# Exercise 13.2.5

$\star$ We say that a binary search tree $T_1$ can be right-converted to a binary search tree $T_2$ if it is possible to obtain $T_2$ from $T_1$ via a series of calls to RIGHT-ROTATE. Give an example of two trees $T_1$ and $T_2$ such that $T_1$ cannot be right-converted to $T_2$. Then, show that if a tree $T_1$ can be right-converted to $T_2$, it can be right-converted using $\O(n^2)$ calls to RIGHT-ROTATE.

Here are two trees, the second of which can't be produced by right-rotating the right:

   1             2
\         /
2     1


We need a left rotation here.

More specifically, right rotations can only decrease the value of the root, but not increase it.

Now, reasoning very informally, and making some unverified assumptions, we can convert $T_1$ to $T_2$ by first doing $\O(n)$ right rotations to get the roots to match, and then performing right rotations on the two sub-trees recursively to position their roots. An upper bound of this approach (assuming it works) will be $\O(n^2)$.