  We consider a branching random walk on $\Z$, where the particles behave
differently in visited and unvisited sites. Informally, each site on the
positive half-line contains initially a cookie. On the first visit of a site
its cookie is removed and particles at positions with a cookie reproduce and
move differently from particles on sites without cookies. Therefore, the
movement and the reproduction of the particles depend on the previous behaviour
of the population of particles. We study the question if the process is
recurrent or transient, i.e., whether infinitely many particles visit the
origin or not.
