= 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.<Graph Theory by Reinhard Diestel>[Third edition2005, section1.5, pp.13–15: unique tree paths and rooted order; full text https://webhomes.maths.ed.ac.uk/~v1ranick/papers/diestel1.pdf]{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 are
$$
(r,a,b),\qquad(r,a,c,d).
$$
Their 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.
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