  We introduce a modified Galton-Watson process using the framework of an
infinite system of particles labeled by $(x,t)$, where $x$ is the rank of the
particle born at time $t$. The key assumption concerning the offspring numbers
of different particles is that they are independent, but their distributions
may depend on the particle label $(x,t)$. For the associated system of coupled
monotone Markov chains, we address the issue of pathwise duality elucidated by
a remarkable graphical representation, with the trajectories of the primary
Markov chains and their duals coalescing together to form forest graphs on a
two-dimensional grid.
