has deleting node in a binary search tree Displacement feature?

Question

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.

Answer 1

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