Testing the Agreement of Trees with Internal Labels

Abstract

The input to the agreement problem is a collection \(\mathcal {P}= \{\mathcal {T}_1, \mathcal {T}_2, \dots , \mathcal {T}_k\}\) of phylogenetic trees, called input trees, over partially overlapping sets of taxa. The question is whether there exists a tree \(\mathcal {T}\), called an agreement tree, whose taxon set is the union of the taxon sets of the input trees, such that for each \(i \in \{1, 2, \dots , k\}\), the restriction of \(\mathcal {T}\) to the taxon set of \(\mathcal {T}_i\) is isomorphic to \(\mathcal {T}_i\). We give a \(\mathcal {O}(n k (\sum _{i \in [k]} d_i + \log ^2(nk)))\) algorithm for a generalization of the agreement problem in which the input trees may have internal labels, where n is the total number of distinct taxa in \(\mathcal {P}\), k is the number of trees in \(\mathcal {P}\), and \(d_i\) is the maximum number of children of a node in \(\mathcal {T}_i\).