  A contemporary and fundamental problem faced by many evolutionary biologists
is how to puzzle together a collection $\mathcal P$ of partial trees
(leaf-labelled trees whose leaves are bijectively labelled by species or, more
generally, taxa, each supported by e. g. a gene) into an overall parental
structure that displays all trees in $\mathcal P$. This already difficult
problem is complicated by the fact that the trees in $\mathcal P$ regularly
support conflicting phylogenetic relationships and are not on the same but only
overlapping taxa sets. A desirable requirement on the sought after parental
structure therefore is that it can accommodate the observed conflicts.
Phylogenetic networks are a popular tool capable of doing precisely this.
However, not much is known about how to construct such networks from partial
trees, a notable exception being the $Z$-closure super-network approach and the
recently introduced $Q$-imputation approach. Here, we propose the usage of
closure rules to obtain such a network. In particular, we introduce the novel
$Y$-closure rule and show that this rule on its own or in combination with one
of Meacham's closure rules (which we call the $M$-rule) has some very desirable
theoretical properties. In addition, we use the $M$- and $Y$-rule to explore
the dependency of Rivera et al.'s ``ring of life'' on the fact that the
underpinning phylogenetic trees are all on the same data set. Our analysis
culminates in the presentation of a collection of induced subtrees from which
this ring can be reconstructed.
