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Let $T$ be an optimum prefix code tree, and let $b$ and $c$ be two siblings at the maximum depth of the tree (must exist because $T$ is full). Assume without loss of generality that $f (b) \le f(c)$ and $f(x) \le f(y)$ (if this is not true, then rename these characters). Since $x$ and $y$ have the two smallest frequencies it follows that $f(x) \le f(b)$ ...

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