A prefix-free encoding of a finite alphabet Γ assigns each symbol in Γ a binary codeword, such that no codeword is a prefix of another codeword. A prefix-free encoding is minimal if it is not possible to arrive at another prefix-free encoding (of the same symbols) by contracting some of the keywords. For instance, the encoding {0, 101} is not minimal since the codeword 101 can be contracted to 1 while still maintaining the prefix-free property.
Show that a minimal prefix-free encoding can be represented by a full binary tree in which each leaf corresponds to a unique element of Γ, whose codeword is generated by the path from the root to that leaf (interpreting a left branch as 0 and a right branch as 1).
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