Definition:Rooted Tree/Branch

Definition

Let $T$ be a rooted tree with root node $r_T$.

A subset $\Gamma$ of $T$ is a branch if and only if all the following conditions hold:

$(1): \quad$ The root node $r_T$ belongs to $\Gamma$

$(2): \quad$ The parent of each node in $\Gamma \setminus \set {r_T}$ is in $\Gamma$

$(3): \quad$ Each node in $\Gamma$ either:

$\text {(a)}: \quad$ is a leaf node of $T$

or:

$\text {(b)}: \quad$ has exactly one child node in $\Gamma$.

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

Let $T$ be a rooted tree with root node $r_T$.

Let $\Gamma$ be a branch of $T$.

Then $\Gamma$ is finite if and only if it has exactly one leaf node.

Let $T$ be a rooted tree with root node $r_T$.

Let $\Gamma$ be a branch of $T$.

Then $\Gamma$ is infinite if and only if it has no leaf node at the end.

Let $T$ be a rooted tree with root node $r_T$.

Let $\Gamma$ be a finite branch of $T$.

The length of $\Gamma$ is defined as the number of ancestors of the leaf at the end of that branch.