Dream and the Multiverse

Dream abstracts the fabric of spacetime as a directed rooted tree (arborescence) with $N$ nodes (numbered $1$ through $N$). Node $1$ is the root and for each $i$ ($1 \le i \le N-1$), the parent of node $i+1$ is $f_i$. All edges of this tree are directed away from the root. Then, Dream employs a magical superpower and adds $M$ directed edges to this tree in such a way that the resulting directed