  We consider the following Markovian dynamic on point processes: at constant
rate and with equal probability, either the rightmost atom of the current
configuration is removed, or a new atom is added at a random distance from the
rightmost atom. Interpreting atoms as limit buy orders, this process was
introduced by Lakner et al. to model a one-sided limit order book. We consider
this model in the regime where the total number of orders converges to a
reflected Brownian motion, and complement the results of Lakner et al. by
showing that, in the case where the mean displacement at which a new order is
added is positive, the measure-valued process describing the whole limit order
book converges to a simple functional of this reflected Brownian motion. Our
results make it possible to derive useful and explicit approximations on
various quantities of interest such as the depth or the total value of the
book. Our approach leverages an unexpected connection with L\'evy trees. More
precisely, the cornerstone of our approach is the regenerative characterization
of L\'evy trees due to Weill, which provides an elegant proof strategy which we
unfold.
