Question

I am reading about AVL trees in Data structures and algorithms by Mark Allen Wesis

Suppose the node to be rebalanced is X. There are 4 cases that we
might have to fix (two are the mirror images of the other two): An
insertion in the left subtree of the left child of X, An insertion in
the right subtree of the left child of X, An insertion in the left
subtree of the right child of X, or An insertion in the right subtree
of the right child of X.

Balance is restored by tree rotations.

Following are questions i have on above text snippet.

1. What does author mean by mirror images of other two?
2. What is symmetric case in single rotation and double rotation?

Thanks

Answer 1

Suppose the node being inserted is I. The book says there are 4 cases. Let’s take the one where I is the left child of the left child of X:

    X
   / \
  ?   ?
 / \ / \
I  ? ?  ?

The “mirror” of this is when I is the right child of the right child of X:

    X
   / \
  ?   ?
 / \ / \
?  ? ?  I

The reason this is a “mirror” is that the rotations you have to do for both cases are the same, just with left and right reversed. The same goes for the other two cases where I is the left child of the right child of X and where I is the right child of the left child of X.

As for your second question, the idea is the same. In the symmetric case, (ie the mirror case), you do the same rotations, just with left and right reversed.