$(20$ pts$.)$ Heaps and Heap Sort.

$($a$)$ What are the minimum and maximum numbers of nodes in a heap of height $h?$

$($b$)$ Is the array with values $\{10,14,19,35,31,42,27,44,26,33\}$ a Min heap?

$($c$)$ Show that in the worst$-$case Heapify$-$UP could make $(logn)$ swaps on a heap with $n$ elements. $($Hint: Give an example heap with $n$ node values that would cause Heapify$-$UP to be called recursively at every node on a simple path up to the root$).$