AVL tree

Question

Consider the following statements
I: The height of any binary search tree with n nodes is O(log n).
II : Inserting into an AVL tree with n nodes requires Θ(log n) rotations.
Which of the following statements is/are true ?

Answer 1

If a tree is left skewed or right skewed so in that case the height will be O(n) not O(logn).Hence statement 1 is false.

For insertion of a  node into an AVL tree, after insertion at most one problem can occur overall .So at most 2 rotations will be required (for LR or RL problem)..Hence statement 2 is also false.In case of deletion yes it is O(logn) 

Hence both of the statements are false.