Isomorphic Suffix Trees

In this problem, your task is to find the number of distinct binary strings, composed of the characters '0' and '1', whose suffix trees are isomorphic to a given rooted tree $T$.

Two rooted trees are considered isomorphic if they have the same number of vertices, $n$, and there exists a way to label the vertices of each tree with unique integers from $1$ to $n$ such that:

• The roots of both trees share the same label.

• An edge exists between vertices labeled $a$ and $b$ in the first tree if and only if an edge exists between vertices labeled $a$ and $b$ in the second tree.

The suffix tree of a string $s$ is a rooted tree that meets the following criteria:

1. Each edge is labeled with a non-empty string of '0's and '1's.

2. Each suffix of the string s+'\$', except for the suffix "$\$$", corresponds to exactly one leaf (a vertex of degree 1 that is not the root). The concatenation of the edge labels along the path from the root to this leaf matches the suffix.

3. Any vertex that is neither a leaf nor the root must have a degree greater than two.

4. The tree has the smallest possible total length of edge labels among all trees satisfying properties 1-3.

For example, the suffix tree for the string "1100" is depicted as follows:

Your task is to determine the number of different binary strings whose suffix trees are isomorphic to the given tree $T$.