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  "body": "\n## Statement\n\nLet $c_n$ be the number of rooted, unlabeled, ordered binary trees with $n$ leaves, for $n \\ge 1$. Here each node that is not a leaf has exactly two children, in a fixed left to right order.\n\nThen the [[generating function|0PHAC833JF]] $C(x) = \\sum_{n \\ge 1} c_n x^n$ is\n$$C(x) = \\frac{1 - \\sqrt{1 - 4x}}{2} = \\sum_{n=1}^{\\infty} \\frac{1}{n} \\binom{2n-2}{n-1} x^n.$$\n\n## Remarks\n\nSo $c_n = \\frac{1}{n}\\binom{2n-2}{n-1}$, which is the [[Catalan number|092DRBFKMG]] $C_{n-1}$.\nExample: $c_1 = 1$, $c_2 = 1$, $c_3 = 2$, $c_4 = 5$.\nSplitting a tree at the root gives $C(x) = x + C(x)^2$, and the square root form is the root of this quadratic that vanishes at $x = 0$.\n",
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