  Phylogenetic networks are a type of directed acyclic graph that represent how
a set $X$ of present-day species are descended from a common ancestor by
processes of speciation and reticulate evolution. In the absence of reticulate
evolution, such networks are simply phylogenetic (evolutionary) trees.
Moreover, phylogenetic networks that are not trees can sometimes be represented
as phylogenetic trees with additional directed edges placed between their
edges. Such networks are called {\em tree based}, and the class of phylogenetic
networks that are tree based has recently been characterised. In this paper, we
establish a number of new characterisations of tree-based networks in terms of
path partitions and antichains (in the spirit of Dilworth's theorem), as well
as via matchings in a bipartite graph. We also show that a temporal network is
tree based if and only if it satisfies an antichain-to-leaf condition. In the
second part of the paper, we define three indices that measure the extent to
which an arbitrary phylogenetic network deviates from being tree based. We
describe how these three indices can be described exactly and computed
efficiently using classical results concerning maximum-sized matchings in
bipartite graphs.
