Recursive schemes, algebraic trees and deterministic languages

The equivalence problems for uninterpreted recursive program schemes and deterministic pushdown automata are reducible to each other. The equivalence class of a scheme is characterized by an infinite tree which is generated by the scheme as a language by a context-free grammar and which satisfies the equations of the system. Such trees are called algebraic. Roughly speaking, a tree is algebraic iff the set of its finite branches is a deterministic language. The interreducibility of the two equivalence problems for schemes and DPDA's follows.