The ground state of an antiferromagnetic Heisenberg model on L X L clusters joined by a single bond and balanced Bethe clusters are investigated with quantum Monte Carlo and mean field theory. The improved Monte Carlo method of Sandvik and Evertz is used and the observables include valence bond and loop valence bond observables introduced by Lin and Sandvik as well as the valence bond entropy and the second Renyi entropy. For the bisecting of the Bethe cluster, in disagreement with our previous results and in agreement with mean field theory, the valence loop entropy and the second Renyi entropy scale as the logarithm of the number of sites in the cluster. For bisecting the L X L - L X L clusters, the valence bond entropy scales as L, however, the loop entropy and the entanglement entropy scale as the ln(L). The calculations suggest that the area law is essentially correct and linking high entanglement objects will not generate much more entanglement.