We consider the set M={ DB, LDB, NDB, LNDB, H, LH, NH} of which the elements are the class of deterministic bottom-up tree transformations (DB), the linear, the nondeleting and the linear, nondeleting subclasses of DB (LDB, NDB and LNDB); the class of homomorphism tree transformations (H), the linear and the nondeleting subclasses of H (LH and NH). The main aim of this paper is to obtain a finite generating system of the equations of the form X 1∘⋯∘ X m = Y 1∘⋯ Y n , where X i , Y j ϵM for 1⩽ i⩽ m, 1⩽ j⩽ n and ∘ is the composition of tree transformation classes. Therefore, we give a finite Thue system over M, where M is now considered as a seven-letter alphabet, such that two words X 1•⋯• X m and Y 1•⋯• Y n of the free monoid M ∗ are congruent modulo the Thue congruence ↔ T ∗ generated by T over M ∗ if and only if X 1∘⋯∘ X m = Y 1∘⋯∘ Y n . Moreover, we give a set N of representatives for the congruence ↔ T ∗ and an inclusion diagram of the tree transformation classes of the form Z 1∘⋯∘ Z k , where Z 1•⋯• Z k ϵN. We also present an algorithm, that to each word X 1•⋯•X mϵM ∗ produces the (unique) representative Z 1•⋯• Z k ϵN such that X 1•⋯• X m ↔ T ∗ Z 1•⋯• Z k . We prove that, using this algorithm and the inclusion diagram of the tree transformation classes represented by the elements of N, we can decide for any given tree transformation classes X 1∘⋯∘ X m and Y 1∘⋯∘ Y n obtained by composition from elements of M if X 1∘⋯∘ X m ⊆ Y 1∘⋯∘ Y n and, hence, if X 1∘⋯∘ X m = Y 1∘⋯∘ Y n .