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Phonon dispersion and thermal conduction properties of hyperbranched nanostructures with unique topological complexity are theoretically and numerically investigated in this research. We present analytical cantilever-in-mass models to analyze and control the inherent resonance hybridization in hyperbranched nanomembranes containing different configurations and cross sections. We show that these local resonances hosted by hyperbranched nanopillars can generate numerous flat bands in the phonon dispersion relation and dramatically lower the group velocities, consequently resulting in a significant reduction of the thermal conductivity. The applicability of the proposed analytical models in thermal conductivity tuning is demonstrated, and a superior performance in reducing the heat flux in nano-structured membranes is exhibited, which can potentially lead to improved thermoelectric energy conversion devices. Published by AIP Publishing.