  One strategy for reconstruction of phylogenetic networks is to solve the
phylogenetic network problem, which involves inferring phylogenetic trees first
and subsequently computing the smallest phylogenetic network that displays all
the trees. This approach capitalizes on exceptional tools available for
inferring phylogenetic trees from biomolecular sequences. Since the vast space
of phylogenetic networks poses difficulties in obtaining comprehensive
sampling, the researchers switch their attention to inferring tree-child
networks from multiple phylogenetic trees, where in a tree-child network each
non-leaf node must have at least one child that is an indegree-one node.
  Two results are obtained: (1) The tree-child network inference problem for
multiple line trees remains NP-hard by a reduction from the shortest common
supersequence problem for permutations and proving that the latter is NP-hard.
(2) The tree-child networks with the least reticulate nodes that display all
the line trees are the same as that display all the binary trees, whose
hybridization number is $\Theta(n^3)$ for $n (> 7)$ taxa.
