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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.
Phylogenetic networks are a generalization of phylogenetic trees allowing for the representation of non-treelike evolutionary events such as hybridization. Typically, such networks have been analyzed based on their `level, i.e. based on the complexit
Phylogenetic networks are generalizations of phylogenetic trees that allow the representation of reticulation events such as horizontal gene transfer or hybridization, and can also represent uncertainty in inference. A subclass of these, tree-based p
A tree-based network $N$ on $X$ is called universal if every phylogenetic tree on $X$ is a base tree for $N$. Recently, binary universal tree-based networks have attracted great attention in the literature and their existence has been analyzed in var
Phylogenetic Diversity (PD) is a prominent quantitative measure of the biodiversity of a collection of present-day species (taxa). This measure is based on the evolutionary distance among the species in the collection. Loosely speaking, if $mathcal{T
Rooted phylogenetic networks provide a way to describe species relationships when evolution departs from the simple model of a tree. However, networks inferred from genomic data can be highly tangled, making it difficult to discern the main reticulat