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Tree in the Poincaré Disk

Why trees fit hyperbolic space: each level of a branching tree sits closer to the rim, yet every edge has the same hyperbolic length.

A tree embedded in the Poincaré diskA branching tree drawn inside a circle. The root sits at the center, and each deeper level sits closer to the boundary circle. Near the rim the nodes look crowded, but in hyperbolic distance every edge has the same length.Boundary circlepoints at infinityLeaves crowd the rimonly in this pictureEvery edge has thesame hyperbolic lengthEdges drawn straight for simplicity; true geodesics are arcs

The rim of the disk is infinitely far away, so the crowding near the edge is an artifact of the picture. Kleinberg (2007) uses exactly this property to give every node in a network a routing address.