Definition:Rooted Tree/Branch

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

A subset $\Gamma$ of $T$ is a branch iff all the following conditions hold:
 * $(1): \quad$ The root node $r_T$ belongs to $\Gamma$
 * $(2): \quad$ The parent of each node in $\Gamma \setminus \left\{{r_T}\right\}$ is in $\Gamma$
 * $(3): \quad$ Each node in $\Gamma$ either:
 * $(a): \quad$ is a leaf node of $T$
 * or:
 * $(b): \quad$ has exactly one child node in $\Gamma$.

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

Also see
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.