Run Build$-$Heap on the array $[12,10,4,7,6,1,3,0,8,5]$ to construct a min heap. Write down the result$-$

ing 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 $$ n

$2,...,$n$1$ of the array.

$($c$)$ In the procedure Build$-$Heap, why do we decrease i from $$ n

$21$ to $0$ instead of increasing from $0$ to

$$ n

$21?$ Give an example array where increasing i fails to create a valid min heap.