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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 node with no children in a rooted tree. This depends on the chosen root: the root of a nontrivial tree can have undirected vertex degree one while still having a child and therefore not being a leaf in the rooted sense. In a single-node rooted tree, the root is a leaf. The distinction matters when counting the complement of the internal vertices of a rooted tree.

With every edge directed away from the root of a tree, a rooted tree leaf has outdegree zero.^[ref]

For an original example, take $E=\{ra,ab,ac\}$ and choose root$r$. The children of$r$ consist of$a$, and the children of$a$ are$b,c$. Thus$b,c$ are the rooted leaves. Vertex$r$ has undirected degree one but has a child, so it is not a rooted leaf in this rooting.

For a nonroot vertex$v$, one incident edge goes to its parent in a rooted tree, and all remaining incident edges go to children. ThereforeThe root instead satisfies $\mathrm{outdeg}(r)=\mathrm{deg}(r)$. A nonroot vertex is consequently a rooted leaf exactly when its undirected degree is one. The root is a rooted leaf exactly when the entire connected tree has one vertex.

Changing the root can change leaf membership without changing any edge. On the three-vertex path$a,b,c$, rooting at$a$ gives one rooted leaf$c$, while rooting at$b$ gives two, $a,c$.

A finite rooted tree always has a leaf: take a vertex maximizing its graph distance from the root. Any child would lie one edge farther away, contradicting maximality. An infinite ray rooted at its endpoint has a child at every vertex and supplies the corresponding counterexample to an unrestricted infinite-tree claim. Rooted terminal points and endpoints of an unrooted graph therefore need their conventions stated explicitly.

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.