Consider a complete binary tree where the left and the right subtrees of the root are max-heaps. The lower bound for the number of operations to convert the tree to a heap is

In a complete binary tree where both the left and right subtrees of the root already satisfy the max-heap property, converting the whole tree into a max-heap requires performing a single max-heapify (or sift-down) operation at the root:

1. The height of a complete binary tree with $n$ nodes is $h = \lfloor \log_2 n \rfloor$.

2. In the worst case (e.g., when the value at the root is smaller than all elements in both subtrees), the root element must be sifted down along a path from the root to a leaf node.

3. The number of level comparisons and swaps along this path is proportional to the tree height $h$, requiring $\Omega(\log n)$ operations.

Hence, the lower bound for the number of operations in the worst case is $\Omega(\log n)$ (Option A).