Definition:Rooted Tree

Definition
A rooted tree is a tree with a countable number of nodes, in which a particular node is distinguished from the others and called the root:


 * RootedTree.png

In some contexts, in which only a rooted tree would make sense, the term tree is often used.

Infinite Tree
A rooted tree is infinite if it contains a countably infinite number of nodes.

Finite Tree
Similarly, a rooted tree is finite if it contains a finite number of nodes.

Parent
Consider a rooted tree $$T$$ whose root is $$r_T$$.

Let $$t$$ be a node of $$T$$.

From Paths in Trees are Unique, there is only one path from $$t$$ to $$r_T$$.

Let $$\pi: T - \left\{{r_T}\right\} \to T$$ be the mapping defined as:
 * $$\pi \left({t}\right) = \text { the node adjacent to } t \text { on the path to } r_T$$

Then $$\pi \left({t}\right)$$ is known as the parent (or parent node) of $$t$$, and $$\pi$$ as the parent function or parent mapping.

Root Node
The root node, or just root, is the one node in a rooted tree which, by definition, has no parent.

Ancestor
An ancestor (or ancestor node) of a node $$t$$ of a rooted tree $$T$$ whose root is $$r_T$$ is a node in the path from $$t$$ to $$r_T$$.

Thus, the root of a rooted tree $$T$$ is the ancestor of every node of $$T$$ (including itself).

Proper Ancestor
A proper ancestor of a node $$t$$ is an ancestor of $$t$$ which is not $$t$$ itself.

Children
The children (or child nodes) of a node $$t$$ in a rooted tree $$T$$ are the elements of the set:
 * $$\left\{{s \in T: \pi \left({s}\right) = t}\right\}$$

That is, the children of $$t$$ are all the nodes of $$T$$ of which $$t$$ is the parent.

The child of a child node of a node $$t$$ is a grandchild node of $$t$$.

Descendant
A descendant (or descendant node) $$s$$ of a node $$t$$ of a rooted tree $$T$$ whose root is $$r_T$$ is a node such that $$t$$ is in the path from $$s$$ to $$r_T$$.

That is, the descendants of $$t$$ are all the nodes of $$T$$ of which $$t$$ is an ancestor.

Proper Descendant
A proper descendant of a node $$t$$ is a descendant of $$t$$ which is not $$t$$ itself.

Sibling
Two children of the same node of a rooted tree are called siblings.

That is, siblings are nodes which both have the same parent.

Leaf Node
A leaf node (or a terminal node, or just leaf) of a rooted tree $$T$$ is a node of $$T$$ which has no children.

Branch
A subset $$\Gamma$$ of a rooted tree $$T$$ is a branch iff:
 * The root node $$r_T$$ belongs to $$\Gamma$$;
 * The parent of each node in $$\Gamma - \left\{{r_T}\right\}$$ is in $$\Gamma$$;
 * Each node in $$\Gamma$$ either:
 * Is a leaf node of $$T$$;
 * Has exactly one child in $$\Gamma$$.

Hence a node in $$T$$ with more than one child will be on more than one branch.

A leaf node will be on exactly one branch.

The length of a branch is defined as the number of ancestors of the leaf at the end of that branch.

Informally, then, a branch of a rooted tree is the path from the root to a leaf.

Note, however, that $$\Gamma$$ is infinite iff it has no leaf node at the end.