How did Arthur Cayley contribute to enumerative graph theory?

Arthur Cayley made significant contributions to enumerative graph theory during the late 19th century. His most notable achievement in this field occurred in 1889, when he demonstrated that the number of labelled trees with $\upsilon$ vertices is given by the formula ${\upsilon }^{\upsilon -2}$.12

This result remains a foundational element of graph theory, reflecting Cayley's broader influence as a leader of the 19th-century British school of pure mathematics. His work in this area helped advance the study of combinatorics, a discipline that explores the properties of structures such as graphs, designs, and arrangements.31