***************************************************************************** * www.FindStat.org - The Combinatorial Statistic Finder * * * * Copyright (C) 2019 The FindStatCrew * * * * This information is distributed in the hope that it will be useful, * * but WITHOUT ANY WARRANTY; without even the implied warranty of * * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. * ***************************************************************************** ----------------------------------------------------------------------------- Statistic identifier: St000201 ----------------------------------------------------------------------------- Collection: Binary trees ----------------------------------------------------------------------------- Description: The number of leaf nodes in a binary tree. Equivalently, the number of cherries [1] in the complete binary tree. The number of binary trees of size $n$, at least $1$, with exactly one leaf node for is $2^{n-1}$, see [2]. The number of binary tree of size $n$, at least $3$, with exactly two leaf nodes is $n(n+1)2^{n-2}$, see [3]. ----------------------------------------------------------------------------- References: [1] Billey, S., Konvalinka, M., Matsen, F. A. I. On the enumeration of tanglegrams and tangled chains [[arXiv:1507.04976]] [2] Powers of 2: a(n) = 2^n. [[OEIS:A000079]] [3] a(n) = n*(n+1)*2^(n-2). [[OEIS:A001788]] ----------------------------------------------------------------------------- Code: def statistic(B): if B == BinaryTree(): return 0 if B == BinaryTree([]): return 1 return sum(statistic(S) for S in B) ----------------------------------------------------------------------------- Statistic values: [.,.] => 1 [.,[.,.]] => 1 [[.,.],.] => 1 [.,[.,[.,.]]] => 1 [.,[[.,.],.]] => 1 [[.,.],[.,.]] => 2 [[.,[.,.]],.] => 1 [[[.,.],.],.] => 1 [.,[.,[.,[.,.]]]] => 1 [.,[.,[[.,.],.]]] => 1 [.,[[.,.],[.,.]]] => 2 [.,[[.,[.,.]],.]] => 1 [.,[[[.,.],.],.]] => 1 [[.,.],[.,[.,.]]] => 2 [[.,.],[[.,.],.]] => 2 [[.,[.,.]],[.,.]] => 2 [[[.,.],.],[.,.]] => 2 [[.,[.,[.,.]]],.] => 1 [[.,[[.,.],.]],.] => 1 [[[.,.],[.,.]],.] => 2 [[[.,[.,.]],.],.] => 1 [[[[.,.],.],.],.] => 1 [.,[.,[.,[.,[.,.]]]]] => 1 [.,[.,[.,[[.,.],.]]]] => 1 [.,[.,[[.,.],[.,.]]]] => 2 [.,[.,[[.,[.,.]],.]]] => 1 [.,[.,[[[.,.],.],.]]] => 1 [.,[[.,.],[.,[.,.]]]] => 2 [.,[[.,.],[[.,.],.]]] => 2 [.,[[.,[.,.]],[.,.]]] => 2 [.,[[[.,.],.],[.,.]]] => 2 [.,[[.,[.,[.,.]]],.]] => 1 [.,[[.,[[.,.],.]],.]] => 1 [.,[[[.,.],[.,.]],.]] => 2 [.,[[[.,[.,.]],.],.]] => 1 [.,[[[[.,.],.],.],.]] => 1 [[.,.],[.,[.,[.,.]]]] => 2 [[.,.],[.,[[.,.],.]]] => 2 [[.,.],[[.,.],[.,.]]] => 3 [[.,.],[[.,[.,.]],.]] => 2 [[.,.],[[[.,.],.],.]] => 2 [[.,[.,.]],[.,[.,.]]] => 2 [[.,[.,.]],[[.,.],.]] => 2 [[[.,.],.],[.,[.,.]]] => 2 [[[.,.],.],[[.,.],.]] => 2 [[.,[.,[.,.]]],[.,.]] => 2 [[.,[[.,.],.]],[.,.]] => 2 [[[.,.],[.,.]],[.,.]] => 3 [[[.,[.,.]],.],[.,.]] => 2 [[[[.,.],.],.],[.,.]] => 2 [[.,[.,[.,[.,.]]]],.] => 1 [[.,[.,[[.,.],.]]],.] => 1 [[.,[[.,.],[.,.]]],.] => 2 [[.,[[.,[.,.]],.]],.] => 1 [[.,[[[.,.],.],.]],.] => 1 [[[.,.],[.,[.,.]]],.] => 2 [[[.,.],[[.,.],.]],.] => 2 [[[.,[.,.]],[.,.]],.] => 2 [[[[.,.],.],[.,.]],.] => 2 [[[.,[.,[.,.]]],.],.] => 1 [[[.,[[.,.],.]],.],.] => 1 [[[[.,.],[.,.]],.],.] => 2 [[[[.,[.,.]],.],.],.] => 1 [[[[[.,.],.],.],.],.] => 1 [.,[.,[.,[.,[.,[.,.]]]]]] => 1 [.,[.,[.,[.,[[.,.],.]]]]] => 1 [.,[.,[.,[[.,.],[.,.]]]]] => 2 [.,[.,[.,[[.,[.,.]],.]]]] => 1 [.,[.,[.,[[[.,.],.],.]]]] => 1 [.,[.,[[.,.],[.,[.,.]]]]] => 2 [.,[.,[[.,.],[[.,.],.]]]] => 2 [.,[.,[[.,[.,.]],[.,.]]]] => 2 [.,[.,[[[.,.],.],[.,.]]]] => 2 [.,[.,[[.,[.,[.,.]]],.]]] => 1 [.,[.,[[.,[[.,.],.]],.]]] => 1 [.,[.,[[[.,.],[.,.]],.]]] => 2 [.,[.,[[[.,[.,.]],.],.]]] => 1 [.,[.,[[[[.,.],.],.],.]]] => 1 [.,[[.,.],[.,[.,[.,.]]]]] => 2 [.,[[.,.],[.,[[.,.],.]]]] => 2 [.,[[.,.],[[.,.],[.,.]]]] => 3 [.,[[.,.],[[.,[.,.]],.]]] => 2 [.,[[.,.],[[[.,.],.],.]]] => 2 [.,[[.,[.,.]],[.,[.,.]]]] => 2 [.,[[.,[.,.]],[[.,.],.]]] => 2 [.,[[[.,.],.],[.,[.,.]]]] => 2 [.,[[[.,.],.],[[.,.],.]]] => 2 [.,[[.,[.,[.,.]]],[.,.]]] => 2 [.,[[.,[[.,.],.]],[.,.]]] => 2 [.,[[[.,.],[.,.]],[.,.]]] => 3 [.,[[[.,[.,.]],.],[.,.]]] => 2 [.,[[[[.,.],.],.],[.,.]]] => 2 [.,[[.,[.,[.,[.,.]]]],.]] => 1 [.,[[.,[.,[[.,.],.]]],.]] => 1 [.,[[.,[[.,.],[.,.]]],.]] => 2 [.,[[.,[[.,[.,.]],.]],.]] => 1 [.,[[.,[[[.,.],.],.]],.]] => 1 [.,[[[.,.],[.,[.,.]]],.]] => 2 [.,[[[.,.],[[.,.],.]],.]] => 2 [.,[[[.,[.,.]],[.,.]],.]] => 2 [.,[[[[.,.],.],[.,.]],.]] => 2 [.,[[[.,[.,[.,.]]],.],.]] => 1 [.,[[[.,[[.,.],.]],.],.]] => 1 [.,[[[[.,.],[.,.]],.],.]] => 2 [.,[[[[.,[.,.]],.],.],.]] => 1 [.,[[[[[.,.],.],.],.],.]] => 1 [[.,.],[.,[.,[.,[.,.]]]]] => 2 [[.,.],[.,[.,[[.,.],.]]]] => 2 [[.,.],[.,[[.,.],[.,.]]]] => 3 [[.,.],[.,[[.,[.,.]],.]]] => 2 [[.,.],[.,[[[.,.],.],.]]] => 2 [[.,.],[[.,.],[.,[.,.]]]] => 3 [[.,.],[[.,.],[[.,.],.]]] => 3 [[.,.],[[.,[.,.]],[.,.]]] => 3 [[.,.],[[[.,.],.],[.,.]]] => 3 [[.,.],[[.,[.,[.,.]]],.]] => 2 [[.,.],[[.,[[.,.],.]],.]] => 2 [[.,.],[[[.,.],[.,.]],.]] => 3 [[.,.],[[[.,[.,.]],.],.]] => 2 [[.,.],[[[[.,.],.],.],.]] => 2 [[.,[.,.]],[.,[.,[.,.]]]] => 2 [[.,[.,.]],[.,[[.,.],.]]] => 2 [[.,[.,.]],[[.,.],[.,.]]] => 3 [[.,[.,.]],[[.,[.,.]],.]] => 2 [[.,[.,.]],[[[.,.],.],.]] => 2 [[[.,.],.],[.,[.,[.,.]]]] => 2 [[[.,.],.],[.,[[.,.],.]]] => 2 [[[.,.],.],[[.,.],[.,.]]] => 3 [[[.,.],.],[[.,[.,.]],.]] => 2 [[[.,.],.],[[[.,.],.],.]] => 2 [[.,[.,[.,.]]],[.,[.,.]]] => 2 [[.,[.,[.,.]]],[[.,.],.]] => 2 [[.,[[.,.],.]],[.,[.,.]]] => 2 [[.,[[.,.],.]],[[.,.],.]] => 2 [[[.,.],[.,.]],[.,[.,.]]] => 3 [[[.,.],[.,.]],[[.,.],.]] => 3 [[[.,[.,.]],.],[.,[.,.]]] => 2 [[[.,[.,.]],.],[[.,.],.]] => 2 [[[[.,.],.],.],[.,[.,.]]] => 2 [[[[.,.],.],.],[[.,.],.]] => 2 [[.,[.,[.,[.,.]]]],[.,.]] => 2 [[.,[.,[[.,.],.]]],[.,.]] => 2 [[.,[[.,.],[.,.]]],[.,.]] => 3 [[.,[[.,[.,.]],.]],[.,.]] => 2 [[.,[[[.,.],.],.]],[.,.]] => 2 [[[.,.],[.,[.,.]]],[.,.]] => 3 [[[.,.],[[.,.],.]],[.,.]] => 3 [[[.,[.,.]],[.,.]],[.,.]] => 3 [[[[.,.],.],[.,.]],[.,.]] => 3 [[[.,[.,[.,.]]],.],[.,.]] => 2 [[[.,[[.,.],.]],.],[.,.]] => 2 [[[[.,.],[.,.]],.],[.,.]] => 3 [[[[.,[.,.]],.],.],[.,.]] => 2 [[[[[.,.],.],.],.],[.,.]] => 2 [[.,[.,[.,[.,[.,.]]]]],.] => 1 [[.,[.,[.,[[.,.],.]]]],.] => 1 [[.,[.,[[.,.],[.,.]]]],.] => 2 [[.,[.,[[.,[.,.]],.]]],.] => 1 [[.,[.,[[[.,.],.],.]]],.] => 1 [[.,[[.,.],[.,[.,.]]]],.] => 2 [[.,[[.,.],[[.,.],.]]],.] => 2 [[.,[[.,[.,.]],[.,.]]],.] => 2 [[.,[[[.,.],.],[.,.]]],.] => 2 [[.,[[.,[.,[.,.]]],.]],.] => 1 [[.,[[.,[[.,.],.]],.]],.] => 1 [[.,[[[.,.],[.,.]],.]],.] => 2 [[.,[[[.,[.,.]],.],.]],.] => 1 [[.,[[[[.,.],.],.],.]],.] => 1 [[[.,.],[.,[.,[.,.]]]],.] => 2 [[[.,.],[.,[[.,.],.]]],.] => 2 [[[.,.],[[.,.],[.,.]]],.] => 3 [[[.,.],[[.,[.,.]],.]],.] => 2 [[[.,.],[[[.,.],.],.]],.] => 2 [[[.,[.,.]],[.,[.,.]]],.] => 2 [[[.,[.,.]],[[.,.],.]],.] => 2 [[[[.,.],.],[.,[.,.]]],.] => 2 [[[[.,.],.],[[.,.],.]],.] => 2 [[[.,[.,[.,.]]],[.,.]],.] => 2 [[[.,[[.,.],.]],[.,.]],.] => 2 [[[[.,.],[.,.]],[.,.]],.] => 3 [[[[.,[.,.]],.],[.,.]],.] => 2 [[[[[.,.],.],.],[.,.]],.] => 2 [[[.,[.,[.,[.,.]]]],.],.] => 1 [[[.,[.,[[.,.],.]]],.],.] => 1 [[[.,[[.,.],[.,.]]],.],.] => 2 [[[.,[[.,[.,.]],.]],.],.] => 1 [[[.,[[[.,.],.],.]],.],.] => 1 [[[[.,.],[.,[.,.]]],.],.] => 2 [[[[.,.],[[.,.],.]],.],.] => 2 [[[[.,[.,.]],[.,.]],.],.] => 2 [[[[[.,.],.],[.,.]],.],.] => 2 [[[[.,[.,[.,.]]],.],.],.] => 1 [[[[.,[[.,.],.]],.],.],.] => 1 [[[[[.,.],[.,.]],.],.],.] => 2 [[[[[.,[.,.]],.],.],.],.] => 1 [[[[[[.,.],.],.],.],.],.] => 1 [.,[.,[.,[.,[.,[.,[.,.]]]]]]] => 1 [.,[.,[.,[.,[.,[[.,.],.]]]]]] => 1 [.,[.,[.,[.,[[.,.],[.,.]]]]]] => 2 [.,[.,[.,[.,[[.,[.,.]],.]]]]] => 1 [.,[.,[.,[.,[[[.,.],.],.]]]]] => 1 [.,[.,[.,[[.,.],[.,[.,.]]]]]] => 2 [.,[.,[.,[[.,.],[[.,.],.]]]]] => 2 [.,[.,[.,[[.,[.,.]],[.,.]]]]] => 2 [.,[.,[.,[[[.,.],.],[.,.]]]]] => 2 [.,[.,[.,[[.,[.,[.,.]]],.]]]] => 1 [.,[.,[.,[[.,[[.,.],.]],.]]]] => 1 [.,[.,[.,[[[.,.],[.,.]],.]]]] => 2 [.,[.,[.,[[[.,[.,.]],.],.]]]] => 1 [.,[.,[.,[[[[.,.],.],.],.]]]] => 1 [.,[.,[[.,.],[.,[.,[.,.]]]]]] => 2 [.,[.,[[.,.],[.,[[.,.],.]]]]] => 2 [.,[.,[[.,.],[[.,.],[.,.]]]]] => 3 [.,[.,[[.,.],[[.,[.,.]],.]]]] => 2 [.,[.,[[.,.],[[[.,.],.],.]]]] => 2 [.,[.,[[.,[.,.]],[.,[.,.]]]]] => 2 [.,[.,[[.,[.,.]],[[.,.],.]]]] => 2 [.,[.,[[[.,.],.],[.,[.,.]]]]] => 2 [.,[.,[[[.,.],.],[[.,.],.]]]] => 2 [.,[.,[[.,[.,[.,.]]],[.,.]]]] => 2 [.,[.,[[.,[[.,.],.]],[.,.]]]] => 2 [.,[.,[[[.,.],[.,.]],[.,.]]]] => 3 [.,[.,[[[.,[.,.]],.],[.,.]]]] => 2 [.,[.,[[[[.,.],.],.],[.,.]]]] => 2 [.,[.,[[.,[.,[.,[.,.]]]],.]]] => 1 [.,[.,[[.,[.,[[.,.],.]]],.]]] => 1 [.,[.,[[.,[[.,.],[.,.]]],.]]] => 2 [.,[.,[[.,[[.,[.,.]],.]],.]]] => 1 [.,[.,[[.,[[[.,.],.],.]],.]]] => 1 [.,[.,[[[.,.],[.,[.,.]]],.]]] => 2 [.,[.,[[[.,.],[[.,.],.]],.]]] => 2 [.,[.,[[[.,[.,.]],[.,.]],.]]] => 2 [.,[.,[[[[.,.],.],[.,.]],.]]] => 2 [.,[.,[[[.,[.,[.,.]]],.],.]]] => 1 [.,[.,[[[.,[[.,.],.]],.],.]]] => 1 [.,[.,[[[[.,.],[.,.]],.],.]]] => 2 [.,[.,[[[[.,[.,.]],.],.],.]]] => 1 [.,[.,[[[[[.,.],.],.],.],.]]] => 1 [.,[[.,.],[.,[.,[.,[.,.]]]]]] => 2 [.,[[.,.],[.,[.,[[.,.],.]]]]] => 2 [.,[[.,.],[.,[[.,.],[.,.]]]]] => 3 [.,[[.,.],[.,[[.,[.,.]],.]]]] => 2 [.,[[.,.],[.,[[[.,.],.],.]]]] => 2 [.,[[.,.],[[.,.],[.,[.,.]]]]] => 3 [.,[[.,.],[[.,.],[[.,.],.]]]] => 3 [.,[[.,.],[[.,[.,.]],[.,.]]]] => 3 [.,[[.,.],[[[.,.],.],[.,.]]]] => 3 [.,[[.,.],[[.,[.,[.,.]]],.]]] => 2 [.,[[.,.],[[.,[[.,.],.]],.]]] => 2 [.,[[.,.],[[[.,.],[.,.]],.]]] => 3 [.,[[.,.],[[[.,[.,.]],.],.]]] => 2 [.,[[.,.],[[[[.,.],.],.],.]]] => 2 [.,[[.,[.,.]],[.,[.,[.,.]]]]] => 2 [.,[[.,[.,.]],[.,[[.,.],.]]]] => 2 [.,[[.,[.,.]],[[.,.],[.,.]]]] => 3 [.,[[.,[.,.]],[[.,[.,.]],.]]] => 2 [.,[[.,[.,.]],[[[.,.],.],.]]] => 2 [.,[[[.,.],.],[.,[.,[.,.]]]]] => 2 [.,[[[.,.],.],[.,[[.,.],.]]]] => 2 [.,[[[.,.],.],[[.,.],[.,.]]]] => 3 [.,[[[.,.],.],[[.,[.,.]],.]]] => 2 [.,[[[.,.],.],[[[.,.],.],.]]] => 2 [.,[[.,[.,[.,.]]],[.,[.,.]]]] => 2 [.,[[.,[.,[.,.]]],[[.,.],.]]] => 2 [.,[[.,[[.,.],.]],[.,[.,.]]]] => 2 [.,[[.,[[.,.],.]],[[.,.],.]]] => 2 [.,[[[.,.],[.,.]],[.,[.,.]]]] => 3 [.,[[[.,.],[.,.]],[[.,.],.]]] => 3 [.,[[[.,[.,.]],.],[.,[.,.]]]] => 2 [.,[[[.,[.,.]],.],[[.,.],.]]] => 2 [.,[[[[.,.],.],.],[.,[.,.]]]] => 2 [.,[[[[.,.],.],.],[[.,.],.]]] => 2 [.,[[.,[.,[.,[.,.]]]],[.,.]]] => 2 [.,[[.,[.,[[.,.],.]]],[.,.]]] => 2 [.,[[.,[[.,.],[.,.]]],[.,.]]] => 3 [.,[[.,[[.,[.,.]],.]],[.,.]]] => 2 [.,[[.,[[[.,.],.],.]],[.,.]]] => 2 [.,[[[.,.],[.,[.,.]]],[.,.]]] => 3 [.,[[[.,.],[[.,.],.]],[.,.]]] => 3 [.,[[[.,[.,.]],[.,.]],[.,.]]] => 3 [.,[[[[.,.],.],[.,.]],[.,.]]] => 3 [.,[[[.,[.,[.,.]]],.],[.,.]]] => 2 [.,[[[.,[[.,.],.]],.],[.,.]]] => 2 [.,[[[[.,.],[.,.]],.],[.,.]]] => 3 [.,[[[[.,[.,.]],.],.],[.,.]]] => 2 [.,[[[[[.,.],.],.],.],[.,.]]] => 2 [.,[[.,[.,[.,[.,[.,.]]]]],.]] => 1 [.,[[.,[.,[.,[[.,.],.]]]],.]] => 1 [.,[[.,[.,[[.,.],[.,.]]]],.]] => 2 [.,[[.,[.,[[.,[.,.]],.]]],.]] => 1 [.,[[.,[.,[[[.,.],.],.]]],.]] => 1 [.,[[.,[[.,.],[.,[.,.]]]],.]] => 2 [.,[[.,[[.,.],[[.,.],.]]],.]] => 2 [.,[[.,[[.,[.,.]],[.,.]]],.]] => 2 [.,[[.,[[[.,.],.],[.,.]]],.]] => 2 [.,[[.,[[.,[.,[.,.]]],.]],.]] => 1 [.,[[.,[[.,[[.,.],.]],.]],.]] => 1 [.,[[.,[[[.,.],[.,.]],.]],.]] => 2 [.,[[.,[[[.,[.,.]],.],.]],.]] => 1 [.,[[.,[[[[.,.],.],.],.]],.]] => 1 [.,[[[.,.],[.,[.,[.,.]]]],.]] => 2 [.,[[[.,.],[.,[[.,.],.]]],.]] => 2 [.,[[[.,.],[[.,.],[.,.]]],.]] => 3 [.,[[[.,.],[[.,[.,.]],.]],.]] => 2 [.,[[[.,.],[[[.,.],.],.]],.]] => 2 [.,[[[.,[.,.]],[.,[.,.]]],.]] => 2 [.,[[[.,[.,.]],[[.,.],.]],.]] => 2 [.,[[[[.,.],.],[.,[.,.]]],.]] => 2 [.,[[[[.,.],.],[[.,.],.]],.]] => 2 [.,[[[.,[.,[.,.]]],[.,.]],.]] => 2 [.,[[[.,[[.,.],.]],[.,.]],.]] => 2 [.,[[[[.,.],[.,.]],[.,.]],.]] => 3 [.,[[[[.,[.,.]],.],[.,.]],.]] => 2 [.,[[[[[.,.],.],.],[.,.]],.]] => 2 [.,[[[.,[.,[.,[.,.]]]],.],.]] => 1 [.,[[[.,[.,[[.,.],.]]],.],.]] => 1 [.,[[[.,[[.,.],[.,.]]],.],.]] => 2 [.,[[[.,[[.,[.,.]],.]],.],.]] => 1 [.,[[[.,[[[.,.],.],.]],.],.]] => 1 [.,[[[[.,.],[.,[.,.]]],.],.]] => 2 [.,[[[[.,.],[[.,.],.]],.],.]] => 2 [.,[[[[.,[.,.]],[.,.]],.],.]] => 2 [.,[[[[[.,.],.],[.,.]],.],.]] => 2 [.,[[[[.,[.,[.,.]]],.],.],.]] => 1 [.,[[[[.,[[.,.],.]],.],.],.]] => 1 [.,[[[[[.,.],[.,.]],.],.],.]] => 2 [.,[[[[[.,[.,.]],.],.],.],.]] => 1 [.,[[[[[[.,.],.],.],.],.],.]] => 1 [[.,.],[.,[.,[.,[.,[.,.]]]]]] => 2 [[.,.],[.,[.,[.,[[.,.],.]]]]] => 2 [[.,.],[.,[.,[[.,.],[.,.]]]]] => 3 [[.,.],[.,[.,[[.,[.,.]],.]]]] => 2 [[.,.],[.,[.,[[[.,.],.],.]]]] => 2 [[.,.],[.,[[.,.],[.,[.,.]]]]] => 3 [[.,.],[.,[[.,.],[[.,.],.]]]] => 3 [[.,.],[.,[[.,[.,.]],[.,.]]]] => 3 [[.,.],[.,[[[.,.],.],[.,.]]]] => 3 [[.,.],[.,[[.,[.,[.,.]]],.]]] => 2 [[.,.],[.,[[.,[[.,.],.]],.]]] => 2 [[.,.],[.,[[[.,.],[.,.]],.]]] => 3 [[.,.],[.,[[[.,[.,.]],.],.]]] => 2 [[.,.],[.,[[[[.,.],.],.],.]]] => 2 [[.,.],[[.,.],[.,[.,[.,.]]]]] => 3 [[.,.],[[.,.],[.,[[.,.],.]]]] => 3 [[.,.],[[.,.],[[.,.],[.,.]]]] => 4 [[.,.],[[.,.],[[.,[.,.]],.]]] => 3 [[.,.],[[.,.],[[[.,.],.],.]]] => 3 [[.,.],[[.,[.,.]],[.,[.,.]]]] => 3 [[.,.],[[.,[.,.]],[[.,.],.]]] => 3 [[.,.],[[[.,.],.],[.,[.,.]]]] => 3 [[.,.],[[[.,.],.],[[.,.],.]]] => 3 [[.,.],[[.,[.,[.,.]]],[.,.]]] => 3 [[.,.],[[.,[[.,.],.]],[.,.]]] => 3 [[.,.],[[[.,.],[.,.]],[.,.]]] => 4 [[.,.],[[[.,[.,.]],.],[.,.]]] => 3 [[.,.],[[[[.,.],.],.],[.,.]]] => 3 [[.,.],[[.,[.,[.,[.,.]]]],.]] => 2 [[.,.],[[.,[.,[[.,.],.]]],.]] => 2 [[.,.],[[.,[[.,.],[.,.]]],.]] => 3 [[.,.],[[.,[[.,[.,.]],.]],.]] => 2 [[.,.],[[.,[[[.,.],.],.]],.]] => 2 [[.,.],[[[.,.],[.,[.,.]]],.]] => 3 [[.,.],[[[.,.],[[.,.],.]],.]] => 3 [[.,.],[[[.,[.,.]],[.,.]],.]] => 3 [[.,.],[[[[.,.],.],[.,.]],.]] => 3 [[.,.],[[[.,[.,[.,.]]],.],.]] => 2 [[.,.],[[[.,[[.,.],.]],.],.]] => 2 [[.,.],[[[[.,.],[.,.]],.],.]] => 3 [[.,.],[[[[.,[.,.]],.],.],.]] => 2 [[.,.],[[[[[.,.],.],.],.],.]] => 2 [[.,[.,.]],[.,[.,[.,[.,.]]]]] => 2 [[.,[.,.]],[.,[.,[[.,.],.]]]] => 2 [[.,[.,.]],[.,[[.,.],[.,.]]]] => 3 [[.,[.,.]],[.,[[.,[.,.]],.]]] => 2 [[.,[.,.]],[.,[[[.,.],.],.]]] => 2 [[.,[.,.]],[[.,.],[.,[.,.]]]] => 3 [[.,[.,.]],[[.,.],[[.,.],.]]] => 3 [[.,[.,.]],[[.,[.,.]],[.,.]]] => 3 [[.,[.,.]],[[[.,.],.],[.,.]]] => 3 [[.,[.,.]],[[.,[.,[.,.]]],.]] => 2 [[.,[.,.]],[[.,[[.,.],.]],.]] => 2 [[.,[.,.]],[[[.,.],[.,.]],.]] => 3 [[.,[.,.]],[[[.,[.,.]],.],.]] => 2 [[.,[.,.]],[[[[.,.],.],.],.]] => 2 [[[.,.],.],[.,[.,[.,[.,.]]]]] => 2 [[[.,.],.],[.,[.,[[.,.],.]]]] => 2 [[[.,.],.],[.,[[.,.],[.,.]]]] => 3 [[[.,.],.],[.,[[.,[.,.]],.]]] => 2 [[[.,.],.],[.,[[[.,.],.],.]]] => 2 [[[.,.],.],[[.,.],[.,[.,.]]]] => 3 [[[.,.],.],[[.,.],[[.,.],.]]] => 3 [[[.,.],.],[[.,[.,.]],[.,.]]] => 3 [[[.,.],.],[[[.,.],.],[.,.]]] => 3 [[[.,.],.],[[.,[.,[.,.]]],.]] => 2 [[[.,.],.],[[.,[[.,.],.]],.]] => 2 [[[.,.],.],[[[.,.],[.,.]],.]] => 3 [[[.,.],.],[[[.,[.,.]],.],.]] => 2 [[[.,.],.],[[[[.,.],.],.],.]] => 2 [[.,[.,[.,.]]],[.,[.,[.,.]]]] => 2 [[.,[.,[.,.]]],[.,[[.,.],.]]] => 2 [[.,[.,[.,.]]],[[.,.],[.,.]]] => 3 [[.,[.,[.,.]]],[[.,[.,.]],.]] => 2 [[.,[.,[.,.]]],[[[.,.],.],.]] => 2 [[.,[[.,.],.]],[.,[.,[.,.]]]] => 2 [[.,[[.,.],.]],[.,[[.,.],.]]] => 2 [[.,[[.,.],.]],[[.,.],[.,.]]] => 3 [[.,[[.,.],.]],[[.,[.,.]],.]] => 2 [[.,[[.,.],.]],[[[.,.],.],.]] => 2 [[[.,.],[.,.]],[.,[.,[.,.]]]] => 3 [[[.,.],[.,.]],[.,[[.,.],.]]] => 3 [[[.,.],[.,.]],[[.,.],[.,.]]] => 4 [[[.,.],[.,.]],[[.,[.,.]],.]] => 3 [[[.,.],[.,.]],[[[.,.],.],.]] => 3 [[[.,[.,.]],.],[.,[.,[.,.]]]] => 2 [[[.,[.,.]],.],[.,[[.,.],.]]] => 2 [[[.,[.,.]],.],[[.,.],[.,.]]] => 3 [[[.,[.,.]],.],[[.,[.,.]],.]] => 2 [[[.,[.,.]],.],[[[.,.],.],.]] => 2 [[[[.,.],.],.],[.,[.,[.,.]]]] => 2 [[[[.,.],.],.],[.,[[.,.],.]]] => 2 [[[[.,.],.],.],[[.,.],[.,.]]] => 3 [[[[.,.],.],.],[[.,[.,.]],.]] => 2 [[[[.,.],.],.],[[[.,.],.],.]] => 2 [[.,[.,[.,[.,.]]]],[.,[.,.]]] => 2 [[.,[.,[.,[.,.]]]],[[.,.],.]] => 2 [[.,[.,[[.,.],.]]],[.,[.,.]]] => 2 [[.,[.,[[.,.],.]]],[[.,.],.]] => 2 [[.,[[.,.],[.,.]]],[.,[.,.]]] => 3 [[.,[[.,.],[.,.]]],[[.,.],.]] => 3 [[.,[[.,[.,.]],.]],[.,[.,.]]] => 2 [[.,[[.,[.,.]],.]],[[.,.],.]] => 2 [[.,[[[.,.],.],.]],[.,[.,.]]] => 2 [[.,[[[.,.],.],.]],[[.,.],.]] => 2 [[[.,.],[.,[.,.]]],[.,[.,.]]] => 3 [[[.,.],[.,[.,.]]],[[.,.],.]] => 3 [[[.,.],[[.,.],.]],[.,[.,.]]] => 3 [[[.,.],[[.,.],.]],[[.,.],.]] => 3 [[[.,[.,.]],[.,.]],[.,[.,.]]] => 3 [[[.,[.,.]],[.,.]],[[.,.],.]] => 3 [[[[.,.],.],[.,.]],[.,[.,.]]] => 3 [[[[.,.],.],[.,.]],[[.,.],.]] => 3 [[[.,[.,[.,.]]],.],[.,[.,.]]] => 2 [[[.,[.,[.,.]]],.],[[.,.],.]] => 2 [[[.,[[.,.],.]],.],[.,[.,.]]] => 2 [[[.,[[.,.],.]],.],[[.,.],.]] => 2 [[[[.,.],[.,.]],.],[.,[.,.]]] => 3 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[.,[[[[[.,[[.,[.,.]],.]],[.,.]],[.,.]],[.,.]],.]] => 4 [.,[[[[[[.,[.,.]],[.,.]],[.,.]],[.,.]],[.,.]],.]] => 5 [[.,[.,.]],[[.,[[.,[[.,[[.,[.,.]],.]],.]],.]],.]] => 2 [[.,[.,.]],[[.,[[.,[[[.,[.,.]],[.,.]],.]],.]],.]] => 3 [[.,[.,.]],[[.,[[[.,[.,.]],[[.,[.,.]],.]],.]],.]] => 3 [[.,[.,.]],[[.,[[[.,[[.,[.,.]],.]],[.,.]],.]],.]] => 3 [[.,[.,.]],[[.,[[[[.,[.,.]],[.,.]],[.,.]],.]],.]] => 4 [[.,[.,.]],[[[.,[.,.]],[[.,[[.,[.,.]],.]],.]],.]] => 3 [[.,[.,.]],[[[.,[.,.]],[[[.,[.,.]],[.,.]],.]],.]] => 4 [[.,[.,.]],[[[.,[[.,[.,.]],.]],[[.,[.,.]],.]],.]] => 3 [[.,[.,.]],[[[[.,[.,.]],[.,.]],[[.,[.,.]],.]],.]] => 4 [[.,[.,.]],[[[.,[[.,[[.,[.,.]],.]],.]],[.,.]],.]] => 3 [[.,[.,.]],[[[.,[[[.,[.,.]],[.,.]],.]],[.,.]],.]] => 4 [[.,[.,.]],[[[[.,[.,.]],[[.,[.,.]],.]],[.,.]],.]] => 4 [[.,[.,.]],[[[[.,[[.,[.,.]],.]],[.,.]],[.,.]],.]] => 4 [[.,[.,.]],[[[[[.,[.,.]],[.,.]],[.,.]],[.,.]],.]] => 5 [[.,[[.,[.,.]],.]],[[.,[[.,[[.,[.,.]],.]],.]],.]] => 2 [[.,[[.,[.,.]],.]],[[.,[[[.,[.,.]],[.,.]],.]],.]] => 3 [[.,[[.,[.,.]],.]],[[[.,[.,.]],[[.,[.,.]],.]],.]] => 3 [[.,[[.,[.,.]],.]],[[[.,[[.,[.,.]],.]],[.,.]],.]] => 3 [[.,[[.,[.,.]],.]],[[[[.,[.,.]],[.,.]],[.,.]],.]] => 4 [[[.,[.,.]],[.,.]],[[.,[[.,[[.,[.,.]],.]],.]],.]] => 3 [[[.,[.,.]],[.,.]],[[.,[[[.,[.,.]],[.,.]],.]],.]] => 4 [[[.,[.,.]],[.,.]],[[[.,[.,.]],[[.,[.,.]],.]],.]] => 4 [[[.,[.,.]],[.,.]],[[[.,[[.,[.,.]],.]],[.,.]],.]] => 4 [[[.,[.,.]],[.,.]],[[[[.,[.,.]],[.,.]],[.,.]],.]] => 5 [[.,[[.,[[.,[.,.]],.]],.]],[[.,[[.,[.,.]],.]],.]] => 2 [[.,[[.,[[.,[.,.]],.]],.]],[[[.,[.,.]],[.,.]],.]] => 3 [[.,[[[.,[.,.]],[.,.]],.]],[[.,[[.,[.,.]],.]],.]] => 3 [[.,[[[.,[.,.]],[.,.]],.]],[[[.,[.,.]],[.,.]],.]] => 4 [[[.,[.,.]],[[.,[.,.]],.]],[[.,[[.,[.,.]],.]],.]] => 3 [[[.,[.,.]],[[.,[.,.]],.]],[[[.,[.,.]],[.,.]],.]] => 4 [[[.,[[.,[.,.]],.]],[.,.]],[[.,[[.,[.,.]],.]],.]] => 3 [[[.,[[.,[.,.]],.]],[.,.]],[[[.,[.,.]],[.,.]],.]] => 4 [[[[.,[.,.]],[.,.]],[.,.]],[[.,[[.,[.,.]],.]],.]] => 4 [[[[.,[.,.]],[.,.]],[.,.]],[[[.,[.,.]],[.,.]],.]] => 5 [[.,[[.,[[.,[[.,[.,.]],.]],.]],.]],[[.,[.,.]],.]] => 2 [[.,[[.,[[[.,[.,.]],[.,.]],.]],.]],[[.,[.,.]],.]] => 3 [[.,[[[.,[.,.]],[[.,[.,.]],.]],.]],[[.,[.,.]],.]] => 3 [[.,[[[.,[[.,[.,.]],.]],[.,.]],.]],[[.,[.,.]],.]] => 3 [[.,[[[[.,[.,.]],[.,.]],[.,.]],.]],[[.,[.,.]],.]] => 4 [[[.,[.,.]],[[.,[[.,[.,.]],.]],.]],[[.,[.,.]],.]] => 3 [[[.,[.,.]],[[[.,[.,.]],[.,.]],.]],[[.,[.,.]],.]] => 4 [[[.,[[.,[.,.]],.]],[[.,[.,.]],.]],[[.,[.,.]],.]] => 3 [[[[.,[.,.]],[.,.]],[[.,[.,.]],.]],[[.,[.,.]],.]] => 4 [[[.,[[.,[[.,[.,.]],.]],.]],[.,.]],[[.,[.,.]],.]] => 3 [[[.,[[[.,[.,.]],[.,.]],.]],[.,.]],[[.,[.,.]],.]] => 4 [[[[.,[.,.]],[[.,[.,.]],.]],[.,.]],[[.,[.,.]],.]] => 4 [[[[.,[[.,[.,.]],.]],[.,.]],[.,.]],[[.,[.,.]],.]] => 4 [[[[[.,[.,.]],[.,.]],[.,.]],[.,.]],[[.,[.,.]],.]] => 5 [[.,[[.,[[.,[[.,[[.,[.,.]],.]],.]],.]],.]],[.,.]] => 2 [[.,[[.,[[.,[[[.,[.,.]],[.,.]],.]],.]],.]],[.,.]] => 3 [[.,[[.,[[[.,[.,.]],[[.,[.,.]],.]],.]],.]],[.,.]] => 3 [[.,[[.,[[[.,[[.,[.,.]],.]],[.,.]],.]],.]],[.,.]] => 3 [[.,[[.,[[[[.,[.,.]],[.,.]],[.,.]],.]],.]],[.,.]] => 4 [[.,[[[.,[.,.]],[[.,[[.,[.,.]],.]],.]],.]],[.,.]] => 3 [[.,[[[.,[.,.]],[[[.,[.,.]],[.,.]],.]],.]],[.,.]] => 4 [[.,[[[.,[[.,[.,.]],.]],[[.,[.,.]],.]],.]],[.,.]] => 3 [[.,[[[[.,[.,.]],[.,.]],[[.,[.,.]],.]],.]],[.,.]] => 4 [[.,[[[.,[[.,[[.,[.,.]],.]],.]],[.,.]],.]],[.,.]] => 3 [[.,[[[.,[[[.,[.,.]],[.,.]],.]],[.,.]],.]],[.,.]] => 4 [[.,[[[[.,[.,.]],[[.,[.,.]],.]],[.,.]],.]],[.,.]] => 4 [[.,[[[[.,[[.,[.,.]],.]],[.,.]],[.,.]],.]],[.,.]] => 4 [[.,[[[[[.,[.,.]],[.,.]],[.,.]],[.,.]],.]],[.,.]] => 5 [[[.,[.,.]],[[.,[[.,[[.,[.,.]],.]],.]],.]],[.,.]] => 3 [[[.,[.,.]],[[.,[[[.,[.,.]],[.,.]],.]],.]],[.,.]] => 4 [[[.,[.,.]],[[[.,[.,.]],[[.,[.,.]],.]],.]],[.,.]] => 4 [[[.,[.,.]],[[[.,[[.,[.,.]],.]],[.,.]],.]],[.,.]] => 4 [[[.,[.,.]],[[[[.,[.,.]],[.,.]],[.,.]],.]],[.,.]] => 5 [[[.,[[.,[.,.]],.]],[[.,[[.,[.,.]],.]],.]],[.,.]] => 3 [[[.,[[.,[.,.]],.]],[[[.,[.,.]],[.,.]],.]],[.,.]] => 4 [[[[.,[.,.]],[.,.]],[[.,[[.,[.,.]],.]],.]],[.,.]] => 4 [[[[.,[.,.]],[.,.]],[[[.,[.,.]],[.,.]],.]],[.,.]] => 5 [[[.,[[.,[[.,[.,.]],.]],.]],[[.,[.,.]],.]],[.,.]] => 3 [[[.,[[[.,[.,.]],[.,.]],.]],[[.,[.,.]],.]],[.,.]] => 4 [[[[.,[.,.]],[[.,[.,.]],.]],[[.,[.,.]],.]],[.,.]] => 4 [[[[.,[[.,[.,.]],.]],[.,.]],[[.,[.,.]],.]],[.,.]] => 4 [[[[[.,[.,.]],[.,.]],[.,.]],[[.,[.,.]],.]],[.,.]] => 5 [[[.,[[.,[[.,[[.,[.,.]],.]],.]],.]],[.,.]],[.,.]] => 3 [[[.,[[.,[[[.,[.,.]],[.,.]],.]],.]],[.,.]],[.,.]] => 4 [[[.,[[[.,[.,.]],[[.,[.,.]],.]],.]],[.,.]],[.,.]] => 4 [[[.,[[[.,[[.,[.,.]],.]],[.,.]],.]],[.,.]],[.,.]] => 4 [[[.,[[[[.,[.,.]],[.,.]],[.,.]],.]],[.,.]],[.,.]] => 5 [[[[.,[.,.]],[[.,[[.,[.,.]],.]],.]],[.,.]],[.,.]] => 4 [[[[.,[.,.]],[[[.,[.,.]],[.,.]],.]],[.,.]],[.,.]] => 5 [[[[.,[[.,[.,.]],.]],[[.,[.,.]],.]],[.,.]],[.,.]] => 4 [[[[[.,[.,.]],[.,.]],[[.,[.,.]],.]],[.,.]],[.,.]] => 5 [[[[.,[[.,[[.,[.,.]],.]],.]],[.,.]],[.,.]],[.,.]] => 4 [[[[.,[[[.,[.,.]],[.,.]],.]],[.,.]],[.,.]],[.,.]] => 5 [[[[[.,[.,.]],[[.,[.,.]],.]],[.,.]],[.,.]],[.,.]] => 5 [[[[[.,[[.,[.,.]],.]],[.,.]],[.,.]],[.,.]],[.,.]] => 5 ----------------------------------------------------------------------------- Created: May 16, 2014 at 09:02 by Joseph Ching ----------------------------------------------------------------------------- Last Updated: Mar 31, 2019 at 22:07 by Martin Rubey