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The complete list of all trees on 2,3,4 labeled vertices: $2^{2-2}=1$ tree with 2 vertices, $3^{3-2}=3$ trees with 3 vertices and $4^{4-2}=16$ trees with 4 vertices.

In mathematics, Cayley's formula is a result in graph theory named after Arthur Cayley. It states that for every positive integer $n$ , the number of trees on $n$ labeled vertices is $n^{n-2}$ .

The formula equivalently counts the number of spanning trees of a complete graph with labeled vertices (sequence A000272 in the OEIS).

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