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For a source with a given number of messages and an unspecified set of probabilities, the number of non-trivially different compact codes that are possible increases in a predictable fashion as increases. Distinct binary compact codes of messages correspond to distinct oriented binary trees with terminal nodes. The theorem of this correspondence shows that, by using a recursion relation, and given that there is one compact code tree for , all compact code trees for any can be automatically constructed. This is done by splitting, for all integers bottom level nodes of all compact code trees which have terminal nodes and which end in or more bottom level nodes.