Phylogenetic networks are a useful model that can represent reticulate evolution and complex biological data. In recent years, mathematical and computational aspects of tree-based networks have been well studied. However, not all phylogenetic networks are tree-based, so it is meaningful to consider how close a given network is to being tree-based; Francis–Steel–Semple (2018) proposed several different indices to measure the degree of deviation of a phylogenetic network from being tree-based. One is the minimum number of leaves that need to be added to convert a given network to tree-based, and another is the number of vertices that are not included in the largest subtree covering its leaf-set. Both values are zero if and only if the network is tree-based. Both deviation indices can be computed efficiently, but the relationship between the above two is unknown, as each has been studied using different approaches. In this study, we derive a tight inequality for the values of the two measures and also give a characterisation of phylogenetic networks such that they coincide.

Tree-based phylogenetic networks provide a powerful model for representing complex data or non-tree-like evolution. Such networks consist of an underlying evolutionary tree called a “support tree” (also known as a “subdivision tree”) together with extra arcs added between the edges of that tree. However, a tree-based network can have exponentially many support trees, and this leads to a variety of computational problems. Recently, Hayamizu established a theory called the structure theorem for rooted binary phylogenetic networks and provided linear-time and linear-delay algorithms for different problems, such as counting, optimization, and enumeration of support trees. However, in practice, it is often more useful to search for both optimal and near-optimal solutions than to calculate only an optimal solution. In the present paper, we thus consider the following problem: Given a tree-based phylogenetic network N where each arc is weighted by its probability, compute the ranking of top-k support trees of N according to likelihood values. We provide a linear-delay (and hence optimal) algorithm for this problem.