  We study rooted cluster algebras and rooted cluster morphisms which were
introduced in \cite{ADS13} recently and cluster structures in $2$-Calabi-Yau
triangulated categories. An example of rooted cluster morphism which is not
ideal is given, this clarifies a doubt in \cite{ADS13}. We introduce the notion
of frozenization of a seed and show that an injective rooted cluster morphism
always arises from a frozenization and a subseed. Moreover, it is a section if
and only if it arises from a subseed. This answers the Problem 7.7 in
\cite{ADS13}. We prove that an inducible rooted cluster morphism is ideal if
and only if it can be decomposed as a surjective rooted cluster morphism and an
injective rooted cluster morphism. We also introduce the tensor decompositions
of a rooted cluster algebra and of a rooted cluster morphism. For rooted
cluster algebras arising from a $2$-Calabi-Yau triangulated category
$\mathcal{C}$ with cluster tilting objects, we give an one-to-one
correspondence between certain pairs of their rooted cluster subalgebras which
we call complete pairs (see Definition \ref{def of complete pairs} for precise
meaning) and cotorsion pairs in $\mathcal{C}$.
