  The firefighter problem with $k$ firefighters on an infinite graph $G$ is an
iterative graph process, defined as follows: Suppose a fire breaks out at a
given vertex $v\in V(G)$ on Turn 1. On each subsequent even turn, $k$
firefighters protect $k$ vertices that are not on fire, and on each subsequent
odd turn, any vertex that is on fire spreads the fire to all adjacent
unprotected vertices. The firefighters' goal is to eventually stop the spread
of the fire. If there exists a strategy for $k$ firefighters to eventually stop
the spread of the fire, then we say $G$ is $k$-containable.
  We consider the firefighter problem on the hexagonal grid, which is the graph
whose vertices and edges are exactly the vertices and edges of a regular
hexagonal tiling of the plane. It is not known if the hexagonal grid is
$1$-containable. In arXiv:1305.7076 [math.CO], it was shown that if the
firefighters have one firefighter per turn and one extra firefighter on two
turns, the firefighters can contain the fire. We improve on this result by
showing that even with only one extra firefighter on one turn, the firefighters
can still contain the fire.
  In addition, we explore $k$-containability for birth sequence trees, which
are infinite rooted trees that have the property that every vertex at the same
level has the same degree. A birth sequence forest is an infinite forest, each
component of which is a birth sequence tree. For birth sequence trees and
forests, the fire always starts at the root of each tree. We provide a
pseudopolynomial time algorithm to decide if all the vertices at a fixed level
can be protected or not.
