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AbstractLet a family S of spaces and a class F of mappings between members of S be given. For two spaces X and Y in S we define $Y{\le }_{F}X$ if there exists a surjection f ∈ F of X onto Y. We investigate the quasi-order ${\le }_F$ in the family of dendrites, where F is one of the following classes of mappings: retractions, monotone, open, confluent or weakly confluent mappings. In particular, we investigate minimal and maximal elements, chains and antichains in the quasi-order ${\le }_F$, and characterize spaces which can be...