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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.

A root-to-leaf path is the unique simple graph path from the root of a tree to a rooted tree leaf. Successive vertices follow parent-to-child relationships. If the rooted tree has only one vertex, its root is also its leaf and the path has zero edges.^[ref]

For an original example, root the tree with $E=\{ra,ab,ac,cd\}$ at$r$. Its rooted leaves are$b,d$, and its complete root-to-leaf paths areTheir lengths are two and three edges. The first two vertices form a shared prefix; the shared route ends at$a$ before the paths enter different descendant branches. Counting only the endpoint labels would omit that shared ancestry.

Uniqueness follows from the absence of a graph cycle. Two different simple routes from the root to one leaf would separate and later rejoin, producing a cycle. Conversely, following a leaf's successive parent in a rooted tree backward must recover its one root path.

If all vertex identities and complete root-to-leaf paths of a finite tree are known, their consecutive vertex pairs recover its edges. Every vertex lies on such a path: from it, keep selecting a child until finiteness forces a leaf. A list of path lengths alone supplies much less information; it discards vertex identities and the locations where paths share prefixes.

The edge count is the endpoint's depth in an unweighted rooted tree. A drawing's Euclidean arc length or a sum of assigned edge lengths is an additional measurement. Two drawings of the same rooted tree can therefore have identical root-to-leaf vertex lists and different geometric lengths.