  Species trees represent the historical divergences of populations or species,
while gene trees trace the ancestry of individual gene copies sampled within
those populations. In cases involving rapid speciation, gene trees with
topologies that differ from that of the species tree can be most probable under
the standard multispecies coalescent model, making species tree inference more
difficult. Such anomalous gene trees are not well understood except for some
small cases. In this work, we establish one constraint that applies to trees of
any size: gene trees with "caterpillar" topologies cannot be anomalous. The
proof of this involves a new combinatorial object, called a population history,
which keeps track of the number of coalescent events in each ancestral
population.
