AVL_IDOff_6
Refer to the AVL Tree shown in image below.
Considering that the node with the value 60 has been just inserted,
After inserting the node with the value 60, which node first becomes unbalanced?

Question

BlackTom AI · Accepted Answer

To determine which node first becomes unbalanced after inserting 60, we trace the path from the newly inserted node up toward the root and check each ancestor's balance factor (difference between left and right subtree heights).

Option 76:
- If 76 is an ancestor of 60, we would look at how the insertion of 60 affects the heights of 76's left and right subtrees. Depending on whether 60 lands in 76's left or right subtree and the heights of those subtrees, 76 could become unbalanced, but whether it is the first unbalanced node depends on whether any closer ancestor already exceeded the balance tolerance. If 76 is higher up than 55 and the closer ancestors remain balanced, 76 would not be the first unbalanced node.

Option 21:
- If 21 is an ancestor of 60, the impact on its balance factor would similarly depend on whether the insertion increases the height of the child subtree on one side enough to cause a difference of 2. However, since the path from the inserted node to the root passes through 60’s immediate parent (likely 55), it is unlikely that 21 becomes unbalanced before 55 unless 21’s other subtree was already imbalanced or very tall. Therefore, 21 would more likely remain balanced until a closer ancestor (like 55) becomes unbalanced.

Option 47:
- For 47, the reasoning is analogous: an imbalance would require a dramatic height increase in one of its child subtrees due to the insertion. If 60 is in the subtree of 47’s right side and 47’s left side was heavier, 47 could become unbalanced, but the first unbalanced node is determined by the nearest ancestor to 60 that fails the balance condition. If that nearest ancestor is 55, then 47 cannot be the first unbalanced node.

Option 55:
- This is typically the closest ancestor to the inserted key 60. Since 60 > 55, the insertion increases the height of 55’s right subtree. If the right subtree height increases by enough to make the difference between left and right heights equal to 2, 55 becomes unbalanced. Being the closest ancestor to the inserted node, 55 is the most likely to be the first node whose balance factor violates the AVL condition after inserting 60.

Summary: The first unbalanced node in an AVL insertion is the nearest ancestor to the inserted key whose balance factor becomes outside [-1, 1]. Among the given options, the nearest candidate to the inserted value 60 that can be driven unbalanced by this insertion is 55, due to the right-subtree height increase caused by placing 60 as a child of 55. The other nodes (76, 21, 47) would only become unbalanced if the height changes propagate further up or if they already started in an imbalanced state, which is not implied by the insertion path described.

AVL_IDOff_6 Refer to the AVL Tree shown in image below. Considering that the node with the value 60 has been just inserted, After inserting the node with the value 60, which node first becomes unbalanced?

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Similar Questions

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