Ancient gene duplication events have left many traces in vertebrate genomes. Reconciled trees represent the differences between gene family trees and the species phylogeny those genes are sampled from, allowing us to both infer gene duplication events and estimate a species phylogeny from a sample of gene families. We show that analysis of 118 gene families yields a phylogeny of vertebrates largely in agreement with other data. We formulate the problem of locating episodes of gene duplication as a set cover problem: given a species tree in which each node has a set of gene duplications associated with it, the smallest set of species nodes whose union includes all gene duplications specifies the locations of gene duplication episodes. By generating a unique mapping from this cover set we can determine the minimal number of such episodes at each location. When applied to our data, this method reveals a complex history of gene duplications in vertebrate evolution that does not conform to the "2R" hypothesis.