According to the standard definition of "Entropy of Entanglement"
https://en.wikipedia.org/wiki/Entropy_of_entanglement
one starts from the density matrix of a pure state $$ \rho=|\psi\rangle\langle\psi|, $$ then divides the system into two parts, $A$ and $B$, traces away the degrees of freedom of one of the two subsystems, say subsystem B, and thus obtains the reduced density matrix of the remaining subsystem $$ \rho_A=\mathrm{Tr}_B(\rho). $$ Eventually, the entanglement between subsystem $A$ and subsystem $B$ is given by the Von Neumann entropy of $\rho_a$: $$ S(\rho_a)=-\mathrm{Tr}[\rho_A\,\log\rho_A]. $$
My question is: does the choice of the bipartition play an essential role? I think that the final result strongly depends on how one chooses subsystem $A$ (and, of course, the complementary subsystem $B$). To my knowledge, in fact, books do not emphasize how important is the choice of the partition. Is there a region?