## 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 *grand parent* of $5$ have changed. In particular, the grand parent (the root) is not balanced anymore!

No single rotation will fix the imbalance!

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 arrangement is a (single) right rotation away from balance

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 grand parent of the newly inserted node. From the perspective of the grand parent node, __the problem was caused in left child’s right subtree.__ The solution is a (double) right-left rotation to bring the

*median*value above the high and low values.