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I am developing an academic project about graph and tree theory.I searched a lot but I didn't find a clear answer. In a part of project we want to delete some nodes from tree for example we want to delete nodes A and B.I want to know that if we Delete A and then B does It give us exactly the same tree that We will produce when We Delete B and then A and if yes is there any proof or anything that can make sure us about that.

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    deleting means that for example detele('4') removes node with key 4 from our tree.2012-11-25

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finally i will find the answer! Deletion (in general) is not commutative. Here is a counterexample:

    4    / \   3   7      /     6 

What if we delete 4 and then 3?

When we delete 4, we get 6 as the new root:

   6   / \  3   7 

Deleting 3 doesn't change the tree, but gives us this:

  6    \     7 

What if we delete 3 and then 4?

When we delete 3 the tree doesn't change:

 4   \    7   /  6 

However, when we now delete 4, the new root becomes 7:

  7  / 6 

The two resulting trees are not the same, therefore deletion is not commutative.

reference and more details : https://stackoverflow.com/questions/2990486/deletion-procedure-for-a-binary-search-tree