Problem $2.(36$ points$)$ Answer each of the following problems assuming that Figure $1$ shows an

AVL tree. In Figure $1,$ ni for i $=0,1,,8$ denotes the memory address of a node.

$($a$)(9$ points$)$ Assume that t refers to the AVL tree shown in Figure $1.$ Draw the tree that would

result from executing t$.$add$(1).$ Justify your answer.

$($b$)(9$ points$)$ Assume that t refers to the AVL tree shown in Figure $1.$ Draw the tree that would

result from executing t$.$add$(4).$ Justify your answer.

$($c$)(9$ points$)$ Assume that t refers to the AVL tree shown in Figure $1.$ Draw the tree that would

result from executing t$.$remove$(6).$ Justify your answer.

$($d$)(9$ points$)$ Assume that t refers to the AVL tree shown in Figure $1.$ Draw the tree that would

result from executing t$.$remove$(2)$ and then t$.$remove$(7).$ Justify your answer.