Drazil Likes Heap

Drazil likes heap very much. So he created a problem with heap:

There is a max heap with a height $$$h$$$ implemented on the array. The details of this heap are the following:

This heap contains exactly $$$2^h - 1$$$ distinct positive non-zero integers. All integers are distinct. These numbers are stored in the array $$$a$$$ indexed from $$$1$$$ to $$$2^h-1$$$. For any $$$1 < i < 2^h$$$, $$$a[i] < a[\left \lfloor{\frac{i}{2}}\right \rfloor]$$$.

Now we want to reduce the height of this heap such that the height becomes $$$g$$$ with exactly $$$2^g-1$$$ numbers in heap. To reduce the height, we should perform the following action $$$2^h-2^g$$$ times:

Choose an index $$$i$$$, which contains an element and call the following function $$$f$$$ in index $$$i$$$:

Note that we suppose that if $$$a[i]=0$$$, then index $$$i$$$ don't contain an element.

After all operations, the remaining $$$2^g-1$$$ element must be located in indices from $$$1$$$ to $$$2^g-1$$$. Now Drazil wonders what's the minimum possible sum of the remaining $$$2^g-1$$$ elements. Please find this sum and find a sequence of the function calls to achieve this value.