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The development of hierarchical matrix techniques has been essential for many modern scientific computing frameworks including fast direct solvers for PDEs and integral equations, scalable kernel methods and Gaussian processes, and second-order optimization and inverse problems, etc. These algorithms oftentimes lead to optimal computational and memory complexities. That said, when dealing with higher-dimensional problems, e.g., PDEs and kernels in at least 3D spaces, matrix algorithms can still be expensive and tensor algorithms become more attractive. Despite several recent pioneering works attempting to extend hierarchical matrices to hierarchical tensors, the field of hierarchical tensors remains largely unexploited. This work attempts to tackle the computational challenge of fast tensor algorithms for highly-oscillatory and high-dimensional integral operators. We develop hierarchical tensors with tensor butterfly compression, and demonstrate its efficiency for computing Green's function tensors in volume integral equations and Babich ansatz for high-frequency wave equations. This is a joint work with Jianliang Qian, Yuxiao Wei, Tianyi Shi, Hengrui Luo and Paul Michael Kielstra.