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*                                                                           *
*       Copyright (C) 2019 The FindStatCrew <info@findstat.org>             *
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*    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.                   *
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-----------------------------------------------------------------------------
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