Semilinear metric semilattices on $\mathbb R$-trees
  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. 
P.D. Andreev