This paper provides a unifying framework for minimization and learning algorithms for bottom-up tree automata valued in monoidal categories. Our aim is two-fold: encompass existing algorithms for various forms of tree automata (with side-effects) – deterministic bottom-up tree automata, residual finite tree automata, tree automata weighted over a field – and instantiate the abstract framework in order to obtain new results – for tree automata weighted over principal ideal domains (PIDs).