  'Tree-based' phylogenetic networks proposed by Francis and Steel have
attracted much attention of theoretical biologists in the last few years. At
the heart of the definitions of tree-based phylogenetic networks is the notion
of 'support trees', about which there are numerous algorithmic problems that
are important for evolutionary data analysis. Recently, Hayamizu
(arXiv:1811.05849 [math.CO]) proved a structure theorem for tree-based
phylogenetic networks and obtained linear-time and linear-delay algorithms for
many basic problems on support trees, such as counting, optimisation, and
enumeration. In the present paper, we consider the following fundamental
problem in statistical data analysis: given a tree-based phylogenetic network
$N$ whose arcs are associated with probability, create the top-$k$ support tree
ranking for $N$ by their likelihood values. We provide a linear-delay (and
hence optimal) algorithm for the problem and thus reveal the interesting
property of tree-based phylogenetic networks that ranking top-$k$ support trees
is as computationally easy as picking $k$ arbitrary support trees.
