The use of hyperbolic space to encode hierarchies is absolutely genius (in not small part because it should be obvious), given the properties of hyperbolic space to "expand" around fixed points exponentially, exactly like trees do* . It seems like a continuous generalization of trees essentially. Beautiful.
*: https://en.wikipedia.org/wiki/Hyperbolic_space
"Another distinctive property is the amount of space covered by the n-ball in hyperbolic n-space: it increases exponentially with respect to the radius of the ball for large radii, rather than polynomially. (...)"