Asymptotic surviving rate of trees with multiple fire sources

For Hartnell’s firefighter game with ff vertices initially on fire and at most dd defended vertices per round, the surviving rate ρ(G,f,d)ρ(G,f,d) of a graph GG is the average proportion of its vertices that can be saved in the game on GG, where the average is taken over all sets of ff fire sources. Cai et al. (2010) showed that ρ(T,1,1)=1−O(lognn)=1−o(1) for every tree TT of order nn.We study the maximum value c(f,d)c(f,d) such that ρ(T,f,d)≥c(f,d)−o(1)ρ(T,f,d)≥c(f,d)−o(1) for every tree TT of order nn, where the o(1)o(1) term tends to 00 as nn tends to infinity. In this notation, the result of Cai et al. states c(1,1)=1c(1,1)=1. Our main results are that c(f,1)≥(13)f and that 49≤c(2,1)≤34.