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  "body": "\n## Definition\n\nLet $n \\ge 2$ be an integer and let $T$ be a [[tree|0MJ0JS72DZ]] with vertex set $\\{1, \\dots, n\\}$.\nThe Prüfer sequence of $T$ is the sequence $(a_1, \\dots, a_{n-2})$ that this process gives.\nLet $T_0 = T$. For $i = 1, \\dots, n - 2$, let $\\ell_i$ be the leaf of $T_{i-1}$ with the smallest label.\nLet $a_i$ be the unique neighbour of $\\ell_i$ in $T_{i-1}$, and let $T_i = T_{i-1} - \\ell_i$.\n\n## Remarks\n\nThe map $T \\mapsto (a_1, \\dots, a_{n-2})$ is a bijection from the labelled trees on $\\{1, \\dots, n\\}$ to $\\{1, \\dots, n\\}^{n-2}$. This proves [[Cayley's formula|0M2SSGSYPE]].\nA vertex $v$ appears $\\deg(v) - 1$ times in the sequence. Thus the leaves are the labels that do not appear.\nExample: the star with centre $1$ on $\\{1, 2, 3, 4\\}$ has the sequence $(1, 1)$.\n\n## Explanations\n\n### L1 human\n\nRemove the smallest leaf again and again, and each time write down the vertex it was attached to.\nStop when two vertices are left. The list of $n - 2$ numbers fixes the tree.\n",
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