A property of prefix codes called strong monotonicity is introduced. Then it is proven that for a prefix code $C$ for a given probability distribution, the following are equivalent: (i) $C$ is expected length minimal; (ii) $C$ is length equivalent to a Huffman code; and (iii) $C$ is complete and strongly monotone. Also, three relations are introduced between prefix code trees called same-parent, same-row, and same-probability swap equivalence, and it is shown that for a given source, all Huffman codes are same-parent, same-probability swap equivalent, and all expected length minimal prefix codes are same-row, same-probability swap equivalent.

翻译：本文引入了前缀码的一种性质，称为强单调性。随后证明，对于给定概率分布下的前缀码$C$，以下条件是等价的：(i) $C$的期望长度最小；(ii) $C$在长度上与霍夫曼码等价；(iii) $C$是完备且强单调的。此外，还引入了前缀码树之间的三种关系，即同父交换等价、同行交换等价和同概率交换等价。结果表明，对于给定信源，所有霍夫曼码均为同父、同概率交换等价，而所有期望长度最小的前缀码均为同行、同概率交换等价。