Rooted trees, ancestors, descendants, depth, height, parents and children

A rooted tree is a pair $(T,r)$ consisting of a tree $T$ and a distinguished vertex $r$, the root (Trees, forests, leaves and isolated vertices). The unique $r$-$v$ path is the root path of $v$ (Equivalent characterisations of a nonempty tree by unique paths, edge count, minimal connectivity and maximal acyclicity).

The depth of $v$ is $d_T(r,v)$, and the height of $(T,r)$ is the maximum vertex depth (Graph distance within a component, eccentricity, diameter and girth, including the acyclic convention). A vertex $u$ is an ancestor of $v$, and $v$ a descendant of $u$, when $u$ lies on the root path of $v$. If $v\ne r$, the neighbour of $v$ immediately preceding it on its root path is its parent; the vertices having parent $v$ are the children of $v$.

The vertices of depth $k$ form level $k$. The root has depth zero and is its own ancestor, but it has no parent.