$1.$ Draw what a binary search tree would look like if the following values were added to an initially empty tree in this order:

$\backslash [$

$50,20,75,98,80,31,150,39,23,11,77.$

$\backslash ]$

$2.$ What are the outputs of a pre$-$order and post$-$order traversal of the final binary search tree drawn in question $1?$

$3.$ If a search was conducted for the value $37$ in the final binary search tree from question $\backslash$#$1,$ which nodes would get visited? $($List them in the order they get visited.$)$

$4.$ Delete $31$ and $98$ from the Tree, insert on the new Tree $92,65,$ then Delete $150.$

$5.$ Add the NIL leaves on the final binary search tree from question $\backslash (\backslash$# $4\backslash )$ and color the nodes such that the black$-$height of the resulting red$-$black tree is $2.$

$6.$ Insert $90$ on the resulting R$-$B tree and fix$-$up the tree if any property is violated. Compute the black$-$height of the new tree.