The fundamental properties of the class QUASI of quasi-relabeling relations are investigated. A quasi-relabeling relation is a tree relation that is defined by a tree bimorphism (φ, L, ψ), where φ and ψ are quasi-relabeling tree homomorphisms and L is a regular tree language. Such relations admit a canonical representation, which immediately also yields that QUASI is closed under finite union. However, QUASI is not closed under intersection and complement. In addition, many standard relations on trees (e.g., branches, subtrees, v-product, v-quotient, and f-top-catenation) are not quasi-relabeling relations. If quasi-relabeling relations are considered as string relations (by taking the yields of the trees), then every Cartesian product of two context-free string languages is a quasi-relabeling relation. Finally, the connections between quasi-relabeling relations, alphabetic relations, and classes of tree relations defined by several types of top-down tree transducers are presented. These connections yield that quasi-relabeling relations preserve the regular and algebraic tree languages.
Deciding Top-Down Determinism of Regular Tree Languages