  In the present paper, we propose a refinement for the notion of quantum
Markov states (QMS) on trees. A structure theorem for QMS on general trees is
proved. We notice that any restriction of QMS in the sense of Ref.
\cite{AccFid03} is not necessarily to be a QMS. It turns out that localized QMS
has the mentioned property which is called \textit{sub-Markov states}, this
allows us to characterize translation invariant QMS on regular trees.
