$(20$ pts$.)$ Heap and Heapsort.

$($a$)$ Run Build$-$Heap on the array $12,10,4,7,6,1,3,0,8,5$ to construct a min heap. Write down the resulting array. Then, run the loop in Heapsort for three iterations. For each iteration write down the array after Heapify$-$Down is finished. You do not need to show intermediate steps.

$($b$)$ Show that the leaves of a heap occupy the positions ${|}_{{?}_{?}}\frac{n}{(2{)}_{{?}_{?}}}|,$dots,$n-1$ of the array.

$($c$)$ In the procedure Build$-$Heap, why do we decrease i from ${|}_{{?}_{?}}\frac{n}{(2{)}_{{?}_{?}}}|-1$ to $0$ instead of increasing from $0$ to ${|}_{{?}_{?}}\frac{n}{(2{)}_{{?}_{?}}}|-1?$ Give an example array where increasing i fails to create a valid min heap.