? trees

Call a rooted tree supercalifragilistic if no node has isomorphic daughters. There must be a standard word for this; does anyone know what it is? A related question. Let T denote the set of binary rooted trees. There is an idempotent retraction that identifies isomorphic sibs and then repeats this process until you produce a supercalifragilistic tree. This retract is compatible with the morphism T --> 1 + T^2/2! (that means the symmetric square or set of one and two element subsets) that takes a bare root to 1 and a proper tree to its set of daughters. Does anyone know if there are any further retracts that are compatible with that morphism? Michael