Multiple Alignment to a Phylogenetic Tree

Definition 0.6   A tree T with a distinct string label (from a set of string 44#44) assigned to each leaf is called a phylogenetic tree on 44#44

Definition 0.7   Given a phylogenetic tree T on 44#44, 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 44#44.

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 4.3: A phylogenetic tree alignment

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Definition 0.8   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 0.9   Let 63#63 be a phylogenetic alignment for a phylogenetic tree T. The distance of 63#63 is given by the sum of all the edge distances over all the edges of T.

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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.