Proof that there are same number rotation moves in any binary tree with both children compulsory
Each rotation involves a node and its left or right child. We can associate this rotation with the child – the rotation is fixed once the child is known. In your case, a child can be rotated as long as it is an internal node. So the number of possible rotations is the number of non-root internal nodes. A simple induction shows that a full binary tree having $n$ leaves has exactly $n-1$ internal nodes, and so $n-2$ non-root internal nodes.