  Bootstrap percolation is a type of cellular automaton which has been used to
model various physical phenomena, such as ferromagnetism. For each natural
number $r$, the $r$-neighbour bootstrap process is an update rule for vertices
of a graph in one of two states: `infected' or `healthy'. In consecutive
rounds, each healthy vertex with at least $r$ infected neighbours becomes
itself infected. Percolation is said to occur if every vertex is eventually
infected.
  Usually, the starting set of infected vertices is chosen at random, with all
vertices initially infected independently with probability $p$. In that case,
given a graph $G$ and infection threshold $r$, a quantity of interest is the
critical probability, $p_c(G,r)$, at which percolation becomes likely to occur.
In this paper, we look at infinite trees and, answering a problem posed by
Balogh, Peres and Pete, we show that for any $b \geq r$ and for any $\epsilon >
0$ there exists a tree $T$ with branching number $\br(T) = b$ and critical
probability $p_c(T,r) < \epsilon$. However, this is false if we limit ourselves
to the well-studied family of Galton--Watson trees. We show that for every $r
\geq 2$ there exists a constant $c_r>0$ such that if $T$ is a Galton--Watson
tree with branching number $\br(T) = b \geq r$ then p_c(T,r) > \frac{c_r}{b}
e^{-\frac{b}{r-1}}.
  We also show that this bound is sharp up to a factor of $O(b)$ by giving an
explicit family of Galton--Watson trees with critical probability bounded from
above by $C_r e^{-\frac{b}{r-1}}$ for some constant $C_r>0$.
