# Spliting in AVL tree in $O(\log n)$ times

Suppose we are given a balanced binary search tree $$T$$, AVL tree, now we want split $$T$$ at a arbitrary node. Can it be done in $$O(\log n)$$?

Now i think, because of height of our tree $$T$$ is $$O(\log n)$$ then after splitting at a arbitrary node, it is sufficient to re-balance our tree $$T$$ start from one of leaf node, in this manner we do rotation and because of each level want constant number of rotation to be balanced, after $$O(\log n)$$ will be in root of $$T$$ and our tree $$T$$ is balanced. Are my argument is true?