A structure theorem for rooted binary phylogenetic networks and its implications for tree-based networks

Attempting to recognize a tree inside a phylogenetic network is a fundamental undertaking in evolutionary analysis. In the last few years, therefore, tree-based phylogenetic networks, which are defined by a spanning tree called a subdivision tree, have attracted attention of theoretical biologists. However, the application of such networks is still not easy, due to many problems whose time complexities are not clearly understood. In this paper, we provide a general framework for solving those various old or new problems from a coherent perspective, rather than analyzing the complexity of each individual problem or developing an algorithm one by one. More precisely, we establish a structure theorem that gives a way to canonically decompose any rooted binary phylogenetic network N into maximal zig-zag trails that are uniquely determined, and use it to characterize the set of subdivision trees of N in the form of a direct product, in a way reminiscent of the structure theorem for finitely generated Abelian groups. From the main results, we derive a series of linear time and linear time delay algorithms for the following problems: given a rooted binary phylogenetic network N, 1) determine whether or not N has a subdivision tree and find one if there exists any; 2) measure the deviation of N from being tree-based; 3) compute the number of subdivision trees of N; 4) list all subdivision trees of N; and 5) 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 a special class of rooted non-binary phylogenetic networks.