  A metric phylogenetic tree relating a collection of taxa induces weighted
rooted triples and weighted quartets for all subsets of three and four taxa,
respectively. New intertaxon distances are defined that can be calculated from
these weights, and shown to exactly fit the same tree topology, but with edge
weights rescaled by certain factors dependent on the associated split size.
These distances are analogs for metric trees of similar ones recently
introduced for topological trees that are based on induced unweighted rooted
triples and quartets. The distances introduced here lead to new statistically
consistent methods of inferring a metric species tree from a collection of
topological gene trees generated under the multispecies coalescent model of
incomplete lineage sorting. Simulations provide insight into their potential.
