  We introduce and study a model of plane random trees generalizing the famous
Bienaym\'e--Galton--Watson model but where births and deaths are locally
correlated. More precisely, given a random variable $(B,H)$ with values in
$\{1,2,3, \dots\}^2$, given the state of the tree at some generation, the next
generation is obtained (informally) by successively deleting $B$ individuals
side-by-side and replacing them with $H$ new particles where the samplings are
i.i.d. We prove that, in the critical case $\mathbb{E}[B]=\mathbb{E}[H]$, and
under a third moment condition on $B$ and $H$, the random trees coding the
genealogy of the population model converges towards the Brownian Continuum
Random Tree. Interestingly, our proof does not use the classical height process
or the {\L}ukasiewicz exploration, but rather the stochastic flow point of view
introduced by Bertoin and Le Gall.
