Multiple Alignment to a Phylogenetic Tree

Definition 5.9   A tree T with a distinct string label (from a set of string $\S$) assigned to each leaf is called a phylogenetic tree on $\S$

Definition 5.10   Given a phylogenetic tree T on $\S$ , a phylogenetic alignment T' for T is an assignment of one string label to each internal node of T. Note that the strings assigned to internal nodes need not be distinct and need not be from the set $\S$.

For example, when T is a star, choosing the sequence to label its central node is a phylogenetic alignment. The phylogenetic tree T is meant to represent the "established" evolutionary history of a set of objects of interest, with the convention that each extant object is represented at a unique leaf of the tree. Each edge (u,v) represents some evolutionary history that transforms the string at u (assuming u is the parent of v) to the string at v. For convenience, when denoting an edge by a pair of nodes, we shall hereafter write the parent node first.

  

Figure 5.2: A phylogenetic tree with lifted alignment. Each internal node is labeled by one of the strings labeling its children.

Definition 5.11   If strings S and S' are assigned to the endpoints of an edge (i,j), then the edge distance of (i,j) is defined to be D(S,S').

Definition 5.12   Let ${{\cal M}}_T$ be a phylogenetic alignment for a phylogenetic tree T. The distance of ${{\cal M}}_T$ is given by the sum of all the edge distances over all the edges of T.

Problem 5.6   Phylogenetic alignment:
INPUT: A set $\S = \left( { S_1, \ldots, S_k } \right)$ of strings and a phylogenetic tree T, on $\S$.
QUESTION: find a phylogenetic alignment ${{\cal M}}_T$ with minimum distance.

We assume that the structure of the tree T is known to us. This usually happens in practice when T was previously reconstructed using solid evolutionary data.