$4.$ Let's say we have $7$ elements $($numbered $0$ to $6).$ Are the following arrays legal

id$-$arrays in Union$-$Finnd? Justify! For those that are legal, they may have come up in

the weighted quick$-$union algorithm? Draw the trees in all cases and explain why each of the arrays can come up in Quick$-$union, or show a sequence of union operations which end in this array.

a$.3,5,6,3,3,2,2$

b$.0,2,2,2,0,2,2$

c$.4,0,2,0,4,4,3$

$5.$

Draw the trees after each of the following operations of the weighted Quick$-$

union has $5$ elements $($numbered $0$ to $4)$: union$(0,1),$ union$(3,1),$ union$(2,4),$ union$(4,1).$

Note that if the weights of the trees are the same, the root of the previous tree is set as the new root.

Return these four drawings and the final value of the array id$.$