  Let $T$ be the regular tree in which every vertex has exactly $d\ge 3$
neighbours. Run a branching random walk on $T$, in which at each time step
every particle gives birth to a random number of children with mean $d$ and
finite variance, and each of these children moves independently to a uniformly
chosen neighbour of its parent. We show that, starting with one particle at
some vertex $0$ and conditionally on survival of the process, the time it takes
for every vertex within distance $r$ of $0$ to be hit by a particle of the
branching random walk is almost surely $r + \frac{2}{\log(3/2)}\log\log r +
o(\log\log r)$.
