How to find the index of an element in sorted set?

Question

I can find an element in a sorted set (backed by BST) in O(logN). Now I would like the index of this element. For example, in set {1, 3, 4, 10}, the index of 4 is 2 and the index of 1 is 0.

Obviously, I can just iterate over the set, so the trivial solution is O(N). Can we do better using probably BST properties and/or auxiliary data structures ?