Question $1(2-3$ Trees$)$

$($a$)(25$ points$)$ Answer this question considering the $2-3$ tree insertion algorithm $($which uses the bottom$-$up approach$)$ that we mentioned in class.

Draw the resulting tree after inserting $6$ into the following $2-3$ tree. Show all steps of this insertion by drawing the tree after each intermediate operation

$($similar to the examples given in class$).$

$($b$)(25$ points$)$ Answer this question considering the $2-3$ tree deletion algorithm $($which uses the bottom$-$up approach$)$ that we mentioned in class.

What is the smallest item value in the following $2-3$ tree whose deletion decreases the tree height? Note that in this tree, there may exist more than one

item value whose deletion decreases the tree height. You should give the smallest of those as your answer.

Then, draw the resulting $2-3$ tree after deleting this smallest item. Show all steps of this deletion by drawing the tree after each intermediate operation

$($similar to the examples given in class$).$