Before revision:  To this end, we focus on a structure called the maximal zig-zag trail decomposition that is inherent in any rooted binary phylogenetic network N and prove a structure theorem that characterizes the collection of all subdivision trees of N . Our theorem does not only imply and unify various results in the literature but also yield linear time (for enumeration, linear delay) algorithms for the following problems:
After revision:  To this end, we focus on a structure called the maximal zig-zag trail decomposition that is inherent in any rooted binary phylogenetic network N and prove a structure theorem that characterizes the collection of all 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: