Does the order of insertion affect the structure of a binary search tree?

[Demonoid]

Homework Statement

I need to prove that different insertion orders of the same keys always gives us a different binary tree.

Homework Equations

All obvious BST properties apply:
The left subtree of a node contains only nodes with keys less than the node's key.
The right subtree of a node contains only nodes with keys greater than the node's key.
Both the left and right subtrees must also be binary search trees.
There must be no duplicate nodes.

The Attempt at a Solution

I don't exactly know how to formally prove this besides just showing a bunch of examples.

I think this is obvious because there aren't any duplicate keys therefore you will always have a unique layout of the tree.

 

[Valkarie]

For example, if you have the keys {1,2,3}, the order of insertion would be different whether you insert 3 first, then 2, then 1 or if you insert 1 first, then 2, then 3. Therefore, you would get a different binary tree every time.