Cayley's formula

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

Another bijective proof, by André Joyal, finds a one-to-one transformation between n-node trees with two distinguished nodes and maximal directed pseudoforests.

[2] In a short 1889 note, Cayley extended the formula in several directions, by taking into account the degrees of the vertices.

[3] Although he referred to Borchardt's original paper, the name "Cayley's formula" became standard in the field.

Cayley's formula immediately gives the number of labelled rooted forests on n vertices, namely (n + 1)n − 1.

The complete list of all trees on 2,3,4 labeled vertices: tree with 2 vertices, trees with 3 vertices and trees with 4 vertices.