More than one rotation needed to balance an AVL Tree?

Question

My best guess is that one rotation is always enough to balance an AVL tree when you insert or delete ONE element from an already balanced AVL tree.

Is one rotation always enough? An example will help where more than one rotations are needed.

PS: I count RL/LR rotations as one rotation only.

score 9 · Answer 1 · answered Oct 10 '15 at 17:18

For insertion I believe one is enough.

but for deletion: consider this tree:

                 50
              /      \
            25       75
           /   \    /   \
          15   40  60    80
               /  /  \    \
              35 55  65   90
                     /
                    62

delete 15 , first the 25's balance factor is destructed, one rotation:

                 50
              /      \
            35       75
           /   \    /   \
          25  40  60    80
                  /  \    \
                 55  65   90
                     /
                    62

but still, we have to check, now the tree's root's balance factor is destructed, have to be rotated again:

                 60
              /      \
            50       75
           /   \     /  \
          35   55   65   80
         /  \       /     \
        25  40     62     90

I know this is an old answer. But can you tell me how did you rotate the tree when 50's(root) balance factor was not proper ? — Zephyr, Aug 21 '17 at 19:46
The root's balance factor is 100% proper. On its left, a subtree of height 2. On its right, height 3. No problem. — Jason, Feb 13 '19 at 19:55

IanWalker0421 · Accepted Answer · 2014-01-03T23:46:37.920

7

For insert 1 rotation is at most.
For delete the number of rotation is bounded by O(log(n)). Log(n) is the height of the tree.
More explanation on AVL deletion. When you delete a node from AVL you might cause the tree unbalanced, which you have to trace back to the point where it is unbalanced. If the unbalanced point is the root. You have to rebalance the tree from top to bottom.

edited Jan 03 '14 at 23:46

answered Jan 03 '14 at 21:11

IanWalker0421

108
1
6

And I can't believe unless you fetch me an example where 2 rotations are needed. – kBisla Jan 03 '14 at 21:20
@BlueFlame I put the link above where you can see many examples. Scroll down you can see the logn deletion. – IanWalker0421 Jan 03 '14 at 23:45
Check again. In none of those examples, the Tree is balanced before the deletion. like I said in the ques, the tree is balanced before the insertion/deletion operation. – kBisla Jan 04 '14 at 00:45
@IanWalker0421 - +1 for "Please proceed, governor" :) – drankin2112 Jan 04 '14 at 20:13
@BlueFlame: Search for "AVL tree after inserting a new node:" – Mooing Duck Mar 12 '14 at 23:29
1

How can I prove that only 1 rotation is needed when inserting? – user3595632 May 14 '17 at 05:28
Echoing @user3595632's concern: has anyone perhaps come up with a proof by induction or something along these lines? – Jason Feb 13 '19 at 19:56

More than one rotation needed to balance an AVL Tree?

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