Bioinformatics Algorithms: Techniques and Applications

SUPERTREES AND SUPERNETWORKS 157
By using tree mappings introduced in Section 7.2.1, we can define methods for
supertree inference that are based on the idea of retaining a largest set of taxa obtained
by removing those taxa that induce conflicts among all trees or contradictory rooted
triples. These methods naturally lead to extend to the case of a supertree the notions
of agreement and compatible subtree discussed in the previous section.
A complementary approach to compute a supertree requires that all taxa appearing
in at least one input tree must necessarily appear also in the output supertree, where
all information encoded in the input trees must be present. Also for this approach, the
notion of tree mapping (especially of tree refinement) is central for formally defining
the idea of information preservation.
7.4.1 Models and Problems
The simplest and more general problem that arises in supertree inference is the construction
of a compatible supertree.
PROBLEM 7.2 Compatible Supertree
Input: a set T ={T1,...,Tk} of phylogenetic trees.
Output: a tree T displaying all trees in T .
This formulation has the drawback that such a supertree is not guaranteed to exist,
even though the problem seems quite easy to solve, as we are looking for a tree T
whose set of clusters contains those of the input trees. Moreover, such a supertree
exists if and only if no two input clusters (possibly in different trees) are overlapping.
Please notice that the problem is much harder on unrooted trees than on rooted trees;
in fact, computing (if it exists) a compatible unrooted supertree displaying all input
trees not only is NP-hard [35] but also cannot be solved by any generic algorithm
(without time constraints!) invariant with respect to permutations of leaves’ labels
[36].
By requiring that clusters of the supertree displaying all trees preserve some strict
relationships between clusters of the input trees, we obtain a variant of the Compatible
supertree problem that is related to the agreement subtree method.
PROBLEM 7.3 Total Agreement Supertree
Input: a set T ={T1,...,Tk} of phylogenetic trees, with Ti leaf-labeled over �(Ti).
Output: a phylogenetic tree T leaf-labeled over S =∪i≤k�(Ti) such that each tree
T |�(Ti) is homeomorphic to Ti.
Observe that in the total agreement supertree problem, the computed tree T is such
that C(T |�(Ti)) = C(Ti) while given the output tree T ′ of the Compatible supertree
problem, it holds that C(Ti) is included in C(T ′ |�(Ti)).