Regular sets over extended tree structures

We investigate notions of decidability and definability for the Monadic Second-Order Logic over labeled tree structures, and its relations with finite automata using oracles to test input prefixes.A general framework is defined allowing to transfer some MSO-properties from a graph-structure to a labeled tree structure. Transferred properties are the decidability of sentences and the existence of a definable model for every satisfiable formula. A class of finite automata with prefix-oracles is also defined, recognizing exactly languages defined by MSO-formulas in any labeled tree-structure.Applying these results, the well-known equivalence between languages recognized by finite automata, sets of vertices MSO definable in a tree-structure and sets of pushdown contexts generated by pushdown-automata is extended to k-iterated pushdown automata.