Cayley's Formula

Theorem
The number of distinct labeled trees with $n$ nodes is $n^{n-2}$.

Proof
Follows directly from Bijection between Prüfer Sequences and Labeled Trees.

This shows that there is a bijection between the set of labeled trees with $n$ nodes and the set of all Prüfer sequences of the form:
 * $\left({\mathbf a_1, \mathbf a_2, \ldots, \mathbf a_{n-2}}\right)$

where each of the $\mathbf a_i$'s is one of the integers $1, 2, \ldots, n$, allowing for repetition.

Since there are exactly $n$ possible values for each integer $\mathbf a_i$, the total number of such sequences is $\displaystyle \prod_{i \mathop = 1}^{n-2} n$.

The result follows from Equivalence of Mappings between Sets of Same Cardinality.

Historical Note
This proof, given by Heinz Prüfer, first appeared in 1918.

Cayley himself first stated this theorem in his A Theorem on Trees in 1889, but his proof was unsatisfactory as he discussed only the case where $n=6$, and his method can not be generalized to larger $n$.