In this paper they prove that the number of rotations after doing $n$ insertions or deletions to a weight-balanced tree is $O(n)$ (when starting from an empty tree).
What isn't clear to me though is how many times each node becomes the new root of a sub-tree after a single or double rotation takes place. Intuitively, I should expect this to be $O(1)$. Is this correct? How can this be shown?