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 overlap** 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 $\mathcal {T}_i$.