= A Sholl profile does not determine rooted tree topology
An original counterexample explains why <a Sholl profile does not determine rooted tree topology>. Construct two embedded <rooted trees> whose distance from the root increases along every edge. Both have one initial segment, three <branch points of a tree> at distances $1,2,3$, and four <rooted tree leaves> at distance $4$. In tree A the second and third branch points lie on different children of the first; in tree B all three branch points lie along one lineage. Their common profile is
$$
N(r)=\begin{cases}
1,&0<r\le1,\\
2,&1<r\le2,\\
3,&2<r\le3,\\
4,&3<r\le4,\\
0,&r>4.
\end{cases}
$$
At a branch-point radius the shared endpoint is counted once. This convention also gives equal profiles at all three branch-point radii, not just between them.
The <multiset> of numbers of <branch points of a tree> along the four <root-to-leaf paths> is $\{2,2,2,2\}$ for tree A and $\{1,2,3,3\}$ for tree B. A <graph isomorphism> preserving the root would preserve those path counts. The different <multisets> prove that the <rooted trees> are different despite identical profiles. \b[The radial summary loses connectivity information.]
\Image[/radial-profile-rooted-tree-ambiguity.png]
{title=Different rooted trees with identical radial intersection profiles}
{description=Original geometric counterexample with checked outward edges, noncrossing embeddings and distinct-intersection counts. Chosen coordinates, not measured anatomy.}
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