A key step in assembling the tree of life is the construction of species-rich phylogenies from multilocus--but often incomplete--sequence data sets. We describe previously unknown structure in the landscape of solutions to the tree reconstruction problem, comprising sometimes vast "terraces" of trees with identical quality, arranged on islands of phylogenetically similar trees. Phylogenetic ambiguity within a terrace can be characterized efficiently and then ameliorated by new algorithms for obtaining a terrace's maximum-agreement subtree or by identifying the smallest set of new targets for additional sequencing. Algorithms to find optimal trees or estimate Bayesian posterior tree distributions may need to navigate strategically in the neighborhood of large terraces in tree space.