  We consider the NP-hard Tree Containment problem that has important
applications in phylogenetics. The problem asks if a given leaf-labeled network
contains a subdivision of a given leaf-labeled tree. We develop a fast
algorithm for the case that the input network is indeed a tree in which
multiple leaves might share a label. By combining this algorithm with a
generalization of a previously known decomposition scheme, we improve the
running time on reticulation visible networks and nearly stable networks to
linear time. While these are special classes of networks, they rank among the
most general of the previously considered classes.
