Refined Buneman Trees

epresentation of evolutionary relationships — but of course, we would like to
do more than just look at the trees.
Definition 1 (Evolutionary tree). An evolutionary tree T is an ordered pair
(T ; φ), where T is a tree with vertex set V and φ : X → V with the property
that, for each v ∈ V of degree at most two, v ∈ φ(X). An evolutionary tree is
also called a semi-labeled tree (on X).
The main thing to notice about Definition 1 is the rule that all leaves must
correspond to species, but not all species have to correspond to leaves. While
working on this thesis, the author experienced some confusion until Definition 2
filled in a gap in the authors understanding of evolutionary and phylogenetic
trees and their relation to graph theoretical trees. One might be tempted to
interpret evolutionary trees as unrooted trees where leaves represent species, but
the small detail that evolutionary trees do not need to be fully resolved is quite
important.
Definition 2 (Phylogenetic tree). A phylogenetic tree T is an evolutionary
tree (T ; φ) with the property that φ is a bijection from X into the set of leaves
of T .
In graph theoretical terms, a phylogenetic tree is a leaf-labeled tree, while
an evolutionary tree is a semi-labeled tree. An illustration of this distinction is
given in Figure 2.1. Here we have an evolutionary tree with three “abnormal”
regions A, B and C, where the tree is not fully resolved. If we look at regions
A and B, we see that a species appears to be the ancestor of other species. It
is of course possible that we have such a dataset, but a more likely explanation
would be that the underlying evolutionary data simple does not tell us how to
resolve these species — which is precisely the situation in region C. Wemight
argue that since we cannot distinguish between these species, they must be the
same species. But it is also very likely that the underlying evolutionary data is
inaccurate.
The important thing to stress is that our tree reconstruction method might
output a tree that is not fully resolved for whatever reason, and additional
analysis might be needed to find the answers we are looking for. An easy way
of obtaining a leaf-labeled tree is to simply add extra edges to those labeled
nodes which have degree two or more. This is illustrated in Figure 2.2. Of
course, by doing so we are loosing information about the original tree, but this
can be remedied by marking the extra edges. The reason for performing this
transformation is that leaf-labeled trees are easier to work with then semi-labeled
trees, for the average computer scientist.
Two graph theoretical results are handy when reasoning about trees and tree
search spaces: a semi-labeled unrooted tree with at most n leaves has at most
n − 2 inner nodes, for a maximum of 2n − 2 nodes in the whole tree. And there
are at most 2n − 3edges.
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