  We investigate the scaling of entanglement entropy in both the multi-scale
entanglement renormalization ansatz (MERA) and in its generalization, the
branching MERA. We provide analytical upper bounds for this scaling, which take
the general form of a boundary law with various types of multiplicative
corrections, including power-law corrections all the way to a bulk law. For
several cases of interest, we also provide numerical results that indicate that
these upper bounds are saturated to leading order. In particular we establish
that, by a suitable choice of holographic tree, the branching MERA can
reproduce the logarithmic multiplicative correction of the boundary law
observed in Fermi liquids and spin-Bose metals in $D\geq 2$ dimensions.
