Refined Buneman Trees

2.6 Splits
The partition of a finite set S into two non-empty parts U and V is denoted a
split σ = U|V .If|U| =1or|V | = 1 the split is called trivial. It is reasonable to
represent a split as a bitvector or binary number, and for convention we shall
say that for a bitvector A representing σ, x i ∈ U if and only if A[i] = 0. Splits
are symmetric, so if w represents the split σ, then¬w also represents σ. Theset
of splits on a set X is denoted σ(X). The size of σ(X) is the number of unique
splits on X, so|σ(X)| = |P(X)|−2
2
= 2n −2
2
=2 n−1 − 1. We exclude symmetric
splits and deduct the two splits where U = ∅ or V = ∅.
The set of quartets associated with a split σ = U|V on a set X is defined
by q(σ) ={uu ′ |vv ′ : u, u ′ ∈ U ∧ v, v ′ ∈ V }. Hereu, u ′ (and similarly v, v ′ ) need
not be distinct. The size of q(U|V ) is in the order of O(|X| 4 ) — recall that an
edge in a tree induces O(|X| 4 ) quartets, and splits are equivalent to edges in
this case.
Definition 3 (Compatibility). Two splits A|B and C|D are said to be compatible
if and only if one of A ∩ C, A ∩ D, B ∩ C or B ∩ D is empty.
Compatible sets of splits are the foundation for the algorithm presented in
this thesis, and they are a perfect tool when dealing with evolutionary trees.
A set of splits is compatible if and only if all splits in the set are pairwise
compatible. And of course, any subset of a compatible set of splits is again
compatible.
There is a close connection between compatible sets of splits and evolutionary
trees. Any edge e in an unrooted tree T splits the set of leaves of T into two
non-empty parts. Let Σ(T ) denote the set of splits associated with the edges of
atreeT . Then Theorem 1 (from [SS03]) gives the relation between compatible
sets of splits and evolutionary trees.
Theorem 1 (Splits-Equivalence Theorem). Let Σ be a collection of splits
on X. Then, there is an evolutionary tree T such that Σ=Σ(T ) if and only if
Σ is a compatible set of splits. If T exists it is unique up to isomorphisms.
From now on we shall use the terms compatible set of splits/ evolutionary
tree and split/ edge interchangeably. They are one and the same: Table 2.1
shows a compatible set of (weighted) splits, and Figure 2.6 shows the equivalent
evolutionary tree. Recall the discussion of evolutionary trees versus phylogenetic
trees; when working with a method such as the refined Buneman tree algorithm
which outputs compatible sets of splits which might or might now correspond
to a fully resolved tree, it is important to be able to such a tree in a precise
manner. When dealing with e.g Neighbor-Joining, we can rely on the more
regular phylogenetic trees since the NJ method always resolves trees completely.
Lemma 1 is due to Dan Gusfield ([Gus91], section 1.2) and gives an important
upper bound for the time required to go from compatible sets of splits to
phylogenetic trees.
Lemma 1. An unrooted tree with n leaves can be constructed from its set of
non-trivial splits in time O(kn), wherek is the number of non-trivial splits.
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