A good loop invariant would be “The entire tree is a heap except that nodei might not be greater or equal to both of its children. As well, the value of i’s parent is at least the value of i and of i’s children.” When i is the root, this is the precondition. The algorithm proceeds as in the recursive algorithm. Node i follows one path down the tree to a leaf. When i is a leaf, the whole tree is a heap. Write algorithm Heapify(r )

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Recurrence relation

A recurrence relation or recursive relation is an equation that represents a function in terms of the values of its smaller inputs. Every recurrence relation T(n) is a recursive function of integer n and consists of a base case and a recursive case.