  The Firefighter problem and a variant of it, known as Resource Minimization
for Fire Containment (RMFC), are natural models for optimal inhibition of
harmful spreading processes. Despite considerable progress on several fronts,
the approximability of these problems is still badly understood. This is the
case even when the underlying graph is a tree, which is one of the most-studied
graph structures in this context and the focus of this paper. In their simplest
version, a fire spreads from one fixed vertex step by step from burning to
adjacent non-burning vertices, and at each time step, $B$ many non-burning
vertices can be protected from catching fire. The Firefighter problem asks, for
a given $B$, to maximize the number of vertices that will not catch fire,
whereas RMFC (on a tree) asks to find the smallest $B$ that allows for saving
all leaves of the tree. Prior to this work, the best known approximation ratios
were an $O(1)$-approximation for the Firefighter problem and an $O(\log^*
n)$-approximation for RMFC, both being LP-based and essentially matching the
integrality gaps of two natural LP relaxations.
  We improve on both approximations by presenting a PTAS for the Firefighter
problem and an $O(1)$-approximation for RMFC, both qualitatively matching the
known hardness results. Our results are obtained through a combination of the
known LPs with several new techniques, which allow for efficiently enumerating
subsets of super-constant size of a good solution to obtain stronger LPs.
