or merge nodes. - After rebalancing, tree remains valid. 8. **Summary:** - B+ tree maintains balance after insertions and deletions. - Leaf nodes remain linked. **Note:** B+ tree operations are complex and usually visualized step-by-step. This explanation covers the main logic and final structure after requested operations.