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The firefighter problem with $k$ firefighters on an infinite graph $G$ is aniterative graph process, defined as follows: Suppose a fire breaks out at agiven 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 subsequentodd turn, any vertex that is on fire spreads the fire to all adjacentunprotected vertices. The firefighters' goal is to eventually stop the spreadof the fire. If there exists a strategy for $k$ firefighters to eventually stopthe spread of the fire, then we say $G$ is $k$-containable.  We consider the firefighter problem on the hexagonal grid, which is the graphwhose vertices and edges are exactly the vertices and edges of a regularhexagonal 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 thefirefighters have one firefighter per turn and one extra firefighter on twoturns, the firefighters can contain the fire. We improve on this result byshowing that even with only one extra firefighter on one turn, the firefighterscan still contain the fire.  In addition, we explore $k$-containability for birth sequence trees, whichare infinite rooted trees that have the property that every vertex at the samelevel has the same degree. A birth sequence forest is an infinite forest, eachcomponent of which is a birth sequence tree. For birth sequence trees andforests, the fire always starts at the root of each tree. We provide apseudopolynomial time algorithm to decide if all the vertices at a fixed levelcan be protected or not.

Firefighting on the Hexagonal Grid and on Infinite Trees