We develop technical tools that enable the use of Bellman functions for BMO defined on a-trees, which are structures that generalize dyadic lattices. As applications, we prove the integral John-Nirenberg inequality and an inequality relating L-1- and L-2-oscillations for BMO on a-trees, with explicit constants. When the tree in question is the collection of all dyadic cubes in R-n, the inequalities proved are sharp. We also reformulate the John-Nirenberg inequality for the continuous BMO in terms of special martingales generated by BMO functions. The tools presented can be used for any function class that corresponds to a non-convex Bellman domain.