Before revision:  In this paper, we provide a structural characterization of tree-based networks that furnishes efficient algorithms for solving the following problems in linear time (for enumeration, in linear delay)  : given a rooted binary phylogenetic network N, 1) determine whether or not N is tree-based and find a subdivision tree  if there exists any (decision/search problem);
After revision:  In this paper, we provide a structural characterization of tree-based networks that furnishes efficient algorithms for solving the following problems in linear time (for enumeration, in linear delay) algorithms for the following problems : given a rooted binary phylogenetic network N, 1) determine whether or not N is tree-based and find a subdivision tree if there exists any (decision/search problem);