 Rooted tree leaf

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.