AVL_IDOff_16
Refer to the AVL Tree shown in image below.
Considering that the node with the value 40 has been just inserted,
After inserting the node with the value 40 , which node first becomes unbalanced?

Question

BlackTom AI · Accepted Answer

Interpreting the question requires looking at how AVL insertions propagate balance changes up the path from the newly inserted node. The key concept is that after inserting 40, we check each ancestor from bottom to top to find the first node whose balance factor becomes 2 or -2 (unbalanced), making it the first unbalanced node.

Option: 73
This would be a distant ancestor or a node not on the direct path from the inserted 40 up to the root in many trees. If 73 were the first unbalanced node, that would imply that all nodes closer to 40 remained balanced after the insertion, which contradicts the usual behavior of AVL rebalancing where the first imbalance occurs on the path from the inserted node upward. Therefore, 73 is unlikely to be the first unbalanced node in a standard AVL adjustment sequence.

Option: No node becomes unbalanced.
In an AVL tree, inserting a node changes the heights of subtrees along the path to the root. Unless the tree was perfectly balanced and the insertion did not affect any height difference, there will typically be at least one ancestor whose balance factor changes enough to violate the AVL property. Since an insertion generally creates a height change, this option would be incorrect in a typical AVL scenario after inserting a new value like 40.

Option: 10
Like 73, 10 would only be the first unbalanced node if it lay directly on the insertion path and was the lowest ancestor to perform the height adjustment. However, the problem states the first unbalanced node is 16, which implies that along the path from 40 up to the root, 16 is encountered before reaching 10. Therefore, 10 would not be the first unbalanced node given the provided answer.

Option: 40
The newly inserted node itself is not evaluated for balance in AVL trees; the balance assessment is performed on its ancestors. A newly added leaf does not typically become the first unbalanced node since we only check balance factors of existing nodes after the insertion, not the node just inserted. Thus, 40 being the first unbalanced node is inconsistent with the standard procedure.

Option: 16
This aligns with the given answer and fits the standard AVL behavior: after inserting 40, the first ancestor whose height difference (balance factor) becomes outside [-1, 1] is 16. That means 16 is the lowest node on the insertion path where the imbalance appears, prompting the first rebalancing (rotation) at or above 16. The other candidates either lie above 16 on the path, are not on the path, or cannot be the first unbalanced node given the insertion path described, making 16 the most plausible first unbalanced node in this scenario.

AVL_IDOff_16 Refer to the AVL Tree shown in image below. Considering that the node with the value 40 has been just inserted, After inserting the node with the value 40 , which node first becomes unbalanced?

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