Many important problems in extremal graph theory can be stated as certifying polynomial inequalities in graph homomorphism numbers, and in particular, many ask to certify pure binomial inequalities. For a fixed collection of graphs U, the tropicalization of the graph profile of U essentially records all valid pure binomial inequalities involving graph ho- momorphism numbers for graphs in U. Building upon ideas and techniques described by Blekherman and Raymond in 2022, we compute the tropicalization of the graph profile for K1 and S2,1k -trees for 0 ≤ k ≤ m−1, that is, stars with k+1 branches, one of which is subdi- vided. We call these almost-stars. This allows pure binomial inequalities in homomorphism numbers (or densities) for these graphs to be verified through a linear program in m + 1 variables and m + 5 constraints. We give a conjecture for the f-vector of this tropicalization. We also present a conjecture for the tropicalization of the graph profile for K1, stars, and almost-stars.