Refined Buneman Trees

evolutionary tree gives an invaluable understanding of the evolutionary data
it represents. One obvious scheme when using an ordinary leaf-labeled tree to
represent a set of splits is to say that all trivial splits are in the tree — but
they have a special marker, if they are not members of the set of splits that the
tree is supposed to represent. In order to be able to mark special edges, and
also keep track of lengths of branches in the evolutionary tree, we will define
the weight of an edge such that a weight of zero is the special marker that
invalidates the edge as a split, and positive weights represent branch lengths.
This is very reasonable since the sets of splits we are going to represent later on
will only have positive weights — edges with non-positive weights are simply not
members of the tree. The graphical representation would closely resemble the
actual evolutionary tree corresponding to the set of splits, when trivial branches
have no extent.
2.7 Buneman tree
Given a split σ for a set of size n, theBuneman Index of a split σ = U|V of X
is defined as:
μ σ (δ) = min β
u,u ′ ∈U,v,v ′ uu
∈V ′ |vv ′ (2.5)
The set of splits B(δ) ={σ : μ σ (δ) > 0} is a compatible set of splits [Bun71].
The Buneman tree corresponding to a given dissimilarity measure δ is defined
to be the weighted unrooted tree whose edges represent the splits σ ∈ B(δ) and
are weighted according to μ σ (δ).
This definition of the refined Buneman tree relates very well to the discussion
in the previous section; a compatible set of splits is represented by a weighted,
unrooted tree.
2.8 Anchored Buneman tree
The anchored Buneman tree is a relaxation of the Buneman tree. The anchored
Buneman tree fixes some species x ∈ X and only considers splits U|V where
x ∈ U. One might say the species x plays the role of outgroup with respect to
the rest of the species.
Let σ = U|V be a split with x ∈ U, then the anchored Buneman index is
defined as follows:
μ x σ (δ) = min β u∈U,v,v ′ xu|vv
∈V ′
The anchored Buneman tree is the tree that represents the compatible set of
splits defined by B x (δ) ={σ : μ x σ > 0}.
A couple of lemmas are worth noting when dealing with anchored Buneman
trees. Firstly, David Bryant and Vincent Moulton ([BM99]) have shown the
connection between anchored Buneman trees and the regular Buneman tree,
given here in Lemma 2. Secondly, Lemma 3 which is due to Vincent Berry
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