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  "body": "\n## Statement\n\nLet $K$ be a field of characteristic $0$.\nLet $\\phi(w) \\in K[\\![w]\\!]$ be a [[formal power series|033WGN00BF]] with $\\phi(0) \\ne 0$.\nThen there is a unique formal power series $T(z) \\in K[\\![z]\\!]$ with $T(0) = 0$ and $T(z) = z\\, \\phi(T(z))$.\nFor every $H(w) \\in K[\\![w]\\!]$ and every integer $n \\ge 1$,\n$$[z^n]\\, H(T(z)) = \\frac{1}{n} [w^{n-1}]\\, H'(w)\\, \\phi(w)^n,$$\nwhere $[z^n] F$ is the [[coefficient|0R0D0XY4HT]] of $z^n$ in $F$.\nIn particular, $[z^n]\\, T(z) = \\frac{1}{n} [w^{n-1}]\\, \\phi(w)^n$.\n\n## Remarks\n\nProof idea: work with formal Laurent series, where the coefficient of $w^{-1}$ in a derivative is always $0$.\nChange the variable from $z$ to $w = T(z)$, so $z = w / \\phi(w)$.\nThe coefficient of $z^{-1}$ in $z^{-n-1} H(T(z))$ becomes a coefficient of $w^{-1}$, and an integration by parts gives the formula.\n\nExample: rooted labelled trees have the exponential [[generating function|0PHAC833JF]] $T(z) = z e^{T(z)}$.\nWith $\\phi(w) = e^w$, the theorem gives $[z^n] T = \\frac{1}{n} \\cdot \\frac{n^{n-1}}{(n-1)!} = \\frac{n^{n-1}}{n!}$.\nSo there are $n^{n-1}$ [[rooted trees|0Y0JGC2K9T]] on $n$ labelled vertices.\n\nExample: with $\\phi(w) = (1 + w)^2$, the series $T$ counts binary trees by internal nodes.\nThe theorem gives $[z^n] T = \\frac{1}{n} \\binom{2n}{n-1}$, which is the [[Catalan number|0YWZJEABB2]] $C_n$.\n\nThe hypothesis $\\phi(0) \\ne 0$ makes the problem meaningful.\nIf $\\phi(0) = 0$, then $T = 0$ is the only solution with $T(0) = 0$, and the equation counts nothing.\nCharacteristic $0$ is used to divide by $n$.\nThere is also an analytic version for holomorphic $\\phi$ near $0$, which then gives the Taylor series of the inverse function.\n\n## Explanations\n\n### L1 human\n\nMany counting problems lead to an equation of the form $T = z\\, \\phi(T)$ for a generating function $T$.\nSuch an equation is usually hard to solve for $T$ in closed form.\nLagrange inversion gives the coefficients of $T$ directly, without solving the equation.\nOne only has to expand the power $\\phi(w)^n$ and read off one coefficient.\n\n### L2 human\n\nThe theorem is the main tool for counting trees and tree-like structures.\nA family of rooted trees defined by the allowed numbers of children has a generating function with $T = z\\, \\phi(T)$, where $\\phi$ records the allowed numbers of children.\nIn the [[symbolic method|JP11D90EJC]] such equations arise at once from the recursive description of the trees.\nThe form with $H$ counts forests, since a sequence of $k$ trees has generating function $T^k$.\nEquivalently, the theorem computes the coefficients of the compositional inverse of $z / \\phi(z)$.\n",
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