  Consider an arbitrary large population at the present time, originated at an
unspecified arbitrary large time in the past, where individuals in the same
generation reproduce independently, forward in time, with the same offspring
distribution but potentially changing among generations. In other words, the
reproduction is driven by a Galton-Watson process in a varying environment. The
genealogy of the current generation backwards in time is uniquely determined by
the coalescent point process $(A_i, i\geq 1)$, where $A_i$ is the coalescent
time between individuals $i$ and $i+1$. In general, this process is not Markov.
In constant environment, Lambert and Popovic (2013) proposed a Markov process
of point measures to reconstruct the coalescent point process. We present a
counterexample where we show that their process does not have the Markov
property. The main contribution of this work is to propose a vector valued
Markov process $(B_i,i\geq 1)$, that reach the goal to reconstruct the
genealogy, with finite information for every $i$. Additionally, when the
offspring distributions are lineal fractional, we show that the variables
$(A_i, i\geq 1)$ are independent and identically distributed.
