Segment Tree Reconstruction

You have a perfect binary tree with $2^N$ leaf nodes. A perfect binary tree is a rooted tree where all non-leaves have exactly $2$ children, and all the leaves are equidistant from the root. It can be shown that there are exactly $2^{N + 1} - 1$ total nodes then. Here is an example with $8$ leaf nodes and $15$ total nodes, i.e. $N = 3$. You want to write binary integers ($0$ or $1$) on each of t