***************************************************************************** * 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: St000204 ----------------------------------------------------------------------------- Collection: Binary trees ----------------------------------------------------------------------------- Description: The number of internal nodes of a binary tree. That is, the total number of nodes of the tree minus [[St000203]]. A counting formula for the total number of internal nodes across all binary trees of size $n$ is given in [1]. This is equivalent to the number of internal triangles in all triangulations of an $(n+1)$-gon. ----------------------------------------------------------------------------- References: [1] Number of internal triangles in all triangulations of an (n+1)-gon. [[OEIS:A002058]] ----------------------------------------------------------------------------- Code: # BinaryTree -> Bool def is_leaf(b): return b == BinaryTree('.') # BinaryTree -> Int def get_external_node(b, flag): count = 0; if is_leaf(b): return 0 else: if (flag): # True counts the left side if not is_leaf(b[0]): return 1 + get_external_node(b[0], True) else: # False counts the right side if not is_leaf(b[1]): return 1 + get_external_node(b[1], False) return count # BinaryTree -> Int def external_node_count(b): return 1 + get_external_node(b, True) + get_external_node(b, False) def size(b): if not b: return 1 else: return size(b[0])+size(b[1]) def statistic(b): return size(b)-1-external_node_count(b) ----------------------------------------------------------------------------- Statistic values: [.,.] => 0 [.,[.,.]] => 0 [[.,.],.] => 0 [.,[.,[.,.]]] => 0 [.,[[.,.],.]] => 1 [[.,.],[.,.]] => 0 [[.,[.,.]],.] => 1 [[[.,.],.],.] => 0 [.,[.,[.,[.,.]]]] => 0 [.,[.,[[.,.],.]]] => 1 [.,[[.,.],[.,.]]] => 1 [.,[[.,[.,.]],.]] => 2 [.,[[[.,.],.],.]] => 2 [[.,.],[.,[.,.]]] => 0 [[.,.],[[.,.],.]] => 1 [[.,[.,.]],[.,.]] => 1 [[[.,.],.],[.,.]] => 0 [[.,[.,[.,.]]],.] => 2 [[.,[[.,.],.]],.] => 2 [[[.,.],[.,.]],.] => 1 [[[.,[.,.]],.],.] => 1 [[[[.,.],.],.],.] => 0 [.,[.,[.,[.,[.,.]]]]] => 0 [.,[.,[.,[[.,.],.]]]] => 1 [.,[.,[[.,.],[.,.]]]] => 1 [.,[.,[[.,[.,.]],.]]] => 2 [.,[.,[[[.,.],.],.]]] => 2 [.,[[.,.],[.,[.,.]]]] => 1 [.,[[.,.],[[.,.],.]]] => 2 [.,[[.,[.,.]],[.,.]]] => 2 [.,[[[.,.],.],[.,.]]] => 2 [.,[[.,[.,[.,.]]],.]] => 3 [.,[[.,[[.,.],.]],.]] => 3 [.,[[[.,.],[.,.]],.]] => 3 [.,[[[.,[.,.]],.],.]] => 3 [.,[[[[.,.],.],.],.]] => 3 [[.,.],[.,[.,[.,.]]]] => 0 [[.,.],[.,[[.,.],.]]] => 1 [[.,.],[[.,.],[.,.]]] => 1 [[.,.],[[.,[.,.]],.]] => 2 [[.,.],[[[.,.],.],.]] => 2 [[.,[.,.]],[.,[.,.]]] => 1 [[.,[.,.]],[[.,.],.]] => 2 [[[.,.],.],[.,[.,.]]] => 0 [[[.,.],.],[[.,.],.]] => 1 [[.,[.,[.,.]]],[.,.]] => 2 [[.,[[.,.],.]],[.,.]] => 2 [[[.,.],[.,.]],[.,.]] => 1 [[[.,[.,.]],.],[.,.]] => 1 [[[[.,.],.],.],[.,.]] => 0 [[.,[.,[.,[.,.]]]],.] => 3 [[.,[.,[[.,.],.]]],.] => 3 [[.,[[.,.],[.,.]]],.] => 3 [[.,[[.,[.,.]],.]],.] => 3 [[.,[[[.,.],.],.]],.] => 3 [[[.,.],[.,[.,.]]],.] => 2 [[[.,.],[[.,.],.]],.] => 2 [[[.,[.,.]],[.,.]],.] => 2 [[[[.,.],.],[.,.]],.] => 1 [[[.,[.,[.,.]]],.],.] => 2 [[[.,[[.,.],.]],.],.] => 2 [[[[.,.],[.,.]],.],.] => 1 [[[[.,[.,.]],.],.],.] => 1 [[[[[.,.],.],.],.],.] => 0 [.,[.,[.,[.,[.,[.,.]]]]]] => 0 [.,[.,[.,[.,[[.,.],.]]]]] => 1 [.,[.,[.,[[.,.],[.,.]]]]] => 1 [.,[.,[.,[[.,[.,.]],.]]]] => 2 [.,[.,[.,[[[.,.],.],.]]]] => 2 [.,[.,[[.,.],[.,[.,.]]]]] => 1 [.,[.,[[.,.],[[.,.],.]]]] => 2 [.,[.,[[.,[.,.]],[.,.]]]] => 2 [.,[.,[[[.,.],.],[.,.]]]] => 2 [.,[.,[[.,[.,[.,.]]],.]]] => 3 [.,[.,[[.,[[.,.],.]],.]]] => 3 [.,[.,[[[.,.],[.,.]],.]]] => 3 [.,[.,[[[.,[.,.]],.],.]]] => 3 [.,[.,[[[[.,.],.],.],.]]] => 3 [.,[[.,.],[.,[.,[.,.]]]]] => 1 [.,[[.,.],[.,[[.,.],.]]]] => 2 [.,[[.,.],[[.,.],[.,.]]]] => 2 [.,[[.,.],[[.,[.,.]],.]]] => 3 [.,[[.,.],[[[.,.],.],.]]] => 3 [.,[[.,[.,.]],[.,[.,.]]]] => 2 [.,[[.,[.,.]],[[.,.],.]]] => 3 [.,[[[.,.],.],[.,[.,.]]]] => 2 [.,[[[.,.],.],[[.,.],.]]] => 3 [.,[[.,[.,[.,.]]],[.,.]]] => 3 [.,[[.,[[.,.],.]],[.,.]]] => 3 [.,[[[.,.],[.,.]],[.,.]]] => 3 [.,[[[.,[.,.]],.],[.,.]]] => 3 [.,[[[[.,.],.],.],[.,.]]] => 3 [.,[[.,[.,[.,[.,.]]]],.]] => 4 [.,[[.,[.,[[.,.],.]]],.]] => 4 [.,[[.,[[.,.],[.,.]]],.]] => 4 [.,[[.,[[.,[.,.]],.]],.]] => 4 [.,[[.,[[[.,.],.],.]],.]] => 4 [.,[[[.,.],[.,[.,.]]],.]] => 4 [.,[[[.,.],[[.,.],.]],.]] => 4 [.,[[[.,[.,.]],[.,.]],.]] => 4 [.,[[[[.,.],.],[.,.]],.]] => 4 [.,[[[.,[.,[.,.]]],.],.]] => 4 [.,[[[.,[[.,.],.]],.],.]] => 4 [.,[[[[.,.],[.,.]],.],.]] => 4 [.,[[[[.,[.,.]],.],.],.]] => 4 [.,[[[[[.,.],.],.],.],.]] => 4 [[.,.],[.,[.,[.,[.,.]]]]] => 0 [[.,.],[.,[.,[[.,.],.]]]] => 1 [[.,.],[.,[[.,.],[.,.]]]] => 1 [[.,.],[.,[[.,[.,.]],.]]] => 2 [[.,.],[.,[[[.,.],.],.]]] => 2 [[.,.],[[.,.],[.,[.,.]]]] => 1 [[.,.],[[.,.],[[.,.],.]]] => 2 [[.,.],[[.,[.,.]],[.,.]]] => 2 [[.,.],[[[.,.],.],[.,.]]] => 2 [[.,.],[[.,[.,[.,.]]],.]] => 3 [[.,.],[[.,[[.,.],.]],.]] => 3 [[.,.],[[[.,.],[.,.]],.]] => 3 [[.,.],[[[.,[.,.]],.],.]] => 3 [[.,.],[[[[.,.],.],.],.]] => 3 [[.,[.,.]],[.,[.,[.,.]]]] => 1 [[.,[.,.]],[.,[[.,.],.]]] => 2 [[.,[.,.]],[[.,.],[.,.]]] => 2 [[.,[.,.]],[[.,[.,.]],.]] => 3 [[.,[.,.]],[[[.,.],.],.]] => 3 [[[.,.],.],[.,[.,[.,.]]]] => 0 [[[.,.],.],[.,[[.,.],.]]] => 1 [[[.,.],.],[[.,.],[.,.]]] => 1 [[[.,.],.],[[.,[.,.]],.]] => 2 [[[.,.],.],[[[.,.],.],.]] => 2 [[.,[.,[.,.]]],[.,[.,.]]] => 2 [[.,[.,[.,.]]],[[.,.],.]] => 3 [[.,[[.,.],.]],[.,[.,.]]] => 2 [[.,[[.,.],.]],[[.,.],.]] => 3 [[[.,.],[.,.]],[.,[.,.]]] => 1 [[[.,.],[.,.]],[[.,.],.]] => 2 [[[.,[.,.]],.],[.,[.,.]]] => 1 [[[.,[.,.]],.],[[.,.],.]] => 2 [[[[.,.],.],.],[.,[.,.]]] => 0 [[[[.,.],.],.],[[.,.],.]] => 1 [[.,[.,[.,[.,.]]]],[.,.]] => 3 [[.,[.,[[.,.],.]]],[.,.]] => 3 [[.,[[.,.],[.,.]]],[.,.]] => 3 [[.,[[.,[.,.]],.]],[.,.]] => 3 [[.,[[[.,.],.],.]],[.,.]] => 3 [[[.,.],[.,[.,.]]],[.,.]] => 2 [[[.,.],[[.,.],.]],[.,.]] => 2 [[[.,[.,.]],[.,.]],[.,.]] => 2 [[[[.,.],.],[.,.]],[.,.]] => 1 [[[.,[.,[.,.]]],.],[.,.]] => 2 [[[.,[[.,.],.]],.],[.,.]] => 2 [[[[.,.],[.,.]],.],[.,.]] => 1 [[[[.,[.,.]],.],.],[.,.]] => 1 [[[[[.,.],.],.],.],[.,.]] => 0 [[.,[.,[.,[.,[.,.]]]]],.] => 4 [[.,[.,[.,[[.,.],.]]]],.] => 4 [[.,[.,[[.,.],[.,.]]]],.] => 4 [[.,[.,[[.,[.,.]],.]]],.] => 4 [[.,[.,[[[.,.],.],.]]],.] => 4 [[.,[[.,.],[.,[.,.]]]],.] => 4 [[.,[[.,.],[[.,.],.]]],.] => 4 [[.,[[.,[.,.]],[.,.]]],.] => 4 [[.,[[[.,.],.],[.,.]]],.] => 4 [[.,[[.,[.,[.,.]]],.]],.] => 4 [[.,[[.,[[.,.],.]],.]],.] => 4 [[.,[[[.,.],[.,.]],.]],.] => 4 [[.,[[[.,[.,.]],.],.]],.] => 4 [[.,[[[[.,.],.],.],.]],.] => 4 [[[.,.],[.,[.,[.,.]]]],.] => 3 [[[.,.],[.,[[.,.],.]]],.] => 3 [[[.,.],[[.,.],[.,.]]],.] => 3 [[[.,.],[[.,[.,.]],.]],.] => 3 [[[.,.],[[[.,.],.],.]],.] => 3 [[[.,[.,.]],[.,[.,.]]],.] => 3 [[[.,[.,.]],[[.,.],.]],.] => 3 [[[[.,.],.],[.,[.,.]]],.] => 2 [[[[.,.],.],[[.,.],.]],.] => 2 [[[.,[.,[.,.]]],[.,.]],.] => 3 [[[.,[[.,.],.]],[.,.]],.] => 3 [[[[.,.],[.,.]],[.,.]],.] => 2 [[[[.,[.,.]],.],[.,.]],.] => 2 [[[[[.,.],.],.],[.,.]],.] => 1 [[[.,[.,[.,[.,.]]]],.],.] => 3 [[[.,[.,[[.,.],.]]],.],.] => 3 [[[.,[[.,.],[.,.]]],.],.] => 3 [[[.,[[.,[.,.]],.]],.],.] => 3 [[[.,[[[.,.],.],.]],.],.] => 3 [[[[.,.],[.,[.,.]]],.],.] => 2 [[[[.,.],[[.,.],.]],.],.] => 2 [[[[.,[.,.]],[.,.]],.],.] => 2 [[[[[.,.],.],[.,.]],.],.] => 1 [[[[.,[.,[.,.]]],.],.],.] => 2 [[[[.,[[.,.],.]],.],.],.] => 2 [[[[[.,.],[.,.]],.],.],.] => 1 [[[[[.,[.,.]],.],.],.],.] => 1 [[[[[[.,.],.],.],.],.],.] => 0 [.,[.,[.,[.,[.,[.,[.,.]]]]]]] => 0 [.,[.,[.,[.,[.,[[.,.],.]]]]]] => 1 [.,[.,[.,[.,[[.,.],[.,.]]]]]] => 1 [.,[.,[.,[.,[[.,[.,.]],.]]]]] => 2 [.,[.,[.,[.,[[[.,.],.],.]]]]] => 2 [.,[.,[.,[[.,.],[.,[.,.]]]]]] => 1 [.,[.,[.,[[.,.],[[.,.],.]]]]] => 2 [.,[.,[.,[[.,[.,.]],[.,.]]]]] => 2 [.,[.,[.,[[[.,.],.],[.,.]]]]] => 2 [.,[.,[.,[[.,[.,[.,.]]],.]]]] => 3 [.,[.,[.,[[.,[[.,.],.]],.]]]] => 3 [.,[.,[.,[[[.,.],[.,.]],.]]]] => 3 [.,[.,[.,[[[.,[.,.]],.],.]]]] => 3 [.,[.,[.,[[[[.,.],.],.],.]]]] => 3 [.,[.,[[.,.],[.,[.,[.,.]]]]]] => 1 [.,[.,[[.,.],[.,[[.,.],.]]]]] => 2 [.,[.,[[.,.],[[.,.],[.,.]]]]] => 2 [.,[.,[[.,.],[[.,[.,.]],.]]]] => 3 [.,[.,[[.,.],[[[.,.],.],.]]]] => 3 [.,[.,[[.,[.,.]],[.,[.,.]]]]] => 2 [.,[.,[[.,[.,.]],[[.,.],.]]]] => 3 [.,[.,[[[.,.],.],[.,[.,.]]]]] => 2 [.,[.,[[[.,.],.],[[.,.],.]]]] => 3 [.,[.,[[.,[.,[.,.]]],[.,.]]]] => 3 [.,[.,[[.,[[.,.],.]],[.,.]]]] => 3 [.,[.,[[[.,.],[.,.]],[.,.]]]] => 3 [.,[.,[[[.,[.,.]],.],[.,.]]]] => 3 [.,[.,[[[[.,.],.],.],[.,.]]]] => 3 [.,[.,[[.,[.,[.,[.,.]]]],.]]] => 4 [.,[.,[[.,[.,[[.,.],.]]],.]]] => 4 [.,[.,[[.,[[.,.],[.,.]]],.]]] => 4 [.,[.,[[.,[[.,[.,.]],.]],.]]] => 4 [.,[.,[[.,[[[.,.],.],.]],.]]] => 4 [.,[.,[[[.,.],[.,[.,.]]],.]]] => 4 [.,[.,[[[.,.],[[.,.],.]],.]]] => 4 [.,[.,[[[.,[.,.]],[.,.]],.]]] => 4 [.,[.,[[[[.,.],.],[.,.]],.]]] => 4 [.,[.,[[[.,[.,[.,.]]],.],.]]] => 4 [.,[.,[[[.,[[.,.],.]],.],.]]] => 4 [.,[.,[[[[.,.],[.,.]],.],.]]] => 4 [.,[.,[[[[.,[.,.]],.],.],.]]] => 4 [.,[.,[[[[[.,.],.],.],.],.]]] => 4 [.,[[.,.],[.,[.,[.,[.,.]]]]]] => 1 [.,[[.,.],[.,[.,[[.,.],.]]]]] => 2 [.,[[.,.],[.,[[.,.],[.,.]]]]] => 2 [.,[[.,.],[.,[[.,[.,.]],.]]]] => 3 [.,[[.,.],[.,[[[.,.],.],.]]]] => 3 [.,[[.,.],[[.,.],[.,[.,.]]]]] => 2 [.,[[.,.],[[.,.],[[.,.],.]]]] => 3 [.,[[.,.],[[.,[.,.]],[.,.]]]] => 3 [.,[[.,.],[[[.,.],.],[.,.]]]] => 3 [.,[[.,.],[[.,[.,[.,.]]],.]]] => 4 [.,[[.,.],[[.,[[.,.],.]],.]]] => 4 [.,[[.,.],[[[.,.],[.,.]],.]]] => 4 [.,[[.,.],[[[.,[.,.]],.],.]]] => 4 [.,[[.,.],[[[[.,.],.],.],.]]] => 4 [.,[[.,[.,.]],[.,[.,[.,.]]]]] => 2 [.,[[.,[.,.]],[.,[[.,.],.]]]] => 3 [.,[[.,[.,.]],[[.,.],[.,.]]]] => 3 [.,[[.,[.,.]],[[.,[.,.]],.]]] => 4 [.,[[.,[.,.]],[[[.,.],.],.]]] => 4 [.,[[[.,.],.],[.,[.,[.,.]]]]] => 2 [.,[[[.,.],.],[.,[[.,.],.]]]] => 3 [.,[[[.,.],.],[[.,.],[.,.]]]] => 3 [.,[[[.,.],.],[[.,[.,.]],.]]] => 4 [.,[[[.,.],.],[[[.,.],.],.]]] => 4 [.,[[.,[.,[.,.]]],[.,[.,.]]]] => 3 [.,[[.,[.,[.,.]]],[[.,.],.]]] => 4 [.,[[.,[[.,.],.]],[.,[.,.]]]] => 3 [.,[[.,[[.,.],.]],[[.,.],.]]] => 4 [.,[[[.,.],[.,.]],[.,[.,.]]]] => 3 [.,[[[.,.],[.,.]],[[.,.],.]]] => 4 [.,[[[.,[.,.]],.],[.,[.,.]]]] => 3 [.,[[[.,[.,.]],.],[[.,.],.]]] => 4 [.,[[[[.,.],.],.],[.,[.,.]]]] => 3 [.,[[[[.,.],.],.],[[.,.],.]]] => 4 [.,[[.,[.,[.,[.,.]]]],[.,.]]] => 4 [.,[[.,[.,[[.,.],.]]],[.,.]]] => 4 [.,[[.,[[.,.],[.,.]]],[.,.]]] => 4 [.,[[.,[[.,[.,.]],.]],[.,.]]] => 4 [.,[[.,[[[.,.],.],.]],[.,.]]] => 4 [.,[[[.,.],[.,[.,.]]],[.,.]]] => 4 [.,[[[.,.],[[.,.],.]],[.,.]]] => 4 [.,[[[.,[.,.]],[.,.]],[.,.]]] => 4 [.,[[[[.,.],.],[.,.]],[.,.]]] => 4 [.,[[[.,[.,[.,.]]],.],[.,.]]] => 4 [.,[[[.,[[.,.],.]],.],[.,.]]] => 4 [.,[[[[.,.],[.,.]],.],[.,.]]] => 4 [.,[[[[.,[.,.]],.],.],[.,.]]] => 4 [.,[[[[[.,.],.],.],.],[.,.]]] => 4 [.,[[.,[.,[.,[.,[.,.]]]]],.]] => 5 [.,[[.,[.,[.,[[.,.],.]]]],.]] => 5 [.,[[.,[.,[[.,.],[.,.]]]],.]] => 5 [.,[[.,[.,[[.,[.,.]],.]]],.]] => 5 [.,[[.,[.,[[[.,.],.],.]]],.]] => 5 [.,[[.,[[.,.],[.,[.,.]]]],.]] => 5 [.,[[.,[[.,.],[[.,.],.]]],.]] => 5 [.,[[.,[[.,[.,.]],[.,.]]],.]] => 5 [.,[[.,[[[.,.],.],[.,.]]],.]] => 5 [.,[[.,[[.,[.,[.,.]]],.]],.]] => 5 [.,[[.,[[.,[[.,.],.]],.]],.]] => 5 [.,[[.,[[[.,.],[.,.]],.]],.]] => 5 [.,[[.,[[[.,[.,.]],.],.]],.]] => 5 [.,[[.,[[[[.,.],.],.],.]],.]] => 5 [.,[[[.,.],[.,[.,[.,.]]]],.]] => 5 [.,[[[.,.],[.,[[.,.],.]]],.]] => 5 [.,[[[.,.],[[.,.],[.,.]]],.]] => 5 [.,[[[.,.],[[.,[.,.]],.]],.]] => 5 [.,[[[.,.],[[[.,.],.],.]],.]] => 5 [.,[[[.,[.,.]],[.,[.,.]]],.]] => 5 [.,[[[.,[.,.]],[[.,.],.]],.]] => 5 [.,[[[[.,.],.],[.,[.,.]]],.]] => 5 [.,[[[[.,.],.],[[.,.],.]],.]] => 5 [.,[[[.,[.,[.,.]]],[.,.]],.]] => 5 [.,[[[.,[[.,.],.]],[.,.]],.]] => 5 [.,[[[[.,.],[.,.]],[.,.]],.]] => 5 [.,[[[[.,[.,.]],.],[.,.]],.]] => 5 [.,[[[[[.,.],.],.],[.,.]],.]] => 5 [.,[[[.,[.,[.,[.,.]]]],.],.]] => 5 [.,[[[.,[.,[[.,.],.]]],.],.]] => 5 [.,[[[.,[[.,.],[.,.]]],.],.]] => 5 [.,[[[.,[[.,[.,.]],.]],.],.]] => 5 [.,[[[.,[[[.,.],.],.]],.],.]] => 5 [.,[[[[.,.],[.,[.,.]]],.],.]] => 5 [.,[[[[.,.],[[.,.],.]],.],.]] => 5 [.,[[[[.,[.,.]],[.,.]],.],.]] => 5 [.,[[[[[.,.],.],[.,.]],.],.]] => 5 [.,[[[[.,[.,[.,.]]],.],.],.]] => 5 [.,[[[[.,[[.,.],.]],.],.],.]] => 5 [.,[[[[[.,.],[.,.]],.],.],.]] => 5 [.,[[[[[.,[.,.]],.],.],.],.]] => 5 [.,[[[[[[.,.],.],.],.],.],.]] => 5 [[.,.],[.,[.,[.,[.,[.,.]]]]]] => 0 [[.,.],[.,[.,[.,[[.,.],.]]]]] => 1 [[.,.],[.,[.,[[.,.],[.,.]]]]] => 1 [[.,.],[.,[.,[[.,[.,.]],.]]]] => 2 [[.,.],[.,[.,[[[.,.],.],.]]]] => 2 [[.,.],[.,[[.,.],[.,[.,.]]]]] => 1 [[.,.],[.,[[.,.],[[.,.],.]]]] => 2 [[.,.],[.,[[.,[.,.]],[.,.]]]] => 2 [[.,.],[.,[[[.,.],.],[.,.]]]] => 2 [[.,.],[.,[[.,[.,[.,.]]],.]]] => 3 [[.,.],[.,[[.,[[.,.],.]],.]]] => 3 [[.,.],[.,[[[.,.],[.,.]],.]]] => 3 [[.,.],[.,[[[.,[.,.]],.],.]]] => 3 [[.,.],[.,[[[[.,.],.],.],.]]] => 3 [[.,.],[[.,.],[.,[.,[.,.]]]]] => 1 [[.,.],[[.,.],[.,[[.,.],.]]]] => 2 [[.,.],[[.,.],[[.,.],[.,.]]]] => 2 [[.,.],[[.,.],[[.,[.,.]],.]]] => 3 [[.,.],[[.,.],[[[.,.],.],.]]] => 3 [[.,.],[[.,[.,.]],[.,[.,.]]]] => 2 [[.,.],[[.,[.,.]],[[.,.],.]]] => 3 [[.,.],[[[.,.],.],[.,[.,.]]]] => 2 [[.,.],[[[.,.],.],[[.,.],.]]] => 3 [[.,.],[[.,[.,[.,.]]],[.,.]]] => 3 [[.,.],[[.,[[.,.],.]],[.,.]]] => 3 [[.,.],[[[.,.],[.,.]],[.,.]]] => 3 [[.,.],[[[.,[.,.]],.],[.,.]]] => 3 [[.,.],[[[[.,.],.],.],[.,.]]] => 3 [[.,.],[[.,[.,[.,[.,.]]]],.]] => 4 [[.,.],[[.,[.,[[.,.],.]]],.]] => 4 [[.,.],[[.,[[.,.],[.,.]]],.]] => 4 [[.,.],[[.,[[.,[.,.]],.]],.]] => 4 [[.,.],[[.,[[[.,.],.],.]],.]] => 4 [[.,.],[[[.,.],[.,[.,.]]],.]] => 4 [[.,.],[[[.,.],[[.,.],.]],.]] => 4 [[.,.],[[[.,[.,.]],[.,.]],.]] => 4 [[.,.],[[[[.,.],.],[.,.]],.]] => 4 [[.,.],[[[.,[.,[.,.]]],.],.]] => 4 [[.,.],[[[.,[[.,.],.]],.],.]] => 4 [[.,.],[[[[.,.],[.,.]],.],.]] => 4 [[.,.],[[[[.,[.,.]],.],.],.]] => 4 [[.,.],[[[[[.,.],.],.],.],.]] => 4 [[.,[.,.]],[.,[.,[.,[.,.]]]]] => 1 [[.,[.,.]],[.,[.,[[.,.],.]]]] => 2 [[.,[.,.]],[.,[[.,.],[.,.]]]] => 2 [[.,[.,.]],[.,[[.,[.,.]],.]]] => 3 [[.,[.,.]],[.,[[[.,.],.],.]]] => 3 [[.,[.,.]],[[.,.],[.,[.,.]]]] => 2 [[.,[.,.]],[[.,.],[[.,.],.]]] => 3 [[.,[.,.]],[[.,[.,.]],[.,.]]] => 3 [[.,[.,.]],[[[.,.],.],[.,.]]] => 3 [[.,[.,.]],[[.,[.,[.,.]]],.]] => 4 [[.,[.,.]],[[.,[[.,.],.]],.]] => 4 [[.,[.,.]],[[[.,.],[.,.]],.]] => 4 [[.,[.,.]],[[[.,[.,.]],.],.]] => 4 [[.,[.,.]],[[[[.,.],.],.],.]] => 4 [[[.,.],.],[.,[.,[.,[.,.]]]]] => 0 [[[.,.],.],[.,[.,[[.,.],.]]]] => 1 [[[.,.],.],[.,[[.,.],[.,.]]]] => 1 [[[.,.],.],[.,[[.,[.,.]],.]]] => 2 [[[.,.],.],[.,[[[.,.],.],.]]] => 2 [[[.,.],.],[[.,.],[.,[.,.]]]] => 1 [[[.,.],.],[[.,.],[[.,.],.]]] => 2 [[[.,.],.],[[.,[.,.]],[.,.]]] => 2 [[[.,.],.],[[[.,.],.],[.,.]]] => 2 [[[.,.],.],[[.,[.,[.,.]]],.]] => 3 [[[.,.],.],[[.,[[.,.],.]],.]] => 3 [[[.,.],.],[[[.,.],[.,.]],.]] => 3 [[[.,.],.],[[[.,[.,.]],.],.]] => 3 [[[.,.],.],[[[[.,.],.],.],.]] => 3 [[.,[.,[.,.]]],[.,[.,[.,.]]]] => 2 [[.,[.,[.,.]]],[.,[[.,.],.]]] => 3 [[.,[.,[.,.]]],[[.,.],[.,.]]] => 3 [[.,[.,[.,.]]],[[.,[.,.]],.]] => 4 [[.,[.,[.,.]]],[[[.,.],.],.]] => 4 [[.,[[.,.],.]],[.,[.,[.,.]]]] => 2 [[.,[[.,.],.]],[.,[[.,.],.]]] => 3 [[.,[[.,.],.]],[[.,.],[.,.]]] => 3 [[.,[[.,.],.]],[[.,[.,.]],.]] => 4 [[.,[[.,.],.]],[[[.,.],.],.]] => 4 [[[.,.],[.,.]],[.,[.,[.,.]]]] => 1 [[[.,.],[.,.]],[.,[[.,.],.]]] => 2 [[[.,.],[.,.]],[[.,.],[.,.]]] => 2 [[[.,.],[.,.]],[[.,[.,.]],.]] => 3 [[[.,.],[.,.]],[[[.,.],.],.]] => 3 [[[.,[.,.]],.],[.,[.,[.,.]]]] => 1 [[[.,[.,.]],.],[.,[[.,.],.]]] => 2 [[[.,[.,.]],.],[[.,.],[.,.]]] => 2 [[[.,[.,.]],.],[[.,[.,.]],.]] => 3 [[[.,[.,.]],.],[[[.,.],.],.]] => 3 [[[[.,.],.],.],[.,[.,[.,.]]]] => 0 [[[[.,.],.],.],[.,[[.,.],.]]] => 1 [[[[.,.],.],.],[[.,.],[.,.]]] => 1 [[[[.,.],.],.],[[.,[.,.]],.]] => 2 [[[[.,.],.],.],[[[.,.],.],.]] => 2 [[.,[.,[.,[.,.]]]],[.,[.,.]]] => 3 [[.,[.,[.,[.,.]]]],[[.,.],.]] => 4 [[.,[.,[[.,.],.]]],[.,[.,.]]] => 3 [[.,[.,[[.,.],.]]],[[.,.],.]] => 4 [[.,[[.,.],[.,.]]],[.,[.,.]]] => 3 [[.,[[.,.],[.,.]]],[[.,.],.]] => 4 [[.,[[.,[.,.]],.]],[.,[.,.]]] => 3 [[.,[[.,[.,.]],.]],[[.,.],.]] => 4 [[.,[[[.,.],.],.]],[.,[.,.]]] => 3 [[.,[[[.,.],.],.]],[[.,.],.]] => 4 [[[.,.],[.,[.,.]]],[.,[.,.]]] => 2 [[[.,.],[.,[.,.]]],[[.,.],.]] => 3 [[[.,.],[[.,.],.]],[.,[.,.]]] => 2 [[[.,.],[[.,.],.]],[[.,.],.]] => 3 [[[.,[.,.]],[.,.]],[.,[.,.]]] => 2 [[[.,[.,.]],[.,.]],[[.,.],.]] => 3 [[[[.,.],.],[.,.]],[.,[.,.]]] => 1 [[[[.,.],.],[.,.]],[[.,.],.]] => 2 [[[.,[.,[.,.]]],.],[.,[.,.]]] => 2 [[[.,[.,[.,.]]],.],[[.,.],.]] => 3 [[[.,[[.,.],.]],.],[.,[.,.]]] => 2 [[[.,[[.,.],.]],.],[[.,.],.]] => 3 [[[[.,.],[.,.]],.],[.,[.,.]]] => 1 [[[[.,.],[.,.]],.],[[.,.],.]] => 2 [[[[.,[.,.]],.],.],[.,[.,.]]] => 1 [[[[.,[.,.]],.],.],[[.,.],.]] => 2 [[[[[.,.],.],.],.],[.,[.,.]]] => 0 [[[[[.,.],.],.],.],[[.,.],.]] => 1 [[.,[.,[.,[.,[.,.]]]]],[.,.]] => 4 [[.,[.,[.,[[.,.],.]]]],[.,.]] => 4 [[.,[.,[[.,.],[.,.]]]],[.,.]] => 4 [[.,[.,[[.,[.,.]],.]]],[.,.]] => 4 [[.,[.,[[[.,.],.],.]]],[.,.]] => 4 [[.,[[.,.],[.,[.,.]]]],[.,.]] => 4 [[.,[[.,.],[[.,.],.]]],[.,.]] => 4 [[.,[[.,[.,.]],[.,.]]],[.,.]] => 4 [[.,[[[.,.],.],[.,.]]],[.,.]] => 4 [[.,[[.,[.,[.,.]]],.]],[.,.]] => 4 [[.,[[.,[[.,.],.]],.]],[.,.]] => 4 [[.,[[[.,.],[.,.]],.]],[.,.]] => 4 [[.,[[[.,[.,.]],.],.]],[.,.]] => 4 [[.,[[[[.,.],.],.],.]],[.,.]] => 4 [[[.,.],[.,[.,[.,.]]]],[.,.]] => 3 [[[.,.],[.,[[.,.],.]]],[.,.]] => 3 [[[.,.],[[.,.],[.,.]]],[.,.]] => 3 [[[.,.],[[.,[.,.]],.]],[.,.]] => 3 [[[.,.],[[[.,.],.],.]],[.,.]] => 3 [[[.,[.,.]],[.,[.,.]]],[.,.]] => 3 [[[.,[.,.]],[[.,.],.]],[.,.]] => 3 [[[[.,.],.],[.,[.,.]]],[.,.]] => 2 [[[[.,.],.],[[.,.],.]],[.,.]] => 2 [[[.,[.,[.,.]]],[.,.]],[.,.]] => 3 [[[.,[[.,.],.]],[.,.]],[.,.]] => 3 [[[[.,.],[.,.]],[.,.]],[.,.]] => 2 [[[[.,[.,.]],.],[.,.]],[.,.]] => 2 [[[[[.,.],.],.],[.,.]],[.,.]] => 1 [[[.,[.,[.,[.,.]]]],.],[.,.]] => 3 [[[.,[.,[[.,.],.]]],.],[.,.]] => 3 [[[.,[[.,.],[.,.]]],.],[.,.]] => 3 [[[.,[[.,[.,.]],.]],.],[.,.]] => 3 [[[.,[[[.,.],.],.]],.],[.,.]] => 3 [[[[.,.],[.,[.,.]]],.],[.,.]] => 2 [[[[.,.],[[.,.],.]],.],[.,.]] => 2 [[[[.,[.,.]],[.,.]],.],[.,.]] => 2 [[[[[.,.],.],[.,.]],.],[.,.]] => 1 [[[[.,[.,[.,.]]],.],.],[.,.]] => 2 [[[[.,[[.,.],.]],.],.],[.,.]] => 2 [[[[[.,.],[.,.]],.],.],[.,.]] => 1 [[[[[.,[.,.]],.],.],.],[.,.]] => 1 [[[[[[.,.],.],.],.],.],[.,.]] => 0 [[.,[.,[.,[.,[.,[.,.]]]]]],.] => 5 [[.,[.,[.,[.,[[.,.],.]]]]],.] => 5 [[.,[.,[.,[[.,.],[.,.]]]]],.] => 5 [[.,[.,[.,[[.,[.,.]],.]]]],.] => 5 [[.,[.,[.,[[[.,.],.],.]]]],.] => 5 [[.,[.,[[.,.],[.,[.,.]]]]],.] => 5 [[.,[.,[[.,.],[[.,.],.]]]],.] => 5 [[.,[.,[[.,[.,.]],[.,.]]]],.] => 5 [[.,[.,[[[.,.],.],[.,.]]]],.] => 5 [[.,[.,[[.,[.,[.,.]]],.]]],.] => 5 [[.,[.,[[.,[[.,.],.]],.]]],.] => 5 [[.,[.,[[[.,.],[.,.]],.]]],.] => 5 [[.,[.,[[[.,[.,.]],.],.]]],.] => 5 [[.,[.,[[[[.,.],.],.],.]]],.] => 5 [[.,[[.,.],[.,[.,[.,.]]]]],.] => 5 [[.,[[.,.],[.,[[.,.],.]]]],.] => 5 [[.,[[.,.],[[.,.],[.,.]]]],.] => 5 [[.,[[.,.],[[.,[.,.]],.]]],.] => 5 [[.,[[.,.],[[[.,.],.],.]]],.] => 5 [[.,[[.,[.,.]],[.,[.,.]]]],.] => 5 [[.,[[.,[.,.]],[[.,.],.]]],.] => 5 [[.,[[[.,.],.],[.,[.,.]]]],.] => 5 [[.,[[[.,.],.],[[.,.],.]]],.] => 5 [[.,[[.,[.,[.,.]]],[.,.]]],.] => 5 [[.,[[.,[[.,.],.]],[.,.]]],.] => 5 [[.,[[[.,.],[.,.]],[.,.]]],.] => 5 [[.,[[[.,[.,.]],.],[.,.]]],.] => 5 [[.,[[[[.,.],.],.],[.,.]]],.] => 5 [[.,[[.,[.,[.,[.,.]]]],.]],.] => 5 [[.,[[.,[.,[[.,.],.]]],.]],.] => 5 [[.,[[.,[[.,.],[.,.]]],.]],.] => 5 [[.,[[.,[[.,[.,.]],.]],.]],.] => 5 [[.,[[.,[[[.,.],.],.]],.]],.] => 5 [[.,[[[.,.],[.,[.,.]]],.]],.] => 5 [[.,[[[.,.],[[.,.],.]],.]],.] => 5 [[.,[[[.,[.,.]],[.,.]],.]],.] => 5 [[.,[[[[.,.],.],[.,.]],.]],.] => 5 [[.,[[[.,[.,[.,.]]],.],.]],.] => 5 [[.,[[[.,[[.,.],.]],.],.]],.] => 5 [[.,[[[[.,.],[.,.]],.],.]],.] => 5 [[.,[[[[.,[.,.]],.],.],.]],.] => 5 [[.,[[[[[.,.],.],.],.],.]],.] => 5 [[[.,.],[.,[.,[.,[.,.]]]]],.] => 4 [[[.,.],[.,[.,[[.,.],.]]]],.] => 4 [[[.,.],[.,[[.,.],[.,.]]]],.] => 4 [[[.,.],[.,[[.,[.,.]],.]]],.] => 4 [[[.,.],[.,[[[.,.],.],.]]],.] => 4 [[[.,.],[[.,.],[.,[.,.]]]],.] => 4 [[[.,.],[[.,.],[[.,.],.]]],.] => 4 [[[.,.],[[.,[.,.]],[.,.]]],.] => 4 [[[.,.],[[[.,.],.],[.,.]]],.] => 4 [[[.,.],[[.,[.,[.,.]]],.]],.] => 4 [[[.,.],[[.,[[.,.],.]],.]],.] => 4 [[[.,.],[[[.,.],[.,.]],.]],.] => 4 [[[.,.],[[[.,[.,.]],.],.]],.] => 4 [[[.,.],[[[[.,.],.],.],.]],.] => 4 [[[.,[.,.]],[.,[.,[.,.]]]],.] => 4 [[[.,[.,.]],[.,[[.,.],.]]],.] => 4 [[[.,[.,.]],[[.,.],[.,.]]],.] => 4 [[[.,[.,.]],[[.,[.,.]],.]],.] => 4 [[[.,[.,.]],[[[.,.],.],.]],.] => 4 [[[[.,.],.],[.,[.,[.,.]]]],.] => 3 [[[[.,.],.],[.,[[.,.],.]]],.] => 3 [[[[.,.],.],[[.,.],[.,.]]],.] => 3 [[[[.,.],.],[[.,[.,.]],.]],.] => 3 [[[[.,.],.],[[[.,.],.],.]],.] => 3 [[[.,[.,[.,.]]],[.,[.,.]]],.] => 4 [[[.,[.,[.,.]]],[[.,.],.]],.] => 4 [[[.,[[.,.],.]],[.,[.,.]]],.] => 4 [[[.,[[.,.],.]],[[.,.],.]],.] => 4 [[[[.,.],[.,.]],[.,[.,.]]],.] => 3 [[[[.,.],[.,.]],[[.,.],.]],.] => 3 [[[[.,[.,.]],.],[.,[.,.]]],.] => 3 [[[[.,[.,.]],.],[[.,.],.]],.] => 3 [[[[[.,.],.],.],[.,[.,.]]],.] => 2 [[[[[.,.],.],.],[[.,.],.]],.] => 2 [[[.,[.,[.,[.,.]]]],[.,.]],.] => 4 [[[.,[.,[[.,.],.]]],[.,.]],.] => 4 [[[.,[[.,.],[.,.]]],[.,.]],.] => 4 [[[.,[[.,[.,.]],.]],[.,.]],.] => 4 [[[.,[[[.,.],.],.]],[.,.]],.] => 4 [[[[.,.],[.,[.,.]]],[.,.]],.] => 3 [[[[.,.],[[.,.],.]],[.,.]],.] => 3 [[[[.,[.,.]],[.,.]],[.,.]],.] => 3 [[[[[.,.],.],[.,.]],[.,.]],.] => 2 [[[[.,[.,[.,.]]],.],[.,.]],.] => 3 [[[[.,[[.,.],.]],.],[.,.]],.] => 3 [[[[[.,.],[.,.]],.],[.,.]],.] => 2 [[[[[.,[.,.]],.],.],[.,.]],.] => 2 [[[[[[.,.],.],.],.],[.,.]],.] => 1 [[[.,[.,[.,[.,[.,.]]]]],.],.] => 4 [[[.,[.,[.,[[.,.],.]]]],.],.] => 4 [[[.,[.,[[.,.],[.,.]]]],.],.] => 4 [[[.,[.,[[.,[.,.]],.]]],.],.] => 4 [[[.,[.,[[[.,.],.],.]]],.],.] => 4 [[[.,[[.,.],[.,[.,.]]]],.],.] => 4 [[[.,[[.,.],[[.,.],.]]],.],.] => 4 [[[.,[[.,[.,.]],[.,.]]],.],.] => 4 [[[.,[[[.,.],.],[.,.]]],.],.] => 4 [[[.,[[.,[.,[.,.]]],.]],.],.] => 4 [[[.,[[.,[[.,.],.]],.]],.],.] => 4 [[[.,[[[.,.],[.,.]],.]],.],.] => 4 [[[.,[[[.,[.,.]],.],.]],.],.] => 4 [[[.,[[[[.,.],.],.],.]],.],.] => 4 [[[[.,.],[.,[.,[.,.]]]],.],.] => 3 [[[[.,.],[.,[[.,.],.]]],.],.] => 3 [[[[.,.],[[.,.],[.,.]]],.],.] => 3 [[[[.,.],[[.,[.,.]],.]],.],.] => 3 [[[[.,.],[[[.,.],.],.]],.],.] => 3 [[[[.,[.,.]],[.,[.,.]]],.],.] => 3 [[[[.,[.,.]],[[.,.],.]],.],.] => 3 [[[[[.,.],.],[.,[.,.]]],.],.] => 2 [[[[[.,.],.],[[.,.],.]],.],.] => 2 [[[[.,[.,[.,.]]],[.,.]],.],.] => 3 [[[[.,[[.,.],.]],[.,.]],.],.] => 3 [[[[[.,.],[.,.]],[.,.]],.],.] => 2 [[[[[.,[.,.]],.],[.,.]],.],.] => 2 [[[[[[.,.],.],.],[.,.]],.],.] => 1 [[[[.,[.,[.,[.,.]]]],.],.],.] => 3 [[[[.,[.,[[.,.],.]]],.],.],.] => 3 [[[[.,[[.,.],[.,.]]],.],.],.] => 3 [[[[.,[[.,[.,.]],.]],.],.],.] => 3 [[[[.,[[[.,.],.],.]],.],.],.] => 3 [[[[[.,.],[.,[.,.]]],.],.],.] => 2 [[[[[.,.],[[.,.],.]],.],.],.] => 2 [[[[[.,[.,.]],[.,.]],.],.],.] => 2 [[[[[[.,.],.],[.,.]],.],.],.] => 1 [[[[[.,[.,[.,.]]],.],.],.],.] => 2 [[[[[.,[[.,.],.]],.],.],.],.] => 2 [[[[[[.,.],[.,.]],.],.],.],.] => 1 [[[[[[.,[.,.]],.],.],.],.],.] => 1 [[[[[[[.,.],.],.],.],.],.],.] => 0 ----------------------------------------------------------------------------- Created: May 16, 2014 at 20:57 by Joseph Ching ----------------------------------------------------------------------------- Last Updated: Dec 30, 2016 at 10:54 by Christian Stump