Before revision:  Against this backdrop, the present paperaims to provide a graph theoretical framework for solving different problems on subdivision trees in a simple and unified manner. 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:  Against this backdrop, the present paperaims to provide a graph theoretical 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 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: