***************************************************************************** * 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: St000203 ----------------------------------------------------------------------------- Collection: Binary trees ----------------------------------------------------------------------------- Description: The number of external nodes of a binary tree. That is, the number of nodes that can be reached from the root by only left steps or only right steps, plus $1$ for the root node itself. A counting formula for the number of external node in all binary trees of size $n$ can be found in [1]. ----------------------------------------------------------------------------- References: [1] Expansion of (1+x*C)*C^3, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108. [[OEIS:A070031]] ----------------------------------------------------------------------------- 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 statistic(b): """ sage: [sum(statistic(b) for b in BinaryTrees(n)) for n in range(1,8)] [1, 4, 13, 42, 138, 462, 1573] """ return 1 + get_external_node(b, True) + get_external_node(b, False) ----------------------------------------------------------------------------- Statistic values: [.,.] => 1 [.,[.,.]] => 2 [[.,.],.] => 2 [.,[.,[.,.]]] => 3 [.,[[.,.],.]] => 2 [[.,.],[.,.]] => 3 [[.,[.,.]],.] => 2 [[[.,.],.],.] => 3 [.,[.,[.,[.,.]]]] => 4 [.,[.,[[.,.],.]]] => 3 [.,[[.,.],[.,.]]] => 3 [.,[[.,[.,.]],.]] => 2 [.,[[[.,.],.],.]] => 2 [[.,.],[.,[.,.]]] => 4 [[.,.],[[.,.],.]] => 3 [[.,[.,.]],[.,.]] => 3 [[[.,.],.],[.,.]] => 4 [[.,[.,[.,.]]],.] => 2 [[.,[[.,.],.]],.] => 2 [[[.,.],[.,.]],.] => 3 [[[.,[.,.]],.],.] => 3 [[[[.,.],.],.],.] => 4 [.,[.,[.,[.,[.,.]]]]] => 5 [.,[.,[.,[[.,.],.]]]] => 4 [.,[.,[[.,.],[.,.]]]] => 4 [.,[.,[[.,[.,.]],.]]] => 3 [.,[.,[[[.,.],.],.]]] => 3 [.,[[.,.],[.,[.,.]]]] => 4 [.,[[.,.],[[.,.],.]]] => 3 [.,[[.,[.,.]],[.,.]]] => 3 [.,[[[.,.],.],[.,.]]] => 3 [.,[[.,[.,[.,.]]],.]] => 2 [.,[[.,[[.,.],.]],.]] => 2 [.,[[[.,.],[.,.]],.]] => 2 [.,[[[.,[.,.]],.],.]] => 2 [.,[[[[.,.],.],.],.]] => 2 [[.,.],[.,[.,[.,.]]]] => 5 [[.,.],[.,[[.,.],.]]] => 4 [[.,.],[[.,.],[.,.]]] => 4 [[.,.],[[.,[.,.]],.]] => 3 [[.,.],[[[.,.],.],.]] => 3 [[.,[.,.]],[.,[.,.]]] => 4 [[.,[.,.]],[[.,.],.]] => 3 [[[.,.],.],[.,[.,.]]] => 5 [[[.,.],.],[[.,.],.]] => 4 [[.,[.,[.,.]]],[.,.]] => 3 [[.,[[.,.],.]],[.,.]] => 3 [[[.,.],[.,.]],[.,.]] => 4 [[[.,[.,.]],.],[.,.]] => 4 [[[[.,.],.],.],[.,.]] => 5 [[.,[.,[.,[.,.]]]],.] => 2 [[.,[.,[[.,.],.]]],.] => 2 [[.,[[.,.],[.,.]]],.] => 2 [[.,[[.,[.,.]],.]],.] => 2 [[.,[[[.,.],.],.]],.] => 2 [[[.,.],[.,[.,.]]],.] => 3 [[[.,.],[[.,.],.]],.] => 3 [[[.,[.,.]],[.,.]],.] => 3 [[[[.,.],.],[.,.]],.] => 4 [[[.,[.,[.,.]]],.],.] => 3 [[[.,[[.,.],.]],.],.] => 3 [[[[.,.],[.,.]],.],.] => 4 [[[[.,[.,.]],.],.],.] => 4 [[[[[.,.],.],.],.],.] => 5 [.,[.,[.,[.,[.,[.,.]]]]]] => 6 [.,[.,[.,[.,[[.,.],.]]]]] => 5 [.,[.,[.,[[.,.],[.,.]]]]] => 5 [.,[.,[.,[[.,[.,.]],.]]]] => 4 [.,[.,[.,[[[.,.],.],.]]]] => 4 [.,[.,[[.,.],[.,[.,.]]]]] => 5 [.,[.,[[.,.],[[.,.],.]]]] => 4 [.,[.,[[.,[.,.]],[.,.]]]] => 4 [.,[.,[[[.,.],.],[.,.]]]] => 4 [.,[.,[[.,[.,[.,.]]],.]]] => 3 [.,[.,[[.,[[.,.],.]],.]]] => 3 [.,[.,[[[.,.],[.,.]],.]]] => 3 [.,[.,[[[.,[.,.]],.],.]]] => 3 [.,[.,[[[[.,.],.],.],.]]] => 3 [.,[[.,.],[.,[.,[.,.]]]]] => 5 [.,[[.,.],[.,[[.,.],.]]]] => 4 [.,[[.,.],[[.,.],[.,.]]]] => 4 [.,[[.,.],[[.,[.,.]],.]]] => 3 [.,[[.,.],[[[.,.],.],.]]] => 3 [.,[[.,[.,.]],[.,[.,.]]]] => 4 [.,[[.,[.,.]],[[.,.],.]]] => 3 [.,[[[.,.],.],[.,[.,.]]]] => 4 [.,[[[.,.],.],[[.,.],.]]] => 3 [.,[[.,[.,[.,.]]],[.,.]]] => 3 [.,[[.,[[.,.],.]],[.,.]]] => 3 [.,[[[.,.],[.,.]],[.,.]]] => 3 [.,[[[.,[.,.]],.],[.,.]]] => 3 [.,[[[[.,.],.],.],[.,.]]] => 3 [.,[[.,[.,[.,[.,.]]]],.]] => 2 [.,[[.,[.,[[.,.],.]]],.]] => 2 [.,[[.,[[.,.],[.,.]]],.]] => 2 [.,[[.,[[.,[.,.]],.]],.]] => 2 [.,[[.,[[[.,.],.],.]],.]] => 2 [.,[[[.,.],[.,[.,.]]],.]] => 2 [.,[[[.,.],[[.,.],.]],.]] => 2 [.,[[[.,[.,.]],[.,.]],.]] => 2 [.,[[[[.,.],.],[.,.]],.]] => 2 [.,[[[.,[.,[.,.]]],.],.]] => 2 [.,[[[.,[[.,.],.]],.],.]] => 2 [.,[[[[.,.],[.,.]],.],.]] => 2 [.,[[[[.,[.,.]],.],.],.]] => 2 [.,[[[[[.,.],.],.],.],.]] => 2 [[.,.],[.,[.,[.,[.,.]]]]] => 6 [[.,.],[.,[.,[[.,.],.]]]] => 5 [[.,.],[.,[[.,.],[.,.]]]] => 5 [[.,.],[.,[[.,[.,.]],.]]] => 4 [[.,.],[.,[[[.,.],.],.]]] => 4 [[.,.],[[.,.],[.,[.,.]]]] => 5 [[.,.],[[.,.],[[.,.],.]]] => 4 [[.,.],[[.,[.,.]],[.,.]]] => 4 [[.,.],[[[.,.],.],[.,.]]] => 4 [[.,.],[[.,[.,[.,.]]],.]] => 3 [[.,.],[[.,[[.,.],.]],.]] => 3 [[.,.],[[[.,.],[.,.]],.]] => 3 [[.,.],[[[.,[.,.]],.],.]] => 3 [[.,.],[[[[.,.],.],.],.]] => 3 [[.,[.,.]],[.,[.,[.,.]]]] => 5 [[.,[.,.]],[.,[[.,.],.]]] => 4 [[.,[.,.]],[[.,.],[.,.]]] => 4 [[.,[.,.]],[[.,[.,.]],.]] => 3 [[.,[.,.]],[[[.,.],.],.]] => 3 [[[.,.],.],[.,[.,[.,.]]]] => 6 [[[.,.],.],[.,[[.,.],.]]] => 5 [[[.,.],.],[[.,.],[.,.]]] => 5 [[[.,.],.],[[.,[.,.]],.]] => 4 [[[.,.],.],[[[.,.],.],.]] => 4 [[.,[.,[.,.]]],[.,[.,.]]] => 4 [[.,[.,[.,.]]],[[.,.],.]] => 3 [[.,[[.,.],.]],[.,[.,.]]] => 4 [[.,[[.,.],.]],[[.,.],.]] => 3 [[[.,.],[.,.]],[.,[.,.]]] => 5 [[[.,.],[.,.]],[[.,.],.]] => 4 [[[.,[.,.]],.],[.,[.,.]]] => 5 [[[.,[.,.]],.],[[.,.],.]] => 4 [[[[.,.],.],.],[.,[.,.]]] => 6 [[[[.,.],.],.],[[.,.],.]] => 5 [[.,[.,[.,[.,.]]]],[.,.]] => 3 [[.,[.,[[.,.],.]]],[.,.]] => 3 [[.,[[.,.],[.,.]]],[.,.]] => 3 [[.,[[.,[.,.]],.]],[.,.]] => 3 [[.,[[[.,.],.],.]],[.,.]] => 3 [[[.,.],[.,[.,.]]],[.,.]] => 4 [[[.,.],[[.,.],.]],[.,.]] => 4 [[[.,[.,.]],[.,.]],[.,.]] => 4 [[[[.,.],.],[.,.]],[.,.]] => 5 [[[.,[.,[.,.]]],.],[.,.]] => 4 [[[.,[[.,.],.]],.],[.,.]] => 4 [[[[.,.],[.,.]],.],[.,.]] => 5 [[[[.,[.,.]],.],.],[.,.]] => 5 [[[[[.,.],.],.],.],[.,.]] => 6 [[.,[.,[.,[.,[.,.]]]]],.] => 2 [[.,[.,[.,[[.,.],.]]]],.] => 2 [[.,[.,[[.,.],[.,.]]]],.] => 2 [[.,[.,[[.,[.,.]],.]]],.] => 2 [[.,[.,[[[.,.],.],.]]],.] => 2 [[.,[[.,.],[.,[.,.]]]],.] => 2 [[.,[[.,.],[[.,.],.]]],.] => 2 [[.,[[.,[.,.]],[.,.]]],.] => 2 [[.,[[[.,.],.],[.,.]]],.] => 2 [[.,[[.,[.,[.,.]]],.]],.] => 2 [[.,[[.,[[.,.],.]],.]],.] => 2 [[.,[[[.,.],[.,.]],.]],.] => 2 [[.,[[[.,[.,.]],.],.]],.] => 2 [[.,[[[[.,.],.],.],.]],.] => 2 [[[.,.],[.,[.,[.,.]]]],.] => 3 [[[.,.],[.,[[.,.],.]]],.] => 3 [[[.,.],[[.,.],[.,.]]],.] => 3 [[[.,.],[[.,[.,.]],.]],.] => 3 [[[.,.],[[[.,.],.],.]],.] => 3 [[[.,[.,.]],[.,[.,.]]],.] => 3 [[[.,[.,.]],[[.,.],.]],.] => 3 [[[[.,.],.],[.,[.,.]]],.] => 4 [[[[.,.],.],[[.,.],.]],.] => 4 [[[.,[.,[.,.]]],[.,.]],.] => 3 [[[.,[[.,.],.]],[.,.]],.] => 3 [[[[.,.],[.,.]],[.,.]],.] => 4 [[[[.,[.,.]],.],[.,.]],.] => 4 [[[[[.,.],.],.],[.,.]],.] => 5 [[[.,[.,[.,[.,.]]]],.],.] => 3 [[[.,[.,[[.,.],.]]],.],.] => 3 [[[.,[[.,.],[.,.]]],.],.] => 3 [[[.,[[.,[.,.]],.]],.],.] => 3 [[[.,[[[.,.],.],.]],.],.] => 3 [[[[.,.],[.,[.,.]]],.],.] => 4 [[[[.,.],[[.,.],.]],.],.] => 4 [[[[.,[.,.]],[.,.]],.],.] => 4 [[[[[.,.],.],[.,.]],.],.] => 5 [[[[.,[.,[.,.]]],.],.],.] => 4 [[[[.,[[.,.],.]],.],.],.] => 4 [[[[[.,.],[.,.]],.],.],.] => 5 [[[[[.,[.,.]],.],.],.],.] => 5 [[[[[[.,.],.],.],.],.],.] => 6 [.,[.,[.,[.,[.,[.,[.,.]]]]]]] => 7 [.,[.,[.,[.,[.,[[.,.],.]]]]]] => 6 [.,[.,[.,[.,[[.,.],[.,.]]]]]] => 6 [.,[.,[.,[.,[[.,[.,.]],.]]]]] => 5 [.,[.,[.,[.,[[[.,.],.],.]]]]] => 5 [.,[.,[.,[[.,.],[.,[.,.]]]]]] => 6 [.,[.,[.,[[.,.],[[.,.],.]]]]] => 5 [.,[.,[.,[[.,[.,.]],[.,.]]]]] => 5 [.,[.,[.,[[[.,.],.],[.,.]]]]] => 5 [.,[.,[.,[[.,[.,[.,.]]],.]]]] => 4 [.,[.,[.,[[.,[[.,.],.]],.]]]] => 4 [.,[.,[.,[[[.,.],[.,.]],.]]]] => 4 [.,[.,[.,[[[.,[.,.]],.],.]]]] => 4 [.,[.,[.,[[[[.,.],.],.],.]]]] => 4 [.,[.,[[.,.],[.,[.,[.,.]]]]]] => 6 [.,[.,[[.,.],[.,[[.,.],.]]]]] => 5 [.,[.,[[.,.],[[.,.],[.,.]]]]] => 5 [.,[.,[[.,.],[[.,[.,.]],.]]]] => 4 [.,[.,[[.,.],[[[.,.],.],.]]]] => 4 [.,[.,[[.,[.,.]],[.,[.,.]]]]] => 5 [.,[.,[[.,[.,.]],[[.,.],.]]]] => 4 [.,[.,[[[.,.],.],[.,[.,.]]]]] => 5 [.,[.,[[[.,.],.],[[.,.],.]]]] => 4 [.,[.,[[.,[.,[.,.]]],[.,.]]]] => 4 [.,[.,[[.,[[.,.],.]],[.,.]]]] => 4 [.,[.,[[[.,.],[.,.]],[.,.]]]] => 4 [.,[.,[[[.,[.,.]],.],[.,.]]]] => 4 [.,[.,[[[[.,.],.],.],[.,.]]]] => 4 [.,[.,[[.,[.,[.,[.,.]]]],.]]] => 3 [.,[.,[[.,[.,[[.,.],.]]],.]]] => 3 [.,[.,[[.,[[.,.],[.,.]]],.]]] => 3 [.,[.,[[.,[[.,[.,.]],.]],.]]] => 3 [.,[.,[[.,[[[.,.],.],.]],.]]] => 3 [.,[.,[[[.,.],[.,[.,.]]],.]]] => 3 [.,[.,[[[.,.],[[.,.],.]],.]]] => 3 [.,[.,[[[.,[.,.]],[.,.]],.]]] => 3 [.,[.,[[[[.,.],.],[.,.]],.]]] => 3 [.,[.,[[[.,[.,[.,.]]],.],.]]] => 3 [.,[.,[[[.,[[.,.],.]],.],.]]] => 3 [.,[.,[[[[.,.],[.,.]],.],.]]] => 3 [.,[.,[[[[.,[.,.]],.],.],.]]] => 3 [.,[.,[[[[[.,.],.],.],.],.]]] => 3 [.,[[.,.],[.,[.,[.,[.,.]]]]]] => 6 [.,[[.,.],[.,[.,[[.,.],.]]]]] => 5 [.,[[.,.],[.,[[.,.],[.,.]]]]] => 5 [.,[[.,.],[.,[[.,[.,.]],.]]]] => 4 [.,[[.,.],[.,[[[.,.],.],.]]]] => 4 [.,[[.,.],[[.,.],[.,[.,.]]]]] => 5 [.,[[.,.],[[.,.],[[.,.],.]]]] => 4 [.,[[.,.],[[.,[.,.]],[.,.]]]] => 4 [.,[[.,.],[[[.,.],.],[.,.]]]] => 4 [.,[[.,.],[[.,[.,[.,.]]],.]]] => 3 [.,[[.,.],[[.,[[.,.],.]],.]]] => 3 [.,[[.,.],[[[.,.],[.,.]],.]]] => 3 [.,[[.,.],[[[.,[.,.]],.],.]]] => 3 [.,[[.,.],[[[[.,.],.],.],.]]] => 3 [.,[[.,[.,.]],[.,[.,[.,.]]]]] => 5 [.,[[.,[.,.]],[.,[[.,.],.]]]] => 4 [.,[[.,[.,.]],[[.,.],[.,.]]]] => 4 [.,[[.,[.,.]],[[.,[.,.]],.]]] => 3 [.,[[.,[.,.]],[[[.,.],.],.]]] => 3 [.,[[[.,.],.],[.,[.,[.,.]]]]] => 5 [.,[[[.,.],.],[.,[[.,.],.]]]] => 4 [.,[[[.,.],.],[[.,.],[.,.]]]] => 4 [.,[[[.,.],.],[[.,[.,.]],.]]] => 3 [.,[[[.,.],.],[[[.,.],.],.]]] => 3 [.,[[.,[.,[.,.]]],[.,[.,.]]]] => 4 [.,[[.,[.,[.,.]]],[[.,.],.]]] => 3 [.,[[.,[[.,.],.]],[.,[.,.]]]] => 4 [.,[[.,[[.,.],.]],[[.,.],.]]] => 3 [.,[[[.,.],[.,.]],[.,[.,.]]]] => 4 [.,[[[.,.],[.,.]],[[.,.],.]]] => 3 [.,[[[.,[.,.]],.],[.,[.,.]]]] => 4 [.,[[[.,[.,.]],.],[[.,.],.]]] => 3 [.,[[[[.,.],.],.],[.,[.,.]]]] => 4 [.,[[[[.,.],.],.],[[.,.],.]]] => 3 [.,[[.,[.,[.,[.,.]]]],[.,.]]] => 3 [.,[[.,[.,[[.,.],.]]],[.,.]]] => 3 [.,[[.,[[.,.],[.,.]]],[.,.]]] => 3 [.,[[.,[[.,[.,.]],.]],[.,.]]] => 3 [.,[[.,[[[.,.],.],.]],[.,.]]] => 3 [.,[[[.,.],[.,[.,.]]],[.,.]]] => 3 [.,[[[.,.],[[.,.],.]],[.,.]]] => 3 [.,[[[.,[.,.]],[.,.]],[.,.]]] => 3 [.,[[[[.,.],.],[.,.]],[.,.]]] => 3 [.,[[[.,[.,[.,.]]],.],[.,.]]] => 3 [.,[[[.,[[.,.],.]],.],[.,.]]] => 3 [.,[[[[.,.],[.,.]],.],[.,.]]] => 3 [.,[[[[.,[.,.]],.],.],[.,.]]] => 3 [.,[[[[[.,.],.],.],.],[.,.]]] => 3 [.,[[.,[.,[.,[.,[.,.]]]]],.]] => 2 [.,[[.,[.,[.,[[.,.],.]]]],.]] => 2 [.,[[.,[.,[[.,.],[.,.]]]],.]] => 2 [.,[[.,[.,[[.,[.,.]],.]]],.]] => 2 [.,[[.,[.,[[[.,.],.],.]]],.]] => 2 [.,[[.,[[.,.],[.,[.,.]]]],.]] => 2 [.,[[.,[[.,.],[[.,.],.]]],.]] => 2 [.,[[.,[[.,[.,.]],[.,.]]],.]] => 2 [.,[[.,[[[.,.],.],[.,.]]],.]] => 2 [.,[[.,[[.,[.,[.,.]]],.]],.]] => 2 [.,[[.,[[.,[[.,.],.]],.]],.]] => 2 [.,[[.,[[[.,.],[.,.]],.]],.]] => 2 [.,[[.,[[[.,[.,.]],.],.]],.]] => 2 [.,[[.,[[[[.,.],.],.],.]],.]] => 2 [.,[[[.,.],[.,[.,[.,.]]]],.]] => 2 [.,[[[.,.],[.,[[.,.],.]]],.]] => 2 [.,[[[.,.],[[.,.],[.,.]]],.]] => 2 [.,[[[.,.],[[.,[.,.]],.]],.]] => 2 [.,[[[.,.],[[[.,.],.],.]],.]] => 2 [.,[[[.,[.,.]],[.,[.,.]]],.]] => 2 [.,[[[.,[.,.]],[[.,.],.]],.]] => 2 [.,[[[[.,.],.],[.,[.,.]]],.]] => 2 [.,[[[[.,.],.],[[.,.],.]],.]] => 2 [.,[[[.,[.,[.,.]]],[.,.]],.]] => 2 [.,[[[.,[[.,.],.]],[.,.]],.]] => 2 [.,[[[[.,.],[.,.]],[.,.]],.]] => 2 [.,[[[[.,[.,.]],.],[.,.]],.]] => 2 [.,[[[[[.,.],.],.],[.,.]],.]] => 2 [.,[[[.,[.,[.,[.,.]]]],.],.]] => 2 [.,[[[.,[.,[[.,.],.]]],.],.]] => 2 [.,[[[.,[[.,.],[.,.]]],.],.]] => 2 [.,[[[.,[[.,[.,.]],.]],.],.]] => 2 [.,[[[.,[[[.,.],.],.]],.],.]] => 2 [.,[[[[.,.],[.,[.,.]]],.],.]] => 2 [.,[[[[.,.],[[.,.],.]],.],.]] => 2 [.,[[[[.,[.,.]],[.,.]],.],.]] => 2 [.,[[[[[.,.],.],[.,.]],.],.]] => 2 [.,[[[[.,[.,[.,.]]],.],.],.]] => 2 [.,[[[[.,[[.,.],.]],.],.],.]] => 2 [.,[[[[[.,.],[.,.]],.],.],.]] => 2 [.,[[[[[.,[.,.]],.],.],.],.]] => 2 [.,[[[[[[.,.],.],.],.],.],.]] => 2 [[.,.],[.,[.,[.,[.,[.,.]]]]]] => 7 [[.,.],[.,[.,[.,[[.,.],.]]]]] => 6 [[.,.],[.,[.,[[.,.],[.,.]]]]] => 6 [[.,.],[.,[.,[[.,[.,.]],.]]]] => 5 [[.,.],[.,[.,[[[.,.],.],.]]]] => 5 [[.,.],[.,[[.,.],[.,[.,.]]]]] => 6 [[.,.],[.,[[.,.],[[.,.],.]]]] => 5 [[.,.],[.,[[.,[.,.]],[.,.]]]] => 5 [[.,.],[.,[[[.,.],.],[.,.]]]] => 5 [[.,.],[.,[[.,[.,[.,.]]],.]]] => 4 [[.,.],[.,[[.,[[.,.],.]],.]]] => 4 [[.,.],[.,[[[.,.],[.,.]],.]]] => 4 [[.,.],[.,[[[.,[.,.]],.],.]]] => 4 [[.,.],[.,[[[[.,.],.],.],.]]] => 4 [[.,.],[[.,.],[.,[.,[.,.]]]]] => 6 [[.,.],[[.,.],[.,[[.,.],.]]]] => 5 [[.,.],[[.,.],[[.,.],[.,.]]]] => 5 [[.,.],[[.,.],[[.,[.,.]],.]]] => 4 [[.,.],[[.,.],[[[.,.],.],.]]] => 4 [[.,.],[[.,[.,.]],[.,[.,.]]]] => 5 [[.,.],[[.,[.,.]],[[.,.],.]]] => 4 [[.,.],[[[.,.],.],[.,[.,.]]]] => 5 [[.,.],[[[.,.],.],[[.,.],.]]] => 4 [[.,.],[[.,[.,[.,.]]],[.,.]]] => 4 [[.,.],[[.,[[.,.],.]],[.,.]]] => 4 [[.,.],[[[.,.],[.,.]],[.,.]]] => 4 [[.,.],[[[.,[.,.]],.],[.,.]]] => 4 [[.,.],[[[[.,.],.],.],[.,.]]] => 4 [[.,.],[[.,[.,[.,[.,.]]]],.]] => 3 [[.,.],[[.,[.,[[.,.],.]]],.]] => 3 [[.,.],[[.,[[.,.],[.,.]]],.]] => 3 [[.,.],[[.,[[.,[.,.]],.]],.]] => 3 [[.,.],[[.,[[[.,.],.],.]],.]] => 3 [[.,.],[[[.,.],[.,[.,.]]],.]] => 3 [[.,.],[[[.,.],[[.,.],.]],.]] => 3 [[.,.],[[[.,[.,.]],[.,.]],.]] => 3 [[.,.],[[[[.,.],.],[.,.]],.]] => 3 [[.,.],[[[.,[.,[.,.]]],.],.]] => 3 [[.,.],[[[.,[[.,.],.]],.],.]] => 3 [[.,.],[[[[.,.],[.,.]],.],.]] => 3 [[.,.],[[[[.,[.,.]],.],.],.]] => 3 [[.,.],[[[[[.,.],.],.],.],.]] => 3 [[.,[.,.]],[.,[.,[.,[.,.]]]]] => 6 [[.,[.,.]],[.,[.,[[.,.],.]]]] => 5 [[.,[.,.]],[.,[[.,.],[.,.]]]] => 5 [[.,[.,.]],[.,[[.,[.,.]],.]]] => 4 [[.,[.,.]],[.,[[[.,.],.],.]]] => 4 [[.,[.,.]],[[.,.],[.,[.,.]]]] => 5 [[.,[.,.]],[[.,.],[[.,.],.]]] => 4 [[.,[.,.]],[[.,[.,.]],[.,.]]] => 4 [[.,[.,.]],[[[.,.],.],[.,.]]] => 4 [[.,[.,.]],[[.,[.,[.,.]]],.]] => 3 [[.,[.,.]],[[.,[[.,.],.]],.]] => 3 [[.,[.,.]],[[[.,.],[.,.]],.]] => 3 [[.,[.,.]],[[[.,[.,.]],.],.]] => 3 [[.,[.,.]],[[[[.,.],.],.],.]] => 3 [[[.,.],.],[.,[.,[.,[.,.]]]]] => 7 [[[.,.],.],[.,[.,[[.,.],.]]]] => 6 [[[.,.],.],[.,[[.,.],[.,.]]]] => 6 [[[.,.],.],[.,[[.,[.,.]],.]]] => 5 [[[.,.],.],[.,[[[.,.],.],.]]] => 5 [[[.,.],.],[[.,.],[.,[.,.]]]] => 6 [[[.,.],.],[[.,.],[[.,.],.]]] => 5 [[[.,.],.],[[.,[.,.]],[.,.]]] => 5 [[[.,.],.],[[[.,.],.],[.,.]]] => 5 [[[.,.],.],[[.,[.,[.,.]]],.]] => 4 [[[.,.],.],[[.,[[.,.],.]],.]] => 4 [[[.,.],.],[[[.,.],[.,.]],.]] => 4 [[[.,.],.],[[[.,[.,.]],.],.]] => 4 [[[.,.],.],[[[[.,.],.],.],.]] => 4 [[.,[.,[.,.]]],[.,[.,[.,.]]]] => 5 [[.,[.,[.,.]]],[.,[[.,.],.]]] => 4 [[.,[.,[.,.]]],[[.,.],[.,.]]] => 4 [[.,[.,[.,.]]],[[.,[.,.]],.]] => 3 [[.,[.,[.,.]]],[[[.,.],.],.]] => 3 [[.,[[.,.],.]],[.,[.,[.,.]]]] => 5 [[.,[[.,.],.]],[.,[[.,.],.]]] => 4 [[.,[[.,.],.]],[[.,.],[.,.]]] => 4 [[.,[[.,.],.]],[[.,[.,.]],.]] => 3 [[.,[[.,.],.]],[[[.,.],.],.]] => 3 [[[.,.],[.,.]],[.,[.,[.,.]]]] => 6 [[[.,.],[.,.]],[.,[[.,.],.]]] => 5 [[[.,.],[.,.]],[[.,.],[.,.]]] => 5 [[[.,.],[.,.]],[[.,[.,.]],.]] => 4 [[[.,.],[.,.]],[[[.,.],.],.]] => 4 [[[.,[.,.]],.],[.,[.,[.,.]]]] => 6 [[[.,[.,.]],.],[.,[[.,.],.]]] => 5 [[[.,[.,.]],.],[[.,.],[.,.]]] => 5 [[[.,[.,.]],.],[[.,[.,.]],.]] => 4 [[[.,[.,.]],.],[[[.,.],.],.]] => 4 [[[[.,.],.],.],[.,[.,[.,.]]]] => 7 [[[[.,.],.],.],[.,[[.,.],.]]] => 6 [[[[.,.],.],.],[[.,.],[.,.]]] => 6 [[[[.,.],.],.],[[.,[.,.]],.]] => 5 [[[[.,.],.],.],[[[.,.],.],.]] => 5 [[.,[.,[.,[.,.]]]],[.,[.,.]]] => 4 [[.,[.,[.,[.,.]]]],[[.,.],.]] => 3 [[.,[.,[[.,.],.]]],[.,[.,.]]] => 4 [[.,[.,[[.,.],.]]],[[.,.],.]] => 3 [[.,[[.,.],[.,.]]],[.,[.,.]]] => 4 [[.,[[.,.],[.,.]]],[[.,.],.]] => 3 [[.,[[.,[.,.]],.]],[.,[.,.]]] => 4 [[.,[[.,[.,.]],.]],[[.,.],.]] => 3 [[.,[[[.,.],.],.]],[.,[.,.]]] => 4 [[.,[[[.,.],.],.]],[[.,.],.]] => 3 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[[.,[[.,[[.,[.,.]],.]],.]],[[[.,[.,.]],[.,.]],.]] => 3 [[.,[[[.,[.,.]],[.,.]],.]],[[.,[[.,[.,.]],.]],.]] => 3 [[.,[[[.,[.,.]],[.,.]],.]],[[[.,[.,.]],[.,.]],.]] => 3 [[[.,[.,.]],[[.,[.,.]],.]],[[.,[[.,[.,.]],.]],.]] => 4 [[[.,[.,.]],[[.,[.,.]],.]],[[[.,[.,.]],[.,.]],.]] => 4 [[[.,[[.,[.,.]],.]],[.,.]],[[.,[[.,[.,.]],.]],.]] => 4 [[[.,[[.,[.,.]],.]],[.,.]],[[[.,[.,.]],[.,.]],.]] => 4 [[[[.,[.,.]],[.,.]],[.,.]],[[.,[[.,[.,.]],.]],.]] => 5 [[[[.,[.,.]],[.,.]],[.,.]],[[[.,[.,.]],[.,.]],.]] => 5 [[.,[[.,[[.,[[.,[.,.]],.]],.]],.]],[[.,[.,.]],.]] => 3 [[.,[[.,[[[.,[.,.]],[.,.]],.]],.]],[[.,[.,.]],.]] => 3 [[.,[[[.,[.,.]],[[.,[.,.]],.]],.]],[[.,[.,.]],.]] => 3 [[.,[[[.,[[.,[.,.]],.]],[.,.]],.]],[[.,[.,.]],.]] => 3 [[.,[[[[.,[.,.]],[.,.]],[.,.]],.]],[[.,[.,.]],.]] => 3 [[[.,[.,.]],[[.,[[.,[.,.]],.]],.]],[[.,[.,.]],.]] => 4 [[[.,[.,.]],[[[.,[.,.]],[.,.]],.]],[[.,[.,.]],.]] => 4 [[[.,[[.,[.,.]],.]],[[.,[.,.]],.]],[[.,[.,.]],.]] => 4 [[[[.,[.,.]],[.,.]],[[.,[.,.]],.]],[[.,[.,.]],.]] => 5 [[[.,[[.,[[.,[.,.]],.]],.]],[.,.]],[[.,[.,.]],.]] => 4 [[[.,[[[.,[.,.]],[.,.]],.]],[.,.]],[[.,[.,.]],.]] => 4 [[[[.,[.,.]],[[.,[.,.]],.]],[.,.]],[[.,[.,.]],.]] => 5 [[[[.,[[.,[.,.]],.]],[.,.]],[.,.]],[[.,[.,.]],.]] => 5 [[[[[.,[.,.]],[.,.]],[.,.]],[.,.]],[[.,[.,.]],.]] => 6 [[.,[[.,[[.,[[.,[[.,[.,.]],.]],.]],.]],.]],[.,.]] => 3 [[.,[[.,[[.,[[[.,[.,.]],[.,.]],.]],.]],.]],[.,.]] => 3 [[.,[[.,[[[.,[.,.]],[[.,[.,.]],.]],.]],.]],[.,.]] => 3 [[.,[[.,[[[.,[[.,[.,.]],.]],[.,.]],.]],.]],[.,.]] => 3 [[.,[[.,[[[[.,[.,.]],[.,.]],[.,.]],.]],.]],[.,.]] => 3 [[.,[[[.,[.,.]],[[.,[[.,[.,.]],.]],.]],.]],[.,.]] => 3 [[.,[[[.,[.,.]],[[[.,[.,.]],[.,.]],.]],.]],[.,.]] => 3 [[.,[[[.,[[.,[.,.]],.]],[[.,[.,.]],.]],.]],[.,.]] => 3 [[.,[[[[.,[.,.]],[.,.]],[[.,[.,.]],.]],.]],[.,.]] => 3 [[.,[[[.,[[.,[[.,[.,.]],.]],.]],[.,.]],.]],[.,.]] => 3 [[.,[[[.,[[[.,[.,.]],[.,.]],.]],[.,.]],.]],[.,.]] => 3 [[.,[[[[.,[.,.]],[[.,[.,.]],.]],[.,.]],.]],[.,.]] => 3 [[.,[[[[.,[[.,[.,.]],.]],[.,.]],[.,.]],.]],[.,.]] => 3 [[.,[[[[[.,[.,.]],[.,.]],[.,.]],[.,.]],.]],[.,.]] => 3 [[[.,[.,.]],[[.,[[.,[[.,[.,.]],.]],.]],.]],[.,.]] => 4 [[[.,[.,.]],[[.,[[[.,[.,.]],[.,.]],.]],.]],[.,.]] => 4 [[[.,[.,.]],[[[.,[.,.]],[[.,[.,.]],.]],.]],[.,.]] => 4 [[[.,[.,.]],[[[.,[[.,[.,.]],.]],[.,.]],.]],[.,.]] => 4 [[[.,[.,.]],[[[[.,[.,.]],[.,.]],[.,.]],.]],[.,.]] => 4 [[[.,[[.,[.,.]],.]],[[.,[[.,[.,.]],.]],.]],[.,.]] => 4 [[[.,[[.,[.,.]],.]],[[[.,[.,.]],[.,.]],.]],[.,.]] => 4 [[[[.,[.,.]],[.,.]],[[.,[[.,[.,.]],.]],.]],[.,.]] => 5 [[[[.,[.,.]],[.,.]],[[[.,[.,.]],[.,.]],.]],[.,.]] => 5 [[[.,[[.,[[.,[.,.]],.]],.]],[[.,[.,.]],.]],[.,.]] => 4 [[[.,[[[.,[.,.]],[.,.]],.]],[[.,[.,.]],.]],[.,.]] => 4 [[[[.,[.,.]],[[.,[.,.]],.]],[[.,[.,.]],.]],[.,.]] => 5 [[[[.,[[.,[.,.]],.]],[.,.]],[[.,[.,.]],.]],[.,.]] => 5 [[[[[.,[.,.]],[.,.]],[.,.]],[[.,[.,.]],.]],[.,.]] => 6 [[[.,[[.,[[.,[[.,[.,.]],.]],.]],.]],[.,.]],[.,.]] => 4 [[[.,[[.,[[[.,[.,.]],[.,.]],.]],.]],[.,.]],[.,.]] => 4 [[[.,[[[.,[.,.]],[[.,[.,.]],.]],.]],[.,.]],[.,.]] => 4 [[[.,[[[.,[[.,[.,.]],.]],[.,.]],.]],[.,.]],[.,.]] => 4 [[[.,[[[[.,[.,.]],[.,.]],[.,.]],.]],[.,.]],[.,.]] => 4 [[[[.,[.,.]],[[.,[[.,[.,.]],.]],.]],[.,.]],[.,.]] => 5 [[[[.,[.,.]],[[[.,[.,.]],[.,.]],.]],[.,.]],[.,.]] => 5 [[[[.,[[.,[.,.]],.]],[[.,[.,.]],.]],[.,.]],[.,.]] => 5 [[[[[.,[.,.]],[.,.]],[[.,[.,.]],.]],[.,.]],[.,.]] => 6 [[[[.,[[.,[[.,[.,.]],.]],.]],[.,.]],[.,.]],[.,.]] => 5 [[[[.,[[[.,[.,.]],[.,.]],.]],[.,.]],[.,.]],[.,.]] => 5 [[[[[.,[.,.]],[[.,[.,.]],.]],[.,.]],[.,.]],[.,.]] => 6 [[[[[.,[[.,[.,.]],.]],[.,.]],[.,.]],[.,.]],[.,.]] => 6 ----------------------------------------------------------------------------- Created: May 16, 2014 at 17:26 by Joseph Ching ----------------------------------------------------------------------------- Last Updated: Jan 17, 2018 at 23:01 by Martin Rubey