Double Rotation: Left-Right

Consider inserting the values $7, 5, 3$ in that order.

We start by inserting $7$:

We have a BST with a single node; it is balanced!

Next, we insert $3$.

Now our BST has two nodes. Notice the height, and balance factor of the root has changed (and it is still balanced).

Next, we insert $5$.

Our BST has three nodes now. Notice the heights and balance factors of the parent and grandparent of $5$ have changed. In particular, the grandparent (the root) is not balanced anymore!

A single rotation will not restore the balance! However, if we were to push $3$ to the left of $5$, we would transform the structure to a pattern we have seen before:

The above is a (single) right rotation away from being balanced!

So we perform a right rotation:

As you have noticed, we needed two rotations, first a left and then a right rotation. This is called a (double) left-right rotation.

Notice the violation of balance property occurred in the grandparent of the newly inserted node. From the perspective of the grandparent node, the problem was caused in left child's right subtree. The solution is a (double) left-right rotation to bring the median value above the high and low values.