What if traversal order of two of pre-order, in-order and post-order is same?
A proper proof uses the recursive definitions of the tree-orders. Here is a sketch.

For inorder the recursive definition is: in(left) - root - in(right).
For postorder we similarly have: post(left) - post(right) - root. 

Look at the position of root in both orders. If we want this to be the same we need to have there are no nodes in in(right) which means the right subtree (at the root) is empty. Same for the root at other levels.