  We consider the tree-valued Fleming-Viot process, $(\mathcal X_t)_{t\geq 0}$,
with mutation and selection as studied in Depperschmidt, Greven, Pfaffelhuber
(2012). This process models the stochastic evolution of the genealogies and
(allelic) types under resampling, mutation and selection in the population
currently alive in the limit of infinitely large populations. Genealogies and
types are described by (isometry classes of) marked metric measure spaces. The
long-time limit of the neutral tree-valued Fleming-Viot dynamics is an
equilibrium given via the marked metric measure space associated with the
Kingman coalescent. In the present paper we pursue two closely linked goals.
First, we show that two well-known properties of the neutral Fleming-Viot
genealogies at fixed time $t$ arising from the properties of the dual, namely
the Kingman coalescent, hold for the whole path. These properties are related
to the geometry of the family tree close to its leaves. In particular we
consider the number and the size of subfamilies whose individuals are not
further than $\ve$ apart in the limit $\ve\to 0$. Second, we answer two open
questions about the sample paths of the tree-valued Fleming-Viot process. We
show that for all $t>0$ almost surely the marked metric measure space $\mathcal
X_t$ has no atoms and admits a mark function. The latter property means that
all individuals in the tree-valued Fleming-Viot process can uniquely be
assigned a type. All main results are proven for the neutral case and then
carried over to selective cases via Girsanov's formula giving absolute
continuity.
