  Building on the spinal decomposition technique in Foutel-Rodier and Schertzer
(2022) we prove a Yaglom limit law for the rescaled size of a nearly critical
branching process in varying environment conditional on survival. In addition,
our spinal approach allows us to prove convergence of the genealogical
structure of the population at a fixed time horizon - where the sequence of
trees are envisioned as a sequence of metric spaces - in the
Gromov-Hausdorff-Prokorov (GHP) topology. We characterize the limiting metric
space as a time-changed version of the Brownian coalescent point process of
Popovic (2004).
  Beyond our specific model, we derive several general results allowing to go
from spinal decompositions to convergence of random trees in the GHP topology.
As a direct application, we show how this type of convergence naturally
condenses the limit of several interesting genealogical quantities: the
population size, the time to the most-recent common ancestor, the reduced tree
and the tree generated by $k$ uniformly sampled individuals. As in a recent
article by the authors (Foutel-Rodier and Schertzer 2022), we hope that our
specific example illustrates a general methodology that could be applied to
more complex branching processes.
