For an AVL tree, decisions about which rotation to perform must be made based on the balance factor $B$($x$)$$$.$ This coefficient is calculated as the difference between the heights of the trees, one way would be to store the height of each tree and keep track of it$.$ We want to implement rotateLeft to maintain the balance factor $B$ stored for each tree. These numbers should only take the values $-1,0,1$ when we have a legal AVL tree and therefore only take $2$ bits of memory.

Below is an implementation of rotateLeft without keeping track of $B$ $,$ and we assume that $B$ is a field in $T$ $.$

Find out how the code below needs to be changed to calculate the new $B$ values. Assume that the tree $T$ is a legal AVL tree, but possibly the new tree will not be a legal AVL tree. Describe in which cases the AVL tree is legal after left rotation $(\backslash$textit$\{$hint$\}$: split into cases according to the values of $T$.$B$ and $T$.$right.B$$)$

$\backslash$begin$\{$minted$\}\{$java$\}$

Tree rotateLeft$($Tree T$)\{$

Tree x $=$ T$.$right;

T$.$right $=$ x$.$left;

x$.$left $=$ T;

return x;

$\}$

$\backslash$end$\{$minted$\}$