Space Complexity of Build Max Heap

Question

Since Heapify is a recursive function, its space complexity is $O(logn)$ because of the stack space required for recursion.
I also read that space complexity of heapsort is $O(1)$ beause of the explanation here - https://gateoverflow.in/79909/space-complexity-of-heap-sort .

So my question is why the space complexity of Build Heap is $O(logn)$ ?

If space complexity of build heap is $O(logn)$ then heapsorts complexity should also be the same . What am I missing here ?

Answer 1

Heapify function can be implemented in recursive as well as iterative way, so if we implement heapify function in iterative way I think space complexity will be O(1)