July 2, 2012, midnight by Rosalind Team

**Topics**:
Combinatorics,
Phylogeny

## Culling the Forest

In “Completing a Tree”, we introduced the tree for the purposes of constructing phylogenies. Yet the definition of tree as a connected graph with no cycles produces a huge class of different graphs, from simple paths and star-like graphs to more familiar arboreal structures (see Figure 1). Which of these graphs are appropriate for phylogenetic study?

Modern evolutionary theory (beginning with Darwin) indicates that a common way for a new species can be created is when it splits off from an existing species after a population is isolated for an extended period of time. This model of species evolution implies a very specific type of phylogeny, in which internal nodes represent branching points of evolution where an ancestor species either evolved into a new species or split into two new species: therefore, one edge of this internal node therefore connects the node to its most recent ancestor, whereas one or two new edges connect it to its immediate descendants. This framework offers a much clearer notion of how to characterize phylogenies.

A binary tree is a tree in which each node has degree equal to at most 3. The binary tree will be our main tool in the construction of phylogenies.

A rooted tree is a tree in which one node (the root) is set aside to serve as the pinnacle of the tree.
A standard graph theory exercise is to verify that for any two nodes of a tree, exactly one path connects the nodes.
In a rooted tree, every node

Even though a binary tree can include nodes having degree 2, an unrooted binary tree is defined more specifically: all internal nodes have degree 3. In turn, a rooted binary tree is such that only the root has degree 2 (all other internal nodes have degree 3).

Given: A positive integer

Return: The number of internal nodes of any unrooted binary tree having

4

2

## Hint

In solving “Completing a Tree”, you may have formed the conjecture that a graph with no cycles and

$n$ nodes is a tree precisely when it has$n-1$ edges. This is indeed a theorem of graph theory.