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  "body": "\n## Statement\n\nLet $n$ be a natural number.\nThen the number of [[binary trees|02A8EMC83C]] with exactly $n$ nodes is the [[Catalan number|092DRBFKMG]] $C_n$.\n\n## Remarks\n\nHere a binary tree is either empty or a root node with an ordered pair of binary trees as its left and right subtrees.\nFor $n = 3$ there are $C_3 = 5$ binary trees.\nProof idea: a nonempty tree with $n + 1$ nodes has a left subtree with $i$ nodes and a right subtree with $n - i$ nodes. This gives the [[Catalan recurrence|0Y1MG9R5VN]].\n\n## Explanations\n\n### L1 human\n\nRemove the root of a binary tree, and you get a left tree and a right tree.\nThe sizes of the two trees add up to one less than the size of the original tree, and this gives the Catalan recurrence.\n",
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