Sholl profile ($N(r)$)

A Sholl profile records intersection counts as a function of distance from a chosen center. For a thin geometric tracing it is an exact counting function; a profile obtained from sampled images also depends on the software's sampling and segmentation conventions.

In an ideal planar model, let $\Gamma$ be a finite embedded rooted tree of line segments in the Euclidean plane with chosen root $c$. Its Sholl profile counts distinct intersections:This is a geometric model for thin tracings; the software documentation separately addresses sampling binary images and handling image segmentation.^[ref]

For an original simple example, take three straight radial segments from the origin, pointing in different directions, with lengths 1, 2 and 4. For radii $r=0.5,1.5,3,5$, the counts are respectively $3,2,1,0$. The segment of length 1 contributes at $r=1$ but not beyond it; record the boundary convention when defining the count. No branch points are needed for this decreasing profile: different lengths of terminal segments already produce it.

The profile retains radial occupancy but discards the angles and the identities of the intersected paths. In particular, a Sholl profile does not determine rooted tree topology. Sampling only a finite list of radii loses still more information; it should not be silently equated with knowledge of the complete continuous-radius counting function.