The difference between a typical tree with nodes and a hyperbolic tree is that it uses hyperbolic space, which allows for more room for the tree. Rather than being a flat Euclidean space which does not utilize a curvature, the hyperbolic tree does. This allows for more spaces for the nodes on the tree.
Exploiting this property allows laying out the tree in hyperbolic space in an uncluttered manner: placing a node far enough from its parent gives the node almost the same amount of space as its parent for laying out its own children
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