Bijection between Prüfer Sequences and Labeled Trees

Theorem
There is a one-to-one correspondence between Prüfer sequences and labeled trees.

That is, every labeled tree has a unique Prüfer sequence that defines it, and every Prüfer sequence defines just one labeled tree.

Proof
Let $$T$$ be the set of all labeled trees of order $n$.

Let $$P$$ be the set of all Prüfer sequence of length $$n-2$$.

Let $$\phi: T \to P$$ be the mapping that maps each tree to its Prüfer sequence.


 * From Prüfer Sequence from Labeled Tree, $$\phi$$ is clearly well-defined, as every element of $$T$$ maps uniquely to an element of $$P$$.


 * However, from Labeled Tree from Prüfer Sequence, $$\phi^{-1}: P \to T$$ is also clearly well-defined, as every element of $$P$$ maps to a unique element of $$T$$.

Hence the result.