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We develop technical tools that enable the use of Bellman functions for BMOdefined on $\alpha$-trees, which are structures that generalize dyadiclattices. As applications, we prove the integral John--Nirenberg inequality andan inequality relating $L^1$- and $L^2$-oscillations for BMO on $\alpha$-trees,with explicit constants. When the tree in question is the collection of alldyadic cubes in $\mathbb{R}^n,$ the inequalities proved are sharp. We alsoreformulate the John--Nirenberg inequality for the continuous BMO in terms ofspecial martingales generated by BMO functions. The tools presented can be usedfor any function class that corresponds to a non-convex Bellman domain.

Inequalities for BMO on $\alpha$-trees