Rebalancing an arbitrary BST?

The DSW algorithm can solve this is O(n) time. The algorithm goes as follows:

1] Using right-rotation operations, turn the tree into a linked list
   (a.k.a. backbone or vine)
2] Rotate every second node of the backbone about its parent to turn
   the backbone into a perfectly balanced BST. 

Reference

"Balanced as possible" = complete (or full) binary tree¹. You cannot get more balanced that it.

The solution is simple - build an "empty" complete binary tree, and iterate the new tree and the input tree (simultaneously) in inorder-traversal to fill the complete tree.

When you are done, you have the most balanced tree you can get, and time complexity of this approach is O(n).

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EDIT:
This should be done following these steps:

1. Build a dummy complete tree with n nodes. All the values to each node will be initialized to some garbage value.
2. Create two iterators: (1) originalIter for the original tree, (2) newIter for the new (initialized with garbage) tree. Both iterators will return elements in in-order traversal.

3. Do the following to fill the tree with the values from the original:

    while (originalIter.hasNext()):
         newIter.next().value = originalIter.next().value
   

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(1) (From Wikipedia): A complete binary tree is a binary tree in which every level, except possibly the last, is completely filled, and all nodes are as far left as possible

You might want to look into the Day-Stout-Warren algorithm, which is an O(n)-time, O(1)-space algorithm for reshaping an arbitrary binary search tree into a complete binary tree. Intuitively, the algorithm works as follows:

1. Using tree rotations, convert the tree into a degenerate linked list.
2. By applying selective rotations to the linked list, convert the list back into a completely balanced tree.

The beauty of this algorithm is that it runs in linear time and requires only constant memory overhead; in fact, it just reshapes the underlying tree, rather than creating a new tree and copying over the old data. It is also relatively simple to code up.

Hope this helps!