Consider the following tree:

This tree resulted from an operation on an AVL tree and is no longer an AVL tree. The critical node is the node containing 42.

What type of imbalance is is exhibited at the critical node? RR

What type of rotation is needed to repair the imbalance? Left

Question

Consider the following tree:

This tree resulted from an operation on an AVL tree and is no longer an AVL tree.    The critical node is the node containing 42.

What type of imbalance is is exhibited at the critical node?  RR

What type of rotation is needed to repair the imbalance? Left

BlackTom AI · Accepted Answer

To tackle the problem, I’ll evaluate the imbalance at the critical node and then decide the appropriate rotation.
Option RR: The notation RR stands for Right-Right imbalance. This occurs when the right subtree of a node is heavier, and the right child itself also has a heavier right subtree, creating a skew to the right. In the given tree, the node 42 has a much deeper right subtree (through 55, then 64, then 62/65) than its left subtree (11). The right subtree of 42 is both deeper and heavier, and the heavy path continues to the right from 42’s child, which matches the RR pattern. Therefore, describing the imbalance at 42 as RR is consistent with the observed structure.
Option Left: If we were to consider a left rotation as the corrective action, we need a scenario where the imbalanced node’s heavy path is leaning to the right (RR) or the left (LL) in a way that a single left or right rotation would fix it. In this case, since the imbalance is Right-Right (RR), a single left rotation around the critical node (42) is the standard remedy. A left rotation repositions the heavier right subtree up and the imbalanced node down to restore balance, aligning with AVL rotation rules for RR imbalances.
Option LL (not shown as a direct choice): If the left-left case were present (left subtree heavier with a heavy left child), the repair would require a right rotation. That does not match the observed right-heavy pattern here.
Option RL or LR (not shown as a direct choice): These would correspond to situations where the heavy path goes in opposite directions (right-left or left-right) and would require a double rotation (e.g., RL would need a right rotation on the right child followed by a left rotation on the imbalanced node). The current configuration does not exhibit the RL/LR pattern at 42, since the imbalance propagates to the right without a left-heavy child on that side.
In summary, the imbalance type is RR because the right subtree of 42 is heavier and the right subtree of that child is also contributing to the depth difference, and the appropriate repair is a single left rotation around node 42 to restore balance.

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