Consider all points of a regular lattice as nodes in a graph. Choose edges to form a tree. The path from any node to the center (root) should be a minimal-length path along the lattice (e.g., Manhattan distance).
For a 2D square lattice, an extension of the swastika works. 4 T-shaped arms (but with many crossbars) may also work.
Perhaps an additional criterion that the path from any node to the root ought not stray too far from the straight "as the crow flies" line.
Things might become more exciting in 3D among the several lattices formed from regular honeycombs (tiling all of space), as we are straddling from 3 dimensions to 1.
Space-filling curves.
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