In a Prüfer code, a vertex label occurs exactly its degree minus one times. A rooted binary tree with vertices has one root of degree two, other interior vertices of degree three, and leaves. Its code of length therefore has one label occurring once, labels occurring twice, and all other labels absent.
Conversely, these multiplicities force exactly those degrees in the tree decoded from the sequence. The unique once-occurring label is its root. This is the complete characterization.

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