Before revision:  and 4 ) find a subdivision tree to maximize or minimize a prescribed objective function (optimization problem). Importantly, the results and algorithms in this paper still hold true for some non-binary phylogenetic networks and this generalization gives a partial answer to an open question from Pons, Semple, and Steel.
After revision:  and 4 ) find a subdivision tree to maximize or minimize a prescribed objective function . All algorithms proposed here are optimal in terms of time complexity. Our results do not only imply and unify various known results, but also answer many open questions and moreover enable novel applications, such as the estimation of a maximum likelihood tree underlying a tree-based network. The results and algorithms in this paper still hold true for some non-binary phylogenetic networks and this generalization gives a partial answer to an open question from Pons, Semple, and Steel.