  Phylogenetic trees and networks are leaf-labelled graphs that are used to
describe evolutionary histories of species. The Tree Containment problem asks
whether a given phylogenetic tree is embedded in a given phylogenetic network.
Given a phylogenetic network and a cluster of species, the Cluster Containment
problem asks whether the given cluster is a cluster of some phylogenetic tree
embedded in the network. Both problems are known to be NP-complete in general.
In this article, we consider the restriction of these problems to several
well-studied classes of phylogenetic networks. We show that Tree Containment is
polynomial-time solvable for normal networks, for binary tree-child networks,
and for level-$k$ networks. On the other hand, we show that, even for
tree-sibling, time-consistent, regular networks, both Tree Containment and
Cluster Containment remain NP-complete.
