  We consider the evolution of the genealogy of the population currently alive
in a Feller branching diffusion model. In contrast to the approach via labeled
trees in the continuum random tree world, the genealogies are modeled as
equivalence classes of ultrametric measure spaces, the elements of the space
$\mathbb{U}$. This space is Polish and has a rich semigroup structure for the
genealogy. We focus on the evolution of the genealogy in time and the large
time asymptotics conditioned both on survival up to present time and on
survival forever. We prove existence, uniqueness and Feller property of
solutions of the martingale problem for this genealogy valued, i.e.,
$\mathbb{U}$-valued Feller diffusion. We give the precise relation to the
time-inhomogeneous $\mathbb{U}_1$-valued Fleming-Viot process. The uniqueness
is shown via Feynman-Kac duality with the distance matrix augmented Kingman
coalescent. Using a semigroup operation on $\mathbb{U}$, called concatenation,
together with the branching property we obtain a L\'evy-Khintchine formula for
$\mathbb{U}$-valued Feller diffusion and we determine explicitly the L\'evy
measure on $\mathbb{U}\setminus\{0\}$. From this we obtain for $h>0$ the
decomposition into depth-$h$ subfamilies, a representation of the process as
concatenation of a Cox point process of genealogies of single ancestor
subfamilies. Furthermore, we will identify the $\mathbb{U}$-valued process
conditioned to survive until a finite time $T$. We study long time asymptotics,
such as generalized quasi-equilibrium and Kolmogorov-Yaglom limit law on the
level of ultrametric measure spaces. We also obtain various representations of
the long time limits.
