Tree tensor network states represent low-energy states faithfully

Extending corresponding results for matrix product states [Verstraete and Cirac, PRB 73, 094423 (2006); Schuch et al. PRL 100, 030504 (2008)], it is shown how the approximation error of tree tensor network states (TTNS) can be bounded using Schmidt spectra or R\'{e}nyi entanglement entropies of the target quantum state. Conversely, one obtains bounds on TTNS bond dimensions needed to achieve a specific approximation accuracy. For tree lattices, the result implies that efficient TTNS approximations exist if $\alpha<1$ R\'{e}nyi entanglement entropies for single-branch cuts obey an area law, as in ground and low-energy states of certain gapped systems.