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We study rooted cluster algebras and rooted cluster mor-phisms which were introduced in [1]recently and cluster struc-tures in 2-Calabi–Yau triangulated categories. An example of rooted cluster morphism which is not ideal is given, this clar-ifyinga doubt in [1]. We introduce the notion of freezing of a seed and show that an injective rooted cluster morphism always arises from a freezing and a subseed. Moreover, it is a section if and only if it arises from a subseed. This answers the Problem 7.7 in [1]. We prove that an inducible rooted clus-ter morphism is ideal if and only if it can be decomposed as a surjective rooted cluster morphism and an injective rooted cluster morphism. For rooted cluster algebras arising from a 2-Calabi–Yau triangulated category Cwith cluster tilting ob-jects, we give an one-to-one correspondence between certain pairs of their rooted cluster subalgebras which we call com-plete pairs (see Definition2.27) and cotorsion pairs in C.

Rooted cluster algebra；(Ideal) Rooted cluster morphism；Rooted cluster subalgebra