Constructing a Tree from Homeomorphic Subtrees, with

We have given ecient algorithms for two natural combinatorial problems, and have demonstrated that these algorithms can have a signicant impact on fundamental problems in computational evolutionary biology. a tree from lowest common ancestors with an application to the optimization of relational expressions, SIAM 11 rooted trees, such each tree is bijectively labelled by the species in a set S. Clearly, no consensus exists according to our denition above unless these trees are identical. Thus, the question of inferring the consensus of a set of dierent trees which have the same leaf set depends upon a model for handling disagreements between trees. This is an important question in computational evolutionary biology because it is frequently the case that the biologist can not determine one single tree for a data set, but instead computes many, each of which is equally plausible. This may arise because dierent methods are used to construct the trees, each based possibly on a dierent data set for the same species set (some could be biomolecular while others could be distances). Even when one method is chosen, there can be many dierent equally good trees for the optimization criterion upon which the method is based, so that the output is not one best tree, but many trees, each of which has a good score for the optimization criterion. The number of dierent trees can be quite large. 1 In [10], a model for problem of computing consensus trees was presented, based upon the concept of a local consensus rule. A local consensus rule determines the exact form of (possibly) each rooted subtree of every triple of species drawn from S , based upon how the trees in the input prole resolve that triple, and a local consensus tree is then required to have the form specied by the local consensus rule for each triple a; b; c of species. When the local consensus rule is entire, so that the form on each triple is specied, it is possible to construct the local consensus tree by using the algorithms of [9]. Because a resolved subtree on a; b; c indicates a denite hypothesis about the evolution of a; b; c from a common ancestor, most of the biologically relevant local consensus rules will not specify the form of the local consensus tree on a triple unless the evolution of that triple from a common ancestor can be determined …