We study the agreement supertree approach for combining rooted phylogenetic trees when the input trees do not fully agree on the relative positions of the taxa. We consider two ways to deal with such conflict. The first is to contract a set of edges in the input trees so that the resulting trees have an agreement supertree. We show that this problem is NP-complete and give a fixed-parameter tractable (FPT) algorithm for the problem parameterized by the number of input trees and the number of edges contracted. The second approach is to remove a set of taxa from the input trees so that the resulting trees have an agreement supertree. Guillemot and Berry (2010) gave an FPT algorithm for this problem when the input trees are all binary. We give an FPT algorithm for the more general case where the input trees are allowed to have arbitrary degree.
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