  Phylogenetic networks model reticulate evolutionary histories. The last two
decades have seen an increased interest in establishing mathematical results
and developing computational methods for inferring and analyzing these
networks. A salient concept underlying a great majority of these developments
has been the notion that a network displays a set of trees and those trees can
be used to infer, analyze, and study the network.
  In this paper, we show that in the presence of coalescence effects, the set
of displayed trees is not sufficient to capture the network. We formally define
the set of parental trees of a network and make three contributions based on
this definition. First, we extend the notion of anomaly zone to phylogenetic
networks and report on anomaly results for different networks. Second, we
demonstrate how coalescence events could negatively affect the ability to infer
a species tree that could be augmented into the correct network. Third, we
demonstrate how a phylogenetic network can be viewed as a mixture model that
lends itself to a novel inference approach via gene tree clustering.
  Our results demonstrate the limitations of focusing on the set of trees
displayed by a network when analyzing and inferring the network. Our findings
can form the basis for achieving higher accuracy when inferring phylogenetic
networks and open up new venues for research in this area, including new
problem formulations based on the notion of a network's parental trees.
