  We consider catalytic branching populations. They consist of a catalyst
population evolving according to a critical binary branching process in
continuous time with a constant branching rate and a reactant population with a
branching rate proportional to the number of catalyst individuals alive. The
reactant forms a process in random medium. We describe asymptotically the
genealogy of catalytic branching populations coded as the induced forest of
$\mathbb{R}$-trees using the many individuals--rapid branching continuum limit.
The limiting continuum genealogical forests are then studied in detail from
both the quenched and annealed points of view. The result is obtained by
constructing a contour process and analyzing the appropriately rescaled version
and its limit. The genealogy of the limiting forest is described by a point
process. We compare geometric properties and statistics of the reactant limit
forest with those of the "classical" forest.
