  Tree-based networks are a class of phylogenetic networks that attempt to
formally capture what is meant by "tree-like" evolution. A given non-tree-based
phylogenetic network, however, might appear to be very close to being
tree-based, or very far. In this paper, we formalise the notion of proximity to
tree-based for unrooted phylogenetic networks, with a range of proximity
measures. These measures also provide characterisations of tree-based networks.
One measure in particular, related to the nearest neighbour interchange
operation, allows us to define the notion of "tree-based rank". This provides a
subclassification within the tree-based networks themselves, identifying those
networks that are "very" tree-based. Finally, we prove results relating
tree-based networks in the settings of rooted and unrooted phylogenetic
networks, showing effectively that an unrooted network is tree-based if and
only if it can be made a rooted tree-based network by rooting it and orienting
the edges appropriately. This leads to a clarification of the contrasting
decision problems for tree-based networks, which are polynomial in the rooted
case but NP complete in the unrooted.
