  We introduce and analyze graph-associated entanglement cost, a generalization
of the entanglement cost of quantum states to multipartite settings. We
identify a necessary and sufficient condition for any multipartite entangled
state to be constructible when quantum communication between the multiple
parties is restricted to a quantum network represented by a tree. The condition
for exact state construction is expressed in terms of the Schmidt ranks of the
state defined with respect to edges of the tree. We also study approximate
state construction and provide a second-order asymptotic analysis.
