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 isAt 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. The radial summary loses connectivity information.