$-$ P$1(5$ pts$)$: $($Proof by induction$)$ Show the maximum number of nodes in a binary tree of height $\backslash ($ h $\backslash )$ is $\backslash (\backslash$left$(2^\{$h$+1\}-1\backslash$right$)\backslash )$

$-$ P$2(5$ pts$)$

$($a$)$ Show the result of inserting $5,6,3,4,2,1,7,9$ into an initially empty binary search tree.

$($b$)$ Show the results of deleting the root of the tree.

$-$ P$3(5$ pts$)$ Draw a binary tree with the nine nodes B$,$ D$,$ E$,$ G$,$ H$,$ I, J$,$ L and M$,$ satisfying the following two conditions at the same time:

$-$ if we print all the nodes in postorder traversal, the output is IJGLMHDEB;

$-$ if we print all the nodes in inorder traversal, the output is IGJDLHMBE.

$-$ P$4(10$ pts$)$ Show the result of inserting $\backslash (6,4,3,1,2,0,9,8\backslash )$ into an initially empty AVL tree $($insert one at a time in the given order$).$

$-$ P$5(5$ pts$)$ Draw a structure of a AVL tree of height $4$ with only $12$ nodes.

$-$ P$6(10$ pts$)$ Use the B$-$tree shown below to answer the following questions: