Is this new sorting algorithm based on Binary Search Tree useful?

Question

If we some how transform a Binary Search Tree into a form where no node other than root may have both right and left child and the nodes the right sub-tree of the root may only have right child, and vice versa, such a configuration of BST is inherently sorted with its root being approximately in the middle (in case of nearly complete BST’s). To to this we need to do reverse rotations. Unlike AVL and red black trees, where roatations are done to make the tree balanced, we would do reversed rotations.

I would like to explain the pseudo code and logical implementation of the algorithm through the following images. The algorithm is to first sort the left subtree with respect to the root and then the right subtree. These two subparts will be opposite to each other, that is, left would interchange with right. For simplicity I have taken a BST with right subtree, with respect to root, sorted.

To improve the complexity as compared to tree sort we can augment the above algorithm. We can add a flag to each node where 0 stands for a normal node while 1 is when the node has non null right child, in the original unsorted BST. The nodes with flag 1 have an entry in a hash table with key being their pointers and the values being the right most node. For example node 23's pointer would map to 30.5's pointer. Then we would not have to traverse all the nodes in between for the iteration. If we have 23's pointer and 30.5's pointer we can do the required operation in O(1). This will bring down time complexity , as compared to tree sort.

Please review the algorithm and give suggestion if this algorithm is usefull.