AVL_BLRT_6
Refer to the AVL Tree shown in image below.
Given that the node with the value 60 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

Question restatement: The AVL tree shown has just inserted the value 60. We are asked to identify which node becomes unbalanced first and describe the rotation sequence needed to restore balance.

Option 1: Single Right rotation on 76.
- Why this is unlikely: a single right rotation on 76 would address a left-left imbalance directly at 76, but after inserting 60, the first unbalanced node and the typical pattern for a deeper right-heavy subtree do not align with a simple right rotation on 76. The scenario described by a single right rotation on 76 does not reflect the common first-step fix for an insertion that propagates balance violations from the inserted node up the tree.

Option 2: Left rotation on 55, then right rotation on 60.
- Why this is incorrect: this sequence involves rotating on 60 after the left rotation on 55, which is unusual since 60 is the newly inserted value and would not typically be the pivot for the second step. The standard AVL double-rotation sequence targets a higher ancestor (the grandparent or an ancestor well above the insertion point), not a rotation centered on the newly inserted node itself. This option also describes an order that does not correspond to the common LR (left-right) or RL (right-left) patterns expected given a single insertion.

Option 3: Left rotation on 55, then right rotation on 76.
- Why this is correct: this describes a double-rotation sequence that commonly resolves a left-right imbalance that can be introduced by inserting a value like 60 in a subtree rooted at 55 with a distant right-leaning ancestor such as 76. First, performing a left rotation on 55 handles the left subtree's misalignment, and then a right rotation on 76 rebalances the overall subtree rooted higher up, bringing the tree back into AVL balance. This two-step LR-type correction matches the typical response pattern for a first unbalanced node formed by an insertion that propagates through multiple levels.

Option 4: Single Left rotation on 47.
- Why this is incorrect: a single left rotation on 47 would address a right-heavy situation at 47, but it doesn’t align with the described scenario where 60 has just been inserted and the unbalance is suggested to propagate through nodes such as 55 and 76. A single rotation on 47 would not properly rebalance the tree after the specific insertion path implied by the question.

Summary observation: Among the four options, only the sequence described in option 3 corresponds to a plausible first-unbalanced-node scenario after inserting 60 and uses the standard double-rotation (left rotation on the lower node followed by a right rotation on the higher ancestor) to restore balance. The other options either propose incorrect rotation directions, apply rotations at inappropriate nodes, or describe rotations that do not match the insertion-driven imbalance pattern.

AVL_BLRT_6 Refer to the AVL Tree shown in image below. Given that the node with the value 60 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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