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Tree-based networks are a class of phylogenetic networks that attempt toformally capture what is meant by "tree-like" evolution. A given non-tree-basedphylogenetic network, however, might appear to be very close to beingtree-based, or very far. In this paper, we formalise the notion of proximity totree-based for unrooted phylogenetic networks, with a range of proximitymeasures. These measures also provide characterisations of tree-based networks.One measure in particular, related to the nearest neighbour interchangeoperation, allows us to define the notion of "tree-based rank". This provides asubclassification within the tree-based networks themselves, identifying thosenetworks that are "very" tree-based. Finally, we prove results relatingtree-based networks in the settings of rooted and unrooted phylogeneticnetworks, showing effectively that an unrooted network is tree-based if andonly if it can be made a rooted tree-based network by rooting it and orientingthe edges appropriately. This leads to a clarification of the contrastingdecision problems for tree-based networks, which are polynomial in the rootedcase but NP complete in the unrooted.

How tree-based is my network? Proximity measures for unrooted phylogenetic networks