  The large deviation function has been known for a long time in the literature
for the displacement of the rightmost particle in a branching random walk
(BRW), or in a branching Brownian motion (BBM). More recently a number of
generalizations of the BBM and of the BRW have been considered where selection
or coalescence mechanisms tend to limit the exponential growth of the number of
particles. Here we try to estimate the large deviation function of the position
of the rightmost particle for several such generalizations: the $L$-BBM, the
$N$-BBM, and the CBRW (coalescing branching random walk) which is closely
related to the noisy FKPP equation. Our approach allows us to obtain only upper
bounds on these large deviation functions. One noticeable feature of our
results is their non analytic dependence on the parameters (such as the
coalescence rate in the CBRW).
