Root-to-leaf path

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.