# How many rotations after AVL insertion and deletion

Is it true that inserting an element to an AVL tree requires $$O(1)$$ rotations?

How many rotations, does deletion from AVL require?

I've searched for these two questions with no luck so far.

picture: originaly the balance at node $$p$$ equals $$-1$$ (left subtree one deeper than right subtree). After insertion in the left subtree the balance is now $$-2$$. Then rebalancing at $$p$$ will restore the balance at the root to $$0$$, while at the same time the heigth of the tree at new root $$q$$ is now the same as for the original tree. This means that all balance factors above will not change.
picture: an AVL tree of "Fibonacci" type. Deleting the node marked "X" results in unbalance at $$11$$. Rebalancing at $$11$$ leads to a tree that is shorter than the original tree, and we get an new imbalance at the root. Another rotation will solve that problem. Note that again the resulting tree is shorter than the original one, so this would propagate to levels above.