  Polyploidization is an evolutionary process by which a species acquires
multiple copies of its complete set of chromosomes. The reticulate nature of
the signal left behind by it means that phylogenetic networks offer themselves
as a framework to reconstruct the evolutionary past of species affected by it.
The main strategy for doing this is to first construct a so called
multiple-labelled tree and to then somehow derive such a network from it. The
following question therefore arises: How much can be said about that past if
such a tree is not readily available? By viewing a polyploid dataset as a
certain vector which we call a ploidy (level) profile we show that, among other
results, there always exists a phylogenetic network in the form of a beaded
phylogenetic tree with additional arcs that realizes a given ploidy profile.
Intriguingly, the two end vertices of almost all of these additional arcs can
be interpreted as having co-existed in time thereby adding biological realism
to our network, a feature that is, in general, not enjoyed by phylogenetic
networks. In addition, we show that our network may be viewed as a generator of
ploidy profile space, a novel concept similar to phylogenetic tree space that
we introduce to be able to compare phylogenetic networks that realize one and
the same ploidy profile. We illustrate our findings in terms of a publicly
available Viola dataset.
