AVL_BLRT_11
Refer to the AVL Tree shown in image below.
Given that the node with the value 21 has just been inserted, after identifying which node becomes unbalanced first, describe the necessary rotation(s) to restore the tree's balance.

Question

BlackTom AI · Accepted Answer

To reason about the necessary rotations after inserting 21, we must consider how AVL rotations fix imbalances and which case the insertion creates.
Option 1: Left rotation at 20, then right rotation at 25.
- This describes a double rotation (left-right) where the left child (20) is rotated left, followed by a rotation right at the unbalanced node (25).
- If the insertion of 21 causes the subtree rooted at 25 to become left-heavy with a right-heavy grandchild (i.e., a left-right case), performing a left rotation on the left child (20) and then a right rotation on the unbalanced node (25) is the standard fix.
- This sequence is consistent with the typical LR (left-right) rotation pattern and commonly arises when a new node is inserted into the right subtree of the left child.

Option 2: Left rotation at 25, then right rotation at 40.
- This suggests performing rotations on nodes 25 and 40, which does not align with the usual single unbalanced node’s corrective steps in an AVL adjustment after inserting 21.
- The standard LR or RL fixes involve the unbalanced node and its immediate child, not rotations arranged at 25 and a distant node like 40, making this sequence unlikely to restore balance in the typical AVL framework for the described insertion.

Option 3: Single right rotation at 25.
- A single right rotation is the fix for a left-left (LL) imbalance, where the left subtree of the left child is heavy. If the insertion caused a straight LL case, this could be valid.
- However, given the insertion of 21 near the left subtree of 25, it is more characteristic of an LR case rather than a pure LL case, so a single right rotation may not suffice to rebalance in this scenario.

Option 4: Single left rotation at 20.
- A single left rotation targets a right-right (RR) case, where the right subtree of the right child is heavy. If the imbalance were purely RR with 21 causing a right-heavy situation, a single left rotation at 20 could be appropriate.
- In the current context, after inserting 21, the structure described by the other options (especially the need to rotate at 25) suggests the imbalance is not a simple RR case, so a lone left rotation at 20 is unlikely to fully restore balance.

Overall, the LR pattern is the most consistent with a scenario where inserting 21 causes an imbalance that requires first rotating the left subtree (on node 20) to realign the heavier side, followed by a rotation on the unbalanced node (25) to restore balance. The combination described in Option 1 aligns with that standard double-rotation approach, while the other options correspond to other, less likely imbalance patterns or misapplied corrective steps.

AVL_BLRT_11 Refer to the AVL Tree shown in image below. Given that the node with the value 21 has just been inserted, after identifying which node becomes unbalanced first, describe the necessary rotation(s) to restore the tree's balance.

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