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Topological phylogenetic trees can be assigned edge weights in severalnatural ways, highlighting different aspects of the tree. Here the rootedtriple and quartet metrizations are introduced, and applied to formulate novelfast methods of inferring large trees from rooted triple and quartet data.These methods can be applied in new statistically consistent procedures forinference of a species tree from gene trees under the multispecies coalescentmodel.

Topological metrizations of trees, and new quartet methods of tree inference