  In the Firefighter problem, introduced by Hartnell in 1995, a fire spreads
through a graph while a player chooses which vertices to protect in order to
contain it. In this paper, we focus on the case of trees and we consider as
well the Fractional Firefighter game where the amount of protection allocated
to a vertex lies between 0 and 1. While most of the work in this area deals
with a constant amount of firefighters available at each turn, we consider
three research questions which arise when including the sequence of
firefighters as part of the instance. We first introduce the online version of
both Firefighter and Fractional Firefighter, in which the number of
firefighters available at each turn is revealed over time. We show that a
greedy algorithm on finite trees is 1/2-competitive for both online versions,
which generalises a result previously known for special cases of Firefighter.
We also show that the optimal competitive ratio of online Firefighter ranges
between 1/2 and the inverse of the golden ratio. Next, given two firefighter
sequences, we discuss sufficient conditions for the existence of an infinite
tree that separates them, in the sense that the fire can be contained with one
sequence but not with the other. To this aim, we study a new purely numerical
game called targeting game. Finally, we give sufficient conditions for the fire
to be contained, expressed as the asymptotic comparison of the number of
firefighters and the size of the tree levels.
