Problem $1.(40$ points$)$ Answer the following problems. For each problem, start with the

B$+-$tree in Figure $1.$

Figure $1$: B$+-$Tree Example

$($a$)(10$ points$)$ Draw the tree that would result from inserting $33$ into the tree in Figure $1.$ When

a node is split into two nodes, ensure that the right node has no more keys$/$pointers than the

left node. You may omit the parts of the tree that do not change.

$($b$)(10$ points$)$ Draw the tree that would result from deleting $58$ from the tree in Figure $1($please

ignore changes in Problem $4($a$)).$ If a node requiring merging$/$redistribution has both left

and right sibling nodes, use the left sibling node for merging$/$redistribution$.$ You may omit

the parts of the tree that do not change.

$($c$)(10$ points$)$ Draw the tree that would result from deleting $1,2,$ and $8$ from the tree in Figure $1.$

If a node requiring merging$/$redistribution has both left and right sibling nodes, use the left

sibling node for merging$/$redistribution$.$ You may omit the parts of the tree that do not

change.

$($d$)(10$ points$)$ Draw the tree that would result from deleting $32,39,99,1,2,$ and $8$ from the tree

in Figure $1.$ You may omit the parts of the tree that do not change.