What is the worst case big-O cost of the insert operation on an AVL tree?

Question

BlackTom AI · Accepted Answer

AVL trees guarantee that the tree height grows only logarithmically with the number of nodes, so the cost of operations that traverse from the root to a leaf is O(log n).

1) Relevant concepts:
- A binary search tree (BST) insertion takes time proportional to the tree height h (you follow a root-to-leaf path).
- An AVL tree enforces a balance condition so that its height h is O(log n). The minimal number of nodes N(h) in an AVL tree of height h satisfies the recurrence N(h) = 1 + N(h−1) + N(h−2), which grows exponentially with h (Fibonacci-like growth). Therefore h = Θ(log n).
- Rebalancing after insertion requires at most a constant number of rotations (one single or one double rotation at the first unbalanced ancestor).

2) Step-by-step reasoning:
- Insert follows BST insertion down a root-to-leaf path: time Θ(h).
- After insertion, walk back up checking balance factors. At the first unbalanced node encountered you perform either one rotation (single) or two rotations (double). Once rebalanced, the AVL invariants are restored and no further rotations are needed for insertion.
- So total extra work for rotations is O(1).

3) Intermediate calculation / height bound:
- Using N(h) = 1 + N(h−1) + N(h−2) with base N(0)=1, N(1)=2, one finds N(h) grows like Fibonacci numbers ≈ φ^h (φ ≈ 1.618). Thus n ≥ c·φ^h implies h ≤ log_φ(n/c) = O(log n). Hence insertion cost Θ(h) = Θ(log n).

4) Why the correct answer is correct:
- Because insertion cost = cost to descend (Θ(h)) + constant rebalancing, and h = O(log n), overall worst-case cost is O(log n).

5) Why other options are incorrect:
- O(1): false because you generally must traverse from root to leaf in a tree with n nodes.
- O(log h): nonsensical here — cost depends on height h, not log of height; since h = Θ(log n), log h = O(log log n) which is too small.
- “No worst-case cost”: incorrect — AVL invariants give a concrete worst-case height and thus a worst-case bound.

Conclusion: The worst-case big-O cost of AVL insert is O(log n) where n is the number of nodes (matches the provided answer).

What is the worst case big-O cost of the insert operation on an AVL tree?Single choice

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