Leaf of Rooted Tree is on One Branch

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

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

Then there exists a unique branch $\Gamma$ of $T$ such that $t \in \Gamma$.


Let $t$ be on the branch $\Gamma$.

By definition of branch, $\Gamma$ is a path from the root node $r_T$ to $t$.

By Path in Tree is Unique, such $\Gamma$ is unique.