  Phylogenetic trees are often constructed by using a metric on the set of taxa
that label the leaves of the tree. While there are a number of methods for
constructing a tree using a given metric, such trees will only display the
metric if it satisfies the so-called "four point condition", established by
Buneman in 1971. While this condition guarantees that a unique tree will
display the metric, meaning that the distance between any two leaves can be
found by adding the distances on arcs in the path between the leaves, it
doesn't exclude the possibility that a phylogenetic network might also display
the metric. This possibility was recently pointed out and "tree-metrized"
networks --- that display a tree metric --- with a single reticulation were
characterized. In this paper, we show that in the case of HGT (horizontal gene
transfer) networks, in fact there are tree-metrized networks containing many
reticulations.
