  We introduce quantum Markov states (QMS) in a general tree graph $G= (V, E)$,
extending the Cayley tree's case. We investigate the Markov property w.r.t. the
finer structure of the considered tree. The main result of this paper concerns
the diagonalizability of a locally faithful QMS $\varphi$ on a UHF-algebra
$\mathcal A_V$ over the considered tree by means of a suitable conditional
expectation into a maximal abelian subalgebra. Namely, we prove the existence
of a Umegaki conditional expectation $\mathfrak E : \mathcal A_V \to \mathcal
D_V$ such that $$\varphi = \varphi_{\lceil \mathcal D_V}\circ \mathfrak E.$$
Moreover, we clarify the Markovian structure of the associated classical
measure on the spectrum of the diagonal algebra $\mathcal D_V$.
