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.

• 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.

  For each labelled node $v \in V(T)$ such that L_v has more than one member:

  (a)

  Define $L^i_v = L_v \cap \{k:k_c = i\}$ for i = 0,1.

  (b)

  If $L^0_v = \emptyset$ or $L^1_v = \emptyset$, go on to the next node.

  (c)

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

  (d)

  Add two edges connecting each vⁱ to v.

  (e)

  Remove the label from v.

  3.

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

  4.

  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 9.2.1.