Insertion and removal from a B-tree

Question

I have come across this question and haven't been able to answer it.

Given a B-tree of order 9 and of 4 levels will insertion and right after it removal of a new item x will always bring the tree to its first structure?

Will removal and insertion of a existing item x always bring the tree to its first structure? Prove it.

So far i tried to disprove it but haven't been able to. Now i honestly can't find the answer, I am not asking for a full proof a general idea on how to prove it will satisfy me.

Answer 1

The answer obviously depends on the implementation of the insert and delete methods but in short: no.

I won't give you a full proof (because you didn't ask for it and because I'm too lazy) but the general idea should be that usually when you delete a node the inner-most node of the opposite side (relative to the parent) takes its place. So in any scenario where that node exists, it will be moved up. It also means that the node was not a leaf, which is a problem because insertion usually puts new nodes on the tree as a leaf. So the original structure will only be maintained if the inner-most node of the opposite side (relative to the parent) is empty.

This is the deletion I'm referring to. If you remove 2 and re-insert it, that's the counter proof.