  The measure-valued Fleming-Viot process is a diffusion which models the
evolution of allele frequencies in a multi-type population. In the neutral
setting the Kingman coalescent is known to generate the genealogies of the
"individuals" in the population at a fixed time. The goal of the present paper
is to replace this static point of view on the genealogies by an analysis of
the evolution of genealogies. We encode the genealogy of the population as an
(isometry class of an) ultra-metric space which is equipped with a probability
measure. The space of ultra-metric measure spaces together with the Gromov-weak
topology serves as state space for tree-valued processes. We use well-posed
martingale problems to construct the tree-valued resampling dynamics of the
evolving genealogies for both the finite population Moran model and the
infinite population Fleming-Viot diffusion. We show that sufficient information
about any ultra-metric measure space is contained in the distribution of the
vector of subtree lengths obtained by sequentially sampled "individuals". We
give explicit formulas for the evolution of the Laplace transform of the
distribution of finite subtrees under the tree-valued Fleming-Viot dynamics.
