 Rooted tree

A rooted tree is a tree with a distinguished vertex, its root of a tree $r$. Choosing $r$ turns connections into a hierarchy: every other vertex has a unique parent in a rooted tree, the preceding vertex on its graph path from $r$. Conversely, the vertices whose parent in a rooted tree is $v$ are the children in a rooted tree of $v$. Choosing the root of a tree changes this description, while the underlying edges remain the same. Arrows towards the root of a tree can encode the relationship, as in MIT's counting notes; the undirected tree itself needs no arrows. ^[ref]

In the seven-vertex tree above, choose $r=1$. The levels, grouped by graph distance from the root of a tree, are $\{1\}$, $\{2,3\}$, $\{4,5,6\}$, $\{7\}$. The children in a rooted tree of $2$ are $4,5$, and the parent in a rooted tree of $7$ is $6$. Choosing $r=7$ instead makes the levels $\{7\}$, $\{6\}$, $\{3\}$, $\{1\}$, $\{2\}$, $\{4,5\}$: now $6$ has parent in a rooted tree $7$, reversing their relationship. All six edges survive.

This hierarchy supports structural recursion: process the smaller rooted trees below each vertex, then combine their results. A depth-first traversal of a tree explores one branch before returning; breadth-first search processes successive levels. The lowest common ancestor identifies where two routes from the root of a tree last coincide. For $4,5$ it is $2$ when $r=1$. A binary tree imposes an additional restriction on children in a rooted tree; a general rooted tree has no such bound.