Counting Unrooted Binary Trees solved by 434

Counting Trees

A natural question is to be able to count the total number of distinct unrooted binary trees having $n$ leaves, where each leaf is labeled by some taxon. Before we can count all these trees, however, we need to have a notion of when two such trees are the same.

Our tool will be the split. Recall from “Creating a Character Table” that removing any edge from a tree $T$ separates its leaves into sets $S$ and $S^{\textrm{c}}$, so that each edge of $T$ can be labeled by this split $S \mid S^{\textrm{c}}$. As a result, an unrooted binary tree can be represented uniquely by its collection of splits.

Problem

Two unrooted binary trees $T_1$ and $T_2$ having the same $n$ labeled leaves are considered to be equivalent if there is some assignment of labels to the internal nodes of $T_1$ and $T_2$ so that the adjacency lists of the two trees coincide. As a result, note that $T_1$ and $T_2$ must have the same splits; conversely, if the two trees do not have the same splits, then they are considered distinct.

Let $b(n)$ denote the total number of distinct unrooted binary trees having $n$ labeled leaves.

Given: A positive integer $n$ ($n \leq 1000$).

Return: The value of $b(n)$ modulo 1,000,000.

Sample Dataset

5

Sample Output

15