Refined Buneman Trees

Algorithm 2 The Neighbor-Joining algorithm
Require: δ is a distance matrix on n species X
Ensure: T is a unrooted binary tree with leaves from X
1: initialise T such that for each species x i ∈ X there is a leaf node n i ∈ T
2: define a set of active nodes L containing the leaves of T
3: while |L| > 2 do
4: find nodes n i ,n j ∈ L such that
D ij = d(i, j) − (r i + r j ) where r i = ∑ k∈L
d(i, k)
|L|−2
is minimal
5: remove nodes n i ,n j from L
6: create a new node n k and set distances from n k to remaining nodes n m ∈ L
such that d(k, m) =(d(i, m)+d(j, m) − d(i, j))/2
7: add n k to L, T and connect n k to n i ,n j such that |e ik | = 1 2 (d(i, j)+r i−r j )
and |e jk | = 1 2 (d(i, j) − r i + r j )
8: end while
9: join the last two nodes n i ,n j ∈ L such that |e ij | = d(i, j)
4.1.3 Quartet based methods
Buneman trees ([Bun71]) are a new form 1 of distance method that relies on
quartets and splits rather than just picking closest pairs. An algorithm with a
running time complexity of O(n 3 ) is available, but not given here (see [BB99],
section3). Another method which relies on quartets is the Q ∗ method proposed
by Berry & Gascuel ([BG00]) which runs in time O(n 4 ).
The latter illustrates the problem for quartet based methods: given a set
of quartets, these quartets do not nescessarily support a tree, e.g. they might
contain contradictory constraints such as quartets wanting to split species in
opposite ways. In the Buneman case, a set of splits is generated from quartets
which are guaranteed to be tree-consistent — in the case of Q ∗ , a tree-consistent
set of quartets is found by weeding out in a larger set of favorable quartets. The
intuition is that by considering quartets, we are looking at the species in a global
sense, determining which species should be separated from other species and
trying to construct a tree under a large number of such (possibly conflicting)
constraints. This is in many ways the opposite of the intuition behind the
clustering methods mentioned earlier.
Another issue for both of these algorithms is that they produce only partially
resolved trees. Compared to the UPGMA and NJ methods we might say that
these quartet based methods only resolve “safe” branches where the clustering
methods resolve trees fully, regardless of data. However, we also have to say that
these methods are perhaps too safe since they might only infer a small fraction
of edges in the evolutionary tree, depending on data ([BG00], [BB99]). The
1 “new form” compared to the clustering methods UPGMA and Neighbor-Joining
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