It is easy to prove that the completion of your space can be any separable metric space where metric spheres are nowhere dense.
Does not it scare you?
Answer to the comment: Not all of your spaces can be embedded into a metric tree.
BTW, there is a nice characterization of subsets in a metric tree: $$ | x - y | + | a - b | \le |x-a| + |x-b| + |y-a| + |y-b|$$ for all points $x,y,a,b$ (here $|x-y|$ denotes the distance from $x$ to $y$). This inequality can be also thought as a discrete analog of CAT[−∞] inequality.