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We study the occurrence of domain branching in a class of $(d+1)$-dimensionalsharp interface models featuring the competition between an interfacial energyand a non-local field energy. Our motivation comes from branching in uniaxialferromagnets corresponding to $d=2$, but our result also covers twinning inshape-memory alloys near an austenite-twinned-martensite interface(corresponding to $d=1$, thereby recovering a result of Conti [Comm. Pure Appl.Math. 53 (2000), 1448-1474. https://doi.org/cgkvm4 ]).  We prove that the energy density of a minimising configuration in a largecuboid domain $Q_{L,T}=[-L,L]^d\times [0,T]$ scales like $T^{-\frac{2}{3}}$(irrespective of the dimension $d$) if $L\gg T^{\frac{2}{3}}$. While thisalready provides a lot of insight into the nature of minimisers, it does notcharacterise their behaviour close to the top and bottom boundaries of thesample, i.e. in the region where the branching is concentrated. Moresignificantly, we show that minimisers exhibit a self-similar behaviour nearthe top and bottom boundaries in a statistical sense through local energybounds: for any minimiser in $Q_{L,T}$, the energy density in a small cuboid$Q_{\ell,t}$ centred at the top or bottom boundaries of the sample, with sidelengths $\ell \gg t^{\frac{2}{3}}$, satisfies the same scaling law, that is, itis of order $t^{-\frac{2}{3}}$.

Domain branching in micromagnetism: scaling law for the global and local energies