  We introduce the notion of metric semilattice on the metric space and prove
the criterion of $\R$-tree as connected geodesic metric space $X$ admitting the
partial order, such that $X$ is semilinear metric semilattice. Also we state
the homeomorphism between topological space of orders defining upper semilinear
metric $\vee$-semilattices on locally compact complete $\mathbb R$-tree $X$ and
its metric compactification $\bar{X}_m$. As an application we construct the
example of locally complete non-homogeneous similarity-homogeneous space
showing essentiality of the condition of locally compactness in V.N.
Berestovski\v\i's conjecture on the structure of such spaces. Constructed
metric space is $\mathbb R$-tree, where every point is a branching point. It is
the metric fibration but is not topological product with factor $\mathbb R$ and
does not satisfy the Berestovski\v\i's conjecture.
