  We propose the following simple stochastic model for phylogenetic trees. New
types are born and die according to a birth and death chain. At each birth we
associate a fitness to the new type sampled from a fixed distribution. At each
death the type with the smallest fitness is killed. We show that if the birth
(i.e. mutation) rate is subcritical we get a phylogenetic tree consistent with
an influenza tree (few types at any given time and one dominating type lasting
a long time). When the birth rate is supercritical we get a phylogenetic tree
consistent with an HIV tree (many types at any given time, none lasting very
long).
