Finding the Tree

After finding the maximal clique in the compatibility graph, which means finding the largest set S^* of jointly compatible characters, we still have to construct the phylogeny itself. This is performed simply by successively splitting the set of species according to each of the characters in S^*:

Constructing the Phylogeny
In this algorithm we will iteratively construct an unrooted phylogeny, given the matrix M and the character set S^*. At each step we use one of the characters in S^* to extend the tree T. The set S holds the characters not yet used. Each node in T will be either unlabeled, or labeled with a set $L_v \subseteq \{1,\ldots,n\}$, representing species. Define, for each character and labeled node v: $L_v^i(c) \equiv L_v \cap \{k:k_c = i\}$.

• Initialization:

  1.

  $T \leftarrow$ a tree containing a single node r, labeled with the set $L_r = \{1 \ldots n\}$.

  2.

  $S \leftarrow S^*$.

• Iteration:

  1.

    Choose any character $c \in S$.

  2.

  Find a labeled node $v \in V(T)$ such that $\vert L_v^i(c)\vert \neq 0$ for i=0,1.

  3.

  Add two new vertices, v⁰ and v¹ to T, labeling vⁱ with Lⁱ_v.

  4.

  Add two edges connecting each vⁱ to v.

  5.

  Remove the label from v.

  6.

  $S \leftarrow S \setminus \{c\}$

  7.

  Go back to step 1 while $S \ne \emptyset$.

The resulting tree T is an unrooted phylogeny with labeled leaves. Finding the labeling of the internal nodes is simple and can be done using Fitch's algorithm described in section 8.2.1.