Species tree estimation is a basic part of many biological research projects, ranging from answering basic evolutionary questions (e.g., how did a group of species adapt to their environments?) to addressing questions in functional biology. Yet, species tree estimation is very challenging, due to processes such as incomplete lineage sorting, gene duplication and loss, horizontal gene transfer, and hybridization, which can make gene trees differ from each other and from the overall evolutionary history of the species. Over the last 10–20 years, there has been tremendous growth in methods and mathematical theory for estimating species trees and phylogenetic networks, and some of these methods are now in wide use. In this survey, we provide an overview of the current state of the art, identify the limitations of existing methods and theory, and propose additional research problems and directions.

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