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  "body": "\n## Statement\n\nLet $n \\ge 2$ be an integer.\nThen the number of [[trees|0MJ0JS72DZ]] with vertex set $\\{1, 2, \\dots, n\\}$ is $n^{n-2}$.\n\n## Remarks\n\nThe trees are labelled: two trees are the same only if they have the same edge set. Thus the formula counts the [[spanning trees|0FQ5B8S1ZC]] of $K_n$.\nFor $n = 3$ there are $3$ trees, and for $n = 4$ there are $16$.\nA proof uses the [[Prüfer sequence|09PV9JSY6E]], a bijection between these trees and the sequences of length $n - 2$ over $\\{1, \\dots, n\\}$.\n\n## History\n\nCarl Wilhelm Borchardt found the formula in 1860.\nArthur Cayley extended it in 1889, and the formula has his name.\n\n## Explanations\n\n### L1 human\n\nLabel $n$ points with the numbers $1$ to $n$ and count the ways to connect them by a tree.\nFor $4$ points there are $4^2 = 16$ ways: $12$ paths and $4$ stars.\n",
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