Identifier
- St001376: Binary trees ⟶ ℤ
Values
[.,.] => 0
[.,[.,.]] => 1
[[.,.],.] => 1
[.,[.,[.,.]]] => 3
[.,[[.,.],.]] => 3
[[.,.],[.,.]] => 0
[[.,[.,.]],.] => 3
[[[.,.],.],.] => 3
[.,[.,[.,[.,.]]]] => 6
[.,[.,[[.,.],.]]] => 6
[.,[[.,.],[.,.]]] => 3
[.,[[.,[.,.]],.]] => 6
[.,[[[.,.],.],.]] => 6
[[.,.],[.,[.,.]]] => 2
[[.,.],[[.,.],.]] => 2
[[.,[.,.]],[.,.]] => 2
[[[.,.],.],[.,.]] => 2
[[.,[.,[.,.]]],.] => 6
[[.,[[.,.],.]],.] => 6
[[[.,.],[.,.]],.] => 3
[[[.,[.,.]],.],.] => 6
[[[[.,.],.],.],.] => 6
[.,[.,[.,[.,[.,.]]]]] => 10
[.,[.,[.,[[.,.],.]]]] => 10
[.,[.,[[.,.],[.,.]]]] => 7
[.,[.,[[.,[.,.]],.]]] => 10
[.,[.,[[[.,.],.],.]]] => 10
[.,[[.,.],[.,[.,.]]]] => 6
[.,[[.,.],[[.,.],.]]] => 6
[.,[[.,[.,.]],[.,.]]] => 6
[.,[[[.,.],.],[.,.]]] => 6
[.,[[.,[.,[.,.]]],.]] => 10
[.,[[.,[[.,.],.]],.]] => 10
[.,[[[.,.],[.,.]],.]] => 7
[.,[[[.,[.,.]],.],.]] => 10
[.,[[[[.,.],.],.],.]] => 10
[[.,.],[.,[.,[.,.]]]] => 5
[[.,.],[.,[[.,.],.]]] => 5
[[.,.],[[.,.],[.,.]]] => 2
[[.,.],[[.,[.,.]],.]] => 5
[[.,.],[[[.,.],.],.]] => 5
[[.,[.,.]],[.,[.,.]]] => 2
[[.,[.,.]],[[.,.],.]] => 2
[[[.,.],.],[.,[.,.]]] => 2
[[[.,.],.],[[.,.],.]] => 2
[[.,[.,[.,.]]],[.,.]] => 5
[[.,[[.,.],.]],[.,.]] => 5
[[[.,.],[.,.]],[.,.]] => 2
[[[.,[.,.]],.],[.,.]] => 5
[[[[.,.],.],.],[.,.]] => 5
[[.,[.,[.,[.,.]]]],.] => 10
[[.,[.,[[.,.],.]]],.] => 10
[[.,[[.,.],[.,.]]],.] => 7
[[.,[[.,[.,.]],.]],.] => 10
[[.,[[[.,.],.],.]],.] => 10
[[[.,.],[.,[.,.]]],.] => 6
[[[.,.],[[.,.],.]],.] => 6
[[[.,[.,.]],[.,.]],.] => 6
[[[[.,.],.],[.,.]],.] => 6
[[[.,[.,[.,.]]],.],.] => 10
[[[.,[[.,.],.]],.],.] => 10
[[[[.,.],[.,.]],.],.] => 7
[[[[.,[.,.]],.],.],.] => 10
[[[[[.,.],.],.],.],.] => 10
[.,[.,[.,[.,[.,[.,.]]]]]] => 15
[.,[.,[.,[.,[[.,.],.]]]]] => 15
[.,[.,[.,[[.,.],[.,.]]]]] => 12
[.,[.,[.,[[.,[.,.]],.]]]] => 15
[.,[.,[.,[[[.,.],.],.]]]] => 15
[.,[.,[[.,.],[.,[.,.]]]]] => 11
[.,[.,[[.,.],[[.,.],.]]]] => 11
[.,[.,[[.,[.,.]],[.,.]]]] => 11
[.,[.,[[[.,.],.],[.,.]]]] => 11
[.,[.,[[.,[.,[.,.]]],.]]] => 15
[.,[.,[[.,[[.,.],.]],.]]] => 15
[.,[.,[[[.,.],[.,.]],.]]] => 12
[.,[.,[[[.,[.,.]],.],.]]] => 15
[.,[.,[[[[.,.],.],.],.]]] => 15
[.,[[.,.],[.,[.,[.,.]]]]] => 10
[.,[[.,.],[.,[[.,.],.]]]] => 10
[.,[[.,.],[[.,.],[.,.]]]] => 7
[.,[[.,.],[[.,[.,.]],.]]] => 10
[.,[[.,.],[[[.,.],.],.]]] => 10
[.,[[.,[.,.]],[.,[.,.]]]] => 7
[.,[[.,[.,.]],[[.,.],.]]] => 7
[.,[[[.,.],.],[.,[.,.]]]] => 7
[.,[[[.,.],.],[[.,.],.]]] => 7
[.,[[.,[.,[.,.]]],[.,.]]] => 10
[.,[[.,[[.,.],.]],[.,.]]] => 10
[.,[[[.,.],[.,.]],[.,.]]] => 7
[.,[[[.,[.,.]],.],[.,.]]] => 10
[.,[[[[.,.],.],.],[.,.]]] => 10
[.,[[.,[.,[.,[.,.]]]],.]] => 15
[.,[[.,[.,[[.,.],.]]],.]] => 15
[.,[[.,[[.,.],[.,.]]],.]] => 12
[.,[[.,[[.,[.,.]],.]],.]] => 15
[.,[[.,[[[.,.],.],.]],.]] => 15
[.,[[[.,.],[.,[.,.]]],.]] => 11
[.,[[[.,.],[[.,.],.]],.]] => 11
[.,[[[.,[.,.]],[.,.]],.]] => 11
[.,[[[[.,.],.],[.,.]],.]] => 11
>>> Load all 1200 entries. <<<[.,[[[.,[.,[.,.]]],.],.]] => 15
[.,[[[.,[[.,.],.]],.],.]] => 15
[.,[[[[.,.],[.,.]],.],.]] => 12
[.,[[[[.,[.,.]],.],.],.]] => 15
[.,[[[[[.,.],.],.],.],.]] => 15
[[.,.],[.,[.,[.,[.,.]]]]] => 9
[[.,.],[.,[.,[[.,.],.]]]] => 9
[[.,.],[.,[[.,.],[.,.]]]] => 6
[[.,.],[.,[[.,[.,.]],.]]] => 9
[[.,.],[.,[[[.,.],.],.]]] => 9
[[.,.],[[.,.],[.,[.,.]]]] => 5
[[.,.],[[.,.],[[.,.],.]]] => 5
[[.,.],[[.,[.,.]],[.,.]]] => 5
[[.,.],[[[.,.],.],[.,.]]] => 5
[[.,.],[[.,[.,[.,.]]],.]] => 9
[[.,.],[[.,[[.,.],.]],.]] => 9
[[.,.],[[[.,.],[.,.]],.]] => 6
[[.,.],[[[.,[.,.]],.],.]] => 9
[[.,.],[[[[.,.],.],.],.]] => 9
[[.,[.,.]],[.,[.,[.,.]]]] => 5
[[.,[.,.]],[.,[[.,.],.]]] => 5
[[.,[.,.]],[[.,.],[.,.]]] => 2
[[.,[.,.]],[[.,[.,.]],.]] => 5
[[.,[.,.]],[[[.,.],.],.]] => 5
[[[.,.],.],[.,[.,[.,.]]]] => 5
[[[.,.],.],[.,[[.,.],.]]] => 5
[[[.,.],.],[[.,.],[.,.]]] => 2
[[[.,.],.],[[.,[.,.]],.]] => 5
[[[.,.],.],[[[.,.],.],.]] => 5
[[.,[.,[.,.]]],[.,[.,.]]] => 5
[[.,[.,[.,.]]],[[.,.],.]] => 5
[[.,[[.,.],.]],[.,[.,.]]] => 5
[[.,[[.,.],.]],[[.,.],.]] => 5
[[[.,.],[.,.]],[.,[.,.]]] => 2
[[[.,.],[.,.]],[[.,.],.]] => 2
[[[.,[.,.]],.],[.,[.,.]]] => 5
[[[.,[.,.]],.],[[.,.],.]] => 5
[[[[.,.],.],.],[.,[.,.]]] => 5
[[[[.,.],.],.],[[.,.],.]] => 5
[[.,[.,[.,[.,.]]]],[.,.]] => 9
[[.,[.,[[.,.],.]]],[.,.]] => 9
[[.,[[.,.],[.,.]]],[.,.]] => 6
[[.,[[.,[.,.]],.]],[.,.]] => 9
[[.,[[[.,.],.],.]],[.,.]] => 9
[[[.,.],[.,[.,.]]],[.,.]] => 5
[[[.,.],[[.,.],.]],[.,.]] => 5
[[[.,[.,.]],[.,.]],[.,.]] => 5
[[[[.,.],.],[.,.]],[.,.]] => 5
[[[.,[.,[.,.]]],.],[.,.]] => 9
[[[.,[[.,.],.]],.],[.,.]] => 9
[[[[.,.],[.,.]],.],[.,.]] => 6
[[[[.,[.,.]],.],.],[.,.]] => 9
[[[[[.,.],.],.],.],[.,.]] => 9
[[.,[.,[.,[.,[.,.]]]]],.] => 15
[[.,[.,[.,[[.,.],.]]]],.] => 15
[[.,[.,[[.,.],[.,.]]]],.] => 12
[[.,[.,[[.,[.,.]],.]]],.] => 15
[[.,[.,[[[.,.],.],.]]],.] => 15
[[.,[[.,.],[.,[.,.]]]],.] => 11
[[.,[[.,.],[[.,.],.]]],.] => 11
[[.,[[.,[.,.]],[.,.]]],.] => 11
[[.,[[[.,.],.],[.,.]]],.] => 11
[[.,[[.,[.,[.,.]]],.]],.] => 15
[[.,[[.,[[.,.],.]],.]],.] => 15
[[.,[[[.,.],[.,.]],.]],.] => 12
[[.,[[[.,[.,.]],.],.]],.] => 15
[[.,[[[[.,.],.],.],.]],.] => 15
[[[.,.],[.,[.,[.,.]]]],.] => 10
[[[.,.],[.,[[.,.],.]]],.] => 10
[[[.,.],[[.,.],[.,.]]],.] => 7
[[[.,.],[[.,[.,.]],.]],.] => 10
[[[.,.],[[[.,.],.],.]],.] => 10
[[[.,[.,.]],[.,[.,.]]],.] => 7
[[[.,[.,.]],[[.,.],.]],.] => 7
[[[[.,.],.],[.,[.,.]]],.] => 7
[[[[.,.],.],[[.,.],.]],.] => 7
[[[.,[.,[.,.]]],[.,.]],.] => 10
[[[.,[[.,.],.]],[.,.]],.] => 10
[[[[.,.],[.,.]],[.,.]],.] => 7
[[[[.,[.,.]],.],[.,.]],.] => 10
[[[[[.,.],.],.],[.,.]],.] => 10
[[[.,[.,[.,[.,.]]]],.],.] => 15
[[[.,[.,[[.,.],.]]],.],.] => 15
[[[.,[[.,.],[.,.]]],.],.] => 12
[[[.,[[.,[.,.]],.]],.],.] => 15
[[[.,[[[.,.],.],.]],.],.] => 15
[[[[.,.],[.,[.,.]]],.],.] => 11
[[[[.,.],[[.,.],.]],.],.] => 11
[[[[.,[.,.]],[.,.]],.],.] => 11
[[[[[.,.],.],[.,.]],.],.] => 11
[[[[.,[.,[.,.]]],.],.],.] => 15
[[[[.,[[.,.],.]],.],.],.] => 15
[[[[[.,.],[.,.]],.],.],.] => 12
[[[[[.,[.,.]],.],.],.],.] => 15
[[[[[[.,.],.],.],.],.],.] => 15
[.,[.,[.,[.,[.,[.,[.,.]]]]]]] => 21
[.,[.,[.,[.,[.,[[.,.],.]]]]]] => 21
[.,[.,[.,[.,[[.,.],[.,.]]]]]] => 18
[.,[.,[.,[.,[[.,[.,.]],.]]]]] => 21
[.,[.,[.,[.,[[[.,.],.],.]]]]] => 21
[.,[.,[.,[[.,.],[.,[.,.]]]]]] => 17
[.,[.,[.,[[.,.],[[.,.],.]]]]] => 17
[.,[.,[.,[[.,[.,.]],[.,.]]]]] => 17
[.,[.,[.,[[[.,.],.],[.,.]]]]] => 17
[.,[.,[.,[[.,[.,[.,.]]],.]]]] => 21
[.,[.,[.,[[.,[[.,.],.]],.]]]] => 21
[.,[.,[.,[[[.,.],[.,.]],.]]]] => 18
[.,[.,[.,[[[.,[.,.]],.],.]]]] => 21
[.,[.,[.,[[[[.,.],.],.],.]]]] => 21
[.,[.,[[.,.],[.,[.,[.,.]]]]]] => 16
[.,[.,[[.,.],[.,[[.,.],.]]]]] => 16
[.,[.,[[.,.],[[.,.],[.,.]]]]] => 13
[.,[.,[[.,.],[[.,[.,.]],.]]]] => 16
[.,[.,[[.,.],[[[.,.],.],.]]]] => 16
[.,[.,[[.,[.,.]],[.,[.,.]]]]] => 13
[.,[.,[[.,[.,.]],[[.,.],.]]]] => 13
[.,[.,[[[.,.],.],[.,[.,.]]]]] => 13
[.,[.,[[[.,.],.],[[.,.],.]]]] => 13
[.,[.,[[.,[.,[.,.]]],[.,.]]]] => 16
[.,[.,[[.,[[.,.],.]],[.,.]]]] => 16
[.,[.,[[[.,.],[.,.]],[.,.]]]] => 13
[.,[.,[[[.,[.,.]],.],[.,.]]]] => 16
[.,[.,[[[[.,.],.],.],[.,.]]]] => 16
[.,[.,[[.,[.,[.,[.,.]]]],.]]] => 21
[.,[.,[[.,[.,[[.,.],.]]],.]]] => 21
[.,[.,[[.,[[.,.],[.,.]]],.]]] => 18
[.,[.,[[.,[[.,[.,.]],.]],.]]] => 21
[.,[.,[[.,[[[.,.],.],.]],.]]] => 21
[.,[.,[[[.,.],[.,[.,.]]],.]]] => 17
[.,[.,[[[.,.],[[.,.],.]],.]]] => 17
[.,[.,[[[.,[.,.]],[.,.]],.]]] => 17
[.,[.,[[[[.,.],.],[.,.]],.]]] => 17
[.,[.,[[[.,[.,[.,.]]],.],.]]] => 21
[.,[.,[[[.,[[.,.],.]],.],.]]] => 21
[.,[.,[[[[.,.],[.,.]],.],.]]] => 18
[.,[.,[[[[.,[.,.]],.],.],.]]] => 21
[.,[.,[[[[[.,.],.],.],.],.]]] => 21
[.,[[.,.],[.,[.,[.,[.,.]]]]]] => 15
[.,[[.,.],[.,[.,[[.,.],.]]]]] => 15
[.,[[.,.],[.,[[.,.],[.,.]]]]] => 12
[.,[[.,.],[.,[[.,[.,.]],.]]]] => 15
[.,[[.,.],[.,[[[.,.],.],.]]]] => 15
[.,[[.,.],[[.,.],[.,[.,.]]]]] => 11
[.,[[.,.],[[.,.],[[.,.],.]]]] => 11
[.,[[.,.],[[.,[.,.]],[.,.]]]] => 11
[.,[[.,.],[[[.,.],.],[.,.]]]] => 11
[.,[[.,.],[[.,[.,[.,.]]],.]]] => 15
[.,[[.,.],[[.,[[.,.],.]],.]]] => 15
[.,[[.,.],[[[.,.],[.,.]],.]]] => 12
[.,[[.,.],[[[.,[.,.]],.],.]]] => 15
[.,[[.,.],[[[[.,.],.],.],.]]] => 15
[.,[[.,[.,.]],[.,[.,[.,.]]]]] => 11
[.,[[.,[.,.]],[.,[[.,.],.]]]] => 11
[.,[[.,[.,.]],[[.,.],[.,.]]]] => 8
[.,[[.,[.,.]],[[.,[.,.]],.]]] => 11
[.,[[.,[.,.]],[[[.,.],.],.]]] => 11
[.,[[[.,.],.],[.,[.,[.,.]]]]] => 11
[.,[[[.,.],.],[.,[[.,.],.]]]] => 11
[.,[[[.,.],.],[[.,.],[.,.]]]] => 8
[.,[[[.,.],.],[[.,[.,.]],.]]] => 11
[.,[[[.,.],.],[[[.,.],.],.]]] => 11
[.,[[.,[.,[.,.]]],[.,[.,.]]]] => 11
[.,[[.,[.,[.,.]]],[[.,.],.]]] => 11
[.,[[.,[[.,.],.]],[.,[.,.]]]] => 11
[.,[[.,[[.,.],.]],[[.,.],.]]] => 11
[.,[[[.,.],[.,.]],[.,[.,.]]]] => 8
[.,[[[.,.],[.,.]],[[.,.],.]]] => 8
[.,[[[.,[.,.]],.],[.,[.,.]]]] => 11
[.,[[[.,[.,.]],.],[[.,.],.]]] => 11
[.,[[[[.,.],.],.],[.,[.,.]]]] => 11
[.,[[[[.,.],.],.],[[.,.],.]]] => 11
[.,[[.,[.,[.,[.,.]]]],[.,.]]] => 15
[.,[[.,[.,[[.,.],.]]],[.,.]]] => 15
[.,[[.,[[.,.],[.,.]]],[.,.]]] => 12
[.,[[.,[[.,[.,.]],.]],[.,.]]] => 15
[.,[[.,[[[.,.],.],.]],[.,.]]] => 15
[.,[[[.,.],[.,[.,.]]],[.,.]]] => 11
[.,[[[.,.],[[.,.],.]],[.,.]]] => 11
[.,[[[.,[.,.]],[.,.]],[.,.]]] => 11
[.,[[[[.,.],.],[.,.]],[.,.]]] => 11
[.,[[[.,[.,[.,.]]],.],[.,.]]] => 15
[.,[[[.,[[.,.],.]],.],[.,.]]] => 15
[.,[[[[.,.],[.,.]],.],[.,.]]] => 12
[.,[[[[.,[.,.]],.],.],[.,.]]] => 15
[.,[[[[[.,.],.],.],.],[.,.]]] => 15
[.,[[.,[.,[.,[.,[.,.]]]]],.]] => 21
[.,[[.,[.,[.,[[.,.],.]]]],.]] => 21
[.,[[.,[.,[[.,.],[.,.]]]],.]] => 18
[.,[[.,[.,[[.,[.,.]],.]]],.]] => 21
[.,[[.,[.,[[[.,.],.],.]]],.]] => 21
[.,[[.,[[.,.],[.,[.,.]]]],.]] => 17
[.,[[.,[[.,.],[[.,.],.]]],.]] => 17
[.,[[.,[[.,[.,.]],[.,.]]],.]] => 17
[.,[[.,[[[.,.],.],[.,.]]],.]] => 17
[.,[[.,[[.,[.,[.,.]]],.]],.]] => 21
[.,[[.,[[.,[[.,.],.]],.]],.]] => 21
[.,[[.,[[[.,.],[.,.]],.]],.]] => 18
[.,[[.,[[[.,[.,.]],.],.]],.]] => 21
[.,[[.,[[[[.,.],.],.],.]],.]] => 21
[.,[[[.,.],[.,[.,[.,.]]]],.]] => 16
[.,[[[.,.],[.,[[.,.],.]]],.]] => 16
[.,[[[.,.],[[.,.],[.,.]]],.]] => 13
[.,[[[.,.],[[.,[.,.]],.]],.]] => 16
[.,[[[.,.],[[[.,.],.],.]],.]] => 16
[.,[[[.,[.,.]],[.,[.,.]]],.]] => 13
[.,[[[.,[.,.]],[[.,.],.]],.]] => 13
[.,[[[[.,.],.],[.,[.,.]]],.]] => 13
[.,[[[[.,.],.],[[.,.],.]],.]] => 13
[.,[[[.,[.,[.,.]]],[.,.]],.]] => 16
[.,[[[.,[[.,.],.]],[.,.]],.]] => 16
[.,[[[[.,.],[.,.]],[.,.]],.]] => 13
[.,[[[[.,[.,.]],.],[.,.]],.]] => 16
[.,[[[[[.,.],.],.],[.,.]],.]] => 16
[.,[[[.,[.,[.,[.,.]]]],.],.]] => 21
[.,[[[.,[.,[[.,.],.]]],.],.]] => 21
[.,[[[.,[[.,.],[.,.]]],.],.]] => 18
[.,[[[.,[[.,[.,.]],.]],.],.]] => 21
[.,[[[.,[[[.,.],.],.]],.],.]] => 21
[.,[[[[.,.],[.,[.,.]]],.],.]] => 17
[.,[[[[.,.],[[.,.],.]],.],.]] => 17
[.,[[[[.,[.,.]],[.,.]],.],.]] => 17
[.,[[[[[.,.],.],[.,.]],.],.]] => 17
[.,[[[[.,[.,[.,.]]],.],.],.]] => 21
[.,[[[[.,[[.,.],.]],.],.],.]] => 21
[.,[[[[[.,.],[.,.]],.],.],.]] => 18
[.,[[[[[.,[.,.]],.],.],.],.]] => 21
[.,[[[[[[.,.],.],.],.],.],.]] => 21
[[.,.],[.,[.,[.,[.,[.,.]]]]]] => 14
[[.,.],[.,[.,[.,[[.,.],.]]]]] => 14
[[.,.],[.,[.,[[.,.],[.,.]]]]] => 11
[[.,.],[.,[.,[[.,[.,.]],.]]]] => 14
[[.,.],[.,[.,[[[.,.],.],.]]]] => 14
[[.,.],[.,[[.,.],[.,[.,.]]]]] => 10
[[.,.],[.,[[.,.],[[.,.],.]]]] => 10
[[.,.],[.,[[.,[.,.]],[.,.]]]] => 10
[[.,.],[.,[[[.,.],.],[.,.]]]] => 10
[[.,.],[.,[[.,[.,[.,.]]],.]]] => 14
[[.,.],[.,[[.,[[.,.],.]],.]]] => 14
[[.,.],[.,[[[.,.],[.,.]],.]]] => 11
[[.,.],[.,[[[.,[.,.]],.],.]]] => 14
[[.,.],[.,[[[[.,.],.],.],.]]] => 14
[[.,.],[[.,.],[.,[.,[.,.]]]]] => 9
[[.,.],[[.,.],[.,[[.,.],.]]]] => 9
[[.,.],[[.,.],[[.,.],[.,.]]]] => 6
[[.,.],[[.,.],[[.,[.,.]],.]]] => 9
[[.,.],[[.,.],[[[.,.],.],.]]] => 9
[[.,.],[[.,[.,.]],[.,[.,.]]]] => 6
[[.,.],[[.,[.,.]],[[.,.],.]]] => 6
[[.,.],[[[.,.],.],[.,[.,.]]]] => 6
[[.,.],[[[.,.],.],[[.,.],.]]] => 6
[[.,.],[[.,[.,[.,.]]],[.,.]]] => 9
[[.,.],[[.,[[.,.],.]],[.,.]]] => 9
[[.,.],[[[.,.],[.,.]],[.,.]]] => 6
[[.,.],[[[.,[.,.]],.],[.,.]]] => 9
[[.,.],[[[[.,.],.],.],[.,.]]] => 9
[[.,.],[[.,[.,[.,[.,.]]]],.]] => 14
[[.,.],[[.,[.,[[.,.],.]]],.]] => 14
[[.,.],[[.,[[.,.],[.,.]]],.]] => 11
[[.,.],[[.,[[.,[.,.]],.]],.]] => 14
[[.,.],[[.,[[[.,.],.],.]],.]] => 14
[[.,.],[[[.,.],[.,[.,.]]],.]] => 10
[[.,.],[[[.,.],[[.,.],.]],.]] => 10
[[.,.],[[[.,[.,.]],[.,.]],.]] => 10
[[.,.],[[[[.,.],.],[.,.]],.]] => 10
[[.,.],[[[.,[.,[.,.]]],.],.]] => 14
[[.,.],[[[.,[[.,.],.]],.],.]] => 14
[[.,.],[[[[.,.],[.,.]],.],.]] => 11
[[.,.],[[[[.,[.,.]],.],.],.]] => 14
[[.,.],[[[[[.,.],.],.],.],.]] => 14
[[.,[.,.]],[.,[.,[.,[.,.]]]]] => 9
[[.,[.,.]],[.,[.,[[.,.],.]]]] => 9
[[.,[.,.]],[.,[[.,.],[.,.]]]] => 6
[[.,[.,.]],[.,[[.,[.,.]],.]]] => 9
[[.,[.,.]],[.,[[[.,.],.],.]]] => 9
[[.,[.,.]],[[.,.],[.,[.,.]]]] => 5
[[.,[.,.]],[[.,.],[[.,.],.]]] => 5
[[.,[.,.]],[[.,[.,.]],[.,.]]] => 5
[[.,[.,.]],[[[.,.],.],[.,.]]] => 5
[[.,[.,.]],[[.,[.,[.,.]]],.]] => 9
[[.,[.,.]],[[.,[[.,.],.]],.]] => 9
[[.,[.,.]],[[[.,.],[.,.]],.]] => 6
[[.,[.,.]],[[[.,[.,.]],.],.]] => 9
[[.,[.,.]],[[[[.,.],.],.],.]] => 9
[[[.,.],.],[.,[.,[.,[.,.]]]]] => 9
[[[.,.],.],[.,[.,[[.,.],.]]]] => 9
[[[.,.],.],[.,[[.,.],[.,.]]]] => 6
[[[.,.],.],[.,[[.,[.,.]],.]]] => 9
[[[.,.],.],[.,[[[.,.],.],.]]] => 9
[[[.,.],.],[[.,.],[.,[.,.]]]] => 5
[[[.,.],.],[[.,.],[[.,.],.]]] => 5
[[[.,.],.],[[.,[.,.]],[.,.]]] => 5
[[[.,.],.],[[[.,.],.],[.,.]]] => 5
[[[.,.],.],[[.,[.,[.,.]]],.]] => 9
[[[.,.],.],[[.,[[.,.],.]],.]] => 9
[[[.,.],.],[[[.,.],[.,.]],.]] => 6
[[[.,.],.],[[[.,[.,.]],.],.]] => 9
[[[.,.],.],[[[[.,.],.],.],.]] => 9
[[.,[.,[.,.]]],[.,[.,[.,.]]]] => 6
[[.,[.,[.,.]]],[.,[[.,.],.]]] => 6
[[.,[.,[.,.]]],[[.,.],[.,.]]] => 3
[[.,[.,[.,.]]],[[.,[.,.]],.]] => 6
[[.,[.,[.,.]]],[[[.,.],.],.]] => 6
[[.,[[.,.],.]],[.,[.,[.,.]]]] => 6
[[.,[[.,.],.]],[.,[[.,.],.]]] => 6
[[.,[[.,.],.]],[[.,.],[.,.]]] => 3
[[.,[[.,.],.]],[[.,[.,.]],.]] => 6
[[.,[[.,.],.]],[[[.,.],.],.]] => 6
[[[.,.],[.,.]],[.,[.,[.,.]]]] => 3
[[[.,.],[.,.]],[.,[[.,.],.]]] => 3
[[[.,.],[.,.]],[[.,.],[.,.]]] => 0
[[[.,.],[.,.]],[[.,[.,.]],.]] => 3
[[[.,.],[.,.]],[[[.,.],.],.]] => 3
[[[.,[.,.]],.],[.,[.,[.,.]]]] => 6
[[[.,[.,.]],.],[.,[[.,.],.]]] => 6
[[[.,[.,.]],.],[[.,.],[.,.]]] => 3
[[[.,[.,.]],.],[[.,[.,.]],.]] => 6
[[[.,[.,.]],.],[[[.,.],.],.]] => 6
[[[[.,.],.],.],[.,[.,[.,.]]]] => 6
[[[[.,.],.],.],[.,[[.,.],.]]] => 6
[[[[.,.],.],.],[[.,.],[.,.]]] => 3
[[[[.,.],.],.],[[.,[.,.]],.]] => 6
[[[[.,.],.],.],[[[.,.],.],.]] => 6
[[.,[.,[.,[.,.]]]],[.,[.,.]]] => 9
[[.,[.,[.,[.,.]]]],[[.,.],.]] => 9
[[.,[.,[[.,.],.]]],[.,[.,.]]] => 9
[[.,[.,[[.,.],.]]],[[.,.],.]] => 9
[[.,[[.,.],[.,.]]],[.,[.,.]]] => 6
[[.,[[.,.],[.,.]]],[[.,.],.]] => 6
[[.,[[.,[.,.]],.]],[.,[.,.]]] => 9
[[.,[[.,[.,.]],.]],[[.,.],.]] => 9
[[.,[[[.,.],.],.]],[.,[.,.]]] => 9
[[.,[[[.,.],.],.]],[[.,.],.]] => 9
[[[.,.],[.,[.,.]]],[.,[.,.]]] => 5
[[[.,.],[.,[.,.]]],[[.,.],.]] => 5
[[[.,.],[[.,.],.]],[.,[.,.]]] => 5
[[[.,.],[[.,.],.]],[[.,.],.]] => 5
[[[.,[.,.]],[.,.]],[.,[.,.]]] => 5
[[[.,[.,.]],[.,.]],[[.,.],.]] => 5
[[[[.,.],.],[.,.]],[.,[.,.]]] => 5
[[[[.,.],.],[.,.]],[[.,.],.]] => 5
[[[.,[.,[.,.]]],.],[.,[.,.]]] => 9
[[[.,[.,[.,.]]],.],[[.,.],.]] => 9
[[[.,[[.,.],.]],.],[.,[.,.]]] => 9
[[[.,[[.,.],.]],.],[[.,.],.]] => 9
[[[[.,.],[.,.]],.],[.,[.,.]]] => 6
[[[[.,.],[.,.]],.],[[.,.],.]] => 6
[[[[.,[.,.]],.],.],[.,[.,.]]] => 9
[[[[.,[.,.]],.],.],[[.,.],.]] => 9
[[[[[.,.],.],.],.],[.,[.,.]]] => 9
[[[[[.,.],.],.],.],[[.,.],.]] => 9
[[.,[.,[.,[.,[.,.]]]]],[.,.]] => 14
[[.,[.,[.,[[.,.],.]]]],[.,.]] => 14
[[.,[.,[[.,.],[.,.]]]],[.,.]] => 11
[[.,[.,[[.,[.,.]],.]]],[.,.]] => 14
[[.,[.,[[[.,.],.],.]]],[.,.]] => 14
[[.,[[.,.],[.,[.,.]]]],[.,.]] => 10
[[.,[[.,.],[[.,.],.]]],[.,.]] => 10
[[.,[[.,[.,.]],[.,.]]],[.,.]] => 10
[[.,[[[.,.],.],[.,.]]],[.,.]] => 10
[[.,[[.,[.,[.,.]]],.]],[.,.]] => 14
[[.,[[.,[[.,.],.]],.]],[.,.]] => 14
[[.,[[[.,.],[.,.]],.]],[.,.]] => 11
[[.,[[[.,[.,.]],.],.]],[.,.]] => 14
[[.,[[[[.,.],.],.],.]],[.,.]] => 14
[[[.,.],[.,[.,[.,.]]]],[.,.]] => 9
[[[.,.],[.,[[.,.],.]]],[.,.]] => 9
[[[.,.],[[.,.],[.,.]]],[.,.]] => 6
[[[.,.],[[.,[.,.]],.]],[.,.]] => 9
[[[.,.],[[[.,.],.],.]],[.,.]] => 9
[[[.,[.,.]],[.,[.,.]]],[.,.]] => 6
[[[.,[.,.]],[[.,.],.]],[.,.]] => 6
[[[[.,.],.],[.,[.,.]]],[.,.]] => 6
[[[[.,.],.],[[.,.],.]],[.,.]] => 6
[[[.,[.,[.,.]]],[.,.]],[.,.]] => 9
[[[.,[[.,.],.]],[.,.]],[.,.]] => 9
[[[[.,.],[.,.]],[.,.]],[.,.]] => 6
[[[[.,[.,.]],.],[.,.]],[.,.]] => 9
[[[[[.,.],.],.],[.,.]],[.,.]] => 9
[[[.,[.,[.,[.,.]]]],.],[.,.]] => 14
[[[.,[.,[[.,.],.]]],.],[.,.]] => 14
[[[.,[[.,.],[.,.]]],.],[.,.]] => 11
[[[.,[[.,[.,.]],.]],.],[.,.]] => 14
[[[.,[[[.,.],.],.]],.],[.,.]] => 14
[[[[.,.],[.,[.,.]]],.],[.,.]] => 10
[[[[.,.],[[.,.],.]],.],[.,.]] => 10
[[[[.,[.,.]],[.,.]],.],[.,.]] => 10
[[[[[.,.],.],[.,.]],.],[.,.]] => 10
[[[[.,[.,[.,.]]],.],.],[.,.]] => 14
[[[[.,[[.,.],.]],.],.],[.,.]] => 14
[[[[[.,.],[.,.]],.],.],[.,.]] => 11
[[[[[.,[.,.]],.],.],.],[.,.]] => 14
[[[[[[.,.],.],.],.],.],[.,.]] => 14
[[.,[.,[.,[.,[.,[.,.]]]]]],.] => 21
[[.,[.,[.,[.,[[.,.],.]]]]],.] => 21
[[.,[.,[.,[[.,.],[.,.]]]]],.] => 18
[[.,[.,[.,[[.,[.,.]],.]]]],.] => 21
[[.,[.,[.,[[[.,.],.],.]]]],.] => 21
[[.,[.,[[.,.],[.,[.,.]]]]],.] => 17
[[.,[.,[[.,.],[[.,.],.]]]],.] => 17
[[.,[.,[[.,[.,.]],[.,.]]]],.] => 17
[[.,[.,[[[.,.],.],[.,.]]]],.] => 17
[[.,[.,[[.,[.,[.,.]]],.]]],.] => 21
[[.,[.,[[.,[[.,.],.]],.]]],.] => 21
[[.,[.,[[[.,.],[.,.]],.]]],.] => 18
[[.,[.,[[[.,[.,.]],.],.]]],.] => 21
[[.,[.,[[[[.,.],.],.],.]]],.] => 21
[[.,[[.,.],[.,[.,[.,.]]]]],.] => 16
[[.,[[.,.],[.,[[.,.],.]]]],.] => 16
[[.,[[.,.],[[.,.],[.,.]]]],.] => 13
[[.,[[.,.],[[.,[.,.]],.]]],.] => 16
[[.,[[.,.],[[[.,.],.],.]]],.] => 16
[[.,[[.,[.,.]],[.,[.,.]]]],.] => 13
[[.,[[.,[.,.]],[[.,.],.]]],.] => 13
[[.,[[[.,.],.],[.,[.,.]]]],.] => 13
[[.,[[[.,.],.],[[.,.],.]]],.] => 13
[[.,[[.,[.,[.,.]]],[.,.]]],.] => 16
[[.,[[.,[[.,.],.]],[.,.]]],.] => 16
[[.,[[[.,.],[.,.]],[.,.]]],.] => 13
[[.,[[[.,[.,.]],.],[.,.]]],.] => 16
[[.,[[[[.,.],.],.],[.,.]]],.] => 16
[[.,[[.,[.,[.,[.,.]]]],.]],.] => 21
[[.,[[.,[.,[[.,.],.]]],.]],.] => 21
[[.,[[.,[[.,.],[.,.]]],.]],.] => 18
[[.,[[.,[[.,[.,.]],.]],.]],.] => 21
[[.,[[.,[[[.,.],.],.]],.]],.] => 21
[[.,[[[.,.],[.,[.,.]]],.]],.] => 17
[[.,[[[.,.],[[.,.],.]],.]],.] => 17
[[.,[[[.,[.,.]],[.,.]],.]],.] => 17
[[.,[[[[.,.],.],[.,.]],.]],.] => 17
[[.,[[[.,[.,[.,.]]],.],.]],.] => 21
[[.,[[[.,[[.,.],.]],.],.]],.] => 21
[[.,[[[[.,.],[.,.]],.],.]],.] => 18
[[.,[[[[.,[.,.]],.],.],.]],.] => 21
[[.,[[[[[.,.],.],.],.],.]],.] => 21
[[[.,.],[.,[.,[.,[.,.]]]]],.] => 15
[[[.,.],[.,[.,[[.,.],.]]]],.] => 15
[[[.,.],[.,[[.,.],[.,.]]]],.] => 12
[[[.,.],[.,[[.,[.,.]],.]]],.] => 15
[[[.,.],[.,[[[.,.],.],.]]],.] => 15
[[[.,.],[[.,.],[.,[.,.]]]],.] => 11
[[[.,.],[[.,.],[[.,.],.]]],.] => 11
[[[.,.],[[.,[.,.]],[.,.]]],.] => 11
[[[.,.],[[[.,.],.],[.,.]]],.] => 11
[[[.,.],[[.,[.,[.,.]]],.]],.] => 15
[[[.,.],[[.,[[.,.],.]],.]],.] => 15
[[[.,.],[[[.,.],[.,.]],.]],.] => 12
[[[.,.],[[[.,[.,.]],.],.]],.] => 15
[[[.,.],[[[[.,.],.],.],.]],.] => 15
[[[.,[.,.]],[.,[.,[.,.]]]],.] => 11
[[[.,[.,.]],[.,[[.,.],.]]],.] => 11
[[[.,[.,.]],[[.,.],[.,.]]],.] => 8
[[[.,[.,.]],[[.,[.,.]],.]],.] => 11
[[[.,[.,.]],[[[.,.],.],.]],.] => 11
[[[[.,.],.],[.,[.,[.,.]]]],.] => 11
[[[[.,.],.],[.,[[.,.],.]]],.] => 11
[[[[.,.],.],[[.,.],[.,.]]],.] => 8
[[[[.,.],.],[[.,[.,.]],.]],.] => 11
[[[[.,.],.],[[[.,.],.],.]],.] => 11
[[[.,[.,[.,.]]],[.,[.,.]]],.] => 11
[[[.,[.,[.,.]]],[[.,.],.]],.] => 11
[[[.,[[.,.],.]],[.,[.,.]]],.] => 11
[[[.,[[.,.],.]],[[.,.],.]],.] => 11
[[[[.,.],[.,.]],[.,[.,.]]],.] => 8
[[[[.,.],[.,.]],[[.,.],.]],.] => 8
[[[[.,[.,.]],.],[.,[.,.]]],.] => 11
[[[[.,[.,.]],.],[[.,.],.]],.] => 11
[[[[[.,.],.],.],[.,[.,.]]],.] => 11
[[[[[.,.],.],.],[[.,.],.]],.] => 11
[[[.,[.,[.,[.,.]]]],[.,.]],.] => 15
[[[.,[.,[[.,.],.]]],[.,.]],.] => 15
[[[.,[[.,.],[.,.]]],[.,.]],.] => 12
[[[.,[[.,[.,.]],.]],[.,.]],.] => 15
[[[.,[[[.,.],.],.]],[.,.]],.] => 15
[[[[.,.],[.,[.,.]]],[.,.]],.] => 11
[[[[.,.],[[.,.],.]],[.,.]],.] => 11
[[[[.,[.,.]],[.,.]],[.,.]],.] => 11
[[[[[.,.],.],[.,.]],[.,.]],.] => 11
[[[[.,[.,[.,.]]],.],[.,.]],.] => 15
[[[[.,[[.,.],.]],.],[.,.]],.] => 15
[[[[[.,.],[.,.]],.],[.,.]],.] => 12
[[[[[.,[.,.]],.],.],[.,.]],.] => 15
[[[[[[.,.],.],.],.],[.,.]],.] => 15
[[[.,[.,[.,[.,[.,.]]]]],.],.] => 21
[[[.,[.,[.,[[.,.],.]]]],.],.] => 21
[[[.,[.,[[.,.],[.,.]]]],.],.] => 18
[[[.,[.,[[.,[.,.]],.]]],.],.] => 21
[[[.,[.,[[[.,.],.],.]]],.],.] => 21
[[[.,[[.,.],[.,[.,.]]]],.],.] => 17
[[[.,[[.,.],[[.,.],.]]],.],.] => 17
[[[.,[[.,[.,.]],[.,.]]],.],.] => 17
[[[.,[[[.,.],.],[.,.]]],.],.] => 17
[[[.,[[.,[.,[.,.]]],.]],.],.] => 21
[[[.,[[.,[[.,.],.]],.]],.],.] => 21
[[[.,[[[.,.],[.,.]],.]],.],.] => 18
[[[.,[[[.,[.,.]],.],.]],.],.] => 21
[[[.,[[[[.,.],.],.],.]],.],.] => 21
[[[[.,.],[.,[.,[.,.]]]],.],.] => 16
[[[[.,.],[.,[[.,.],.]]],.],.] => 16
[[[[.,.],[[.,.],[.,.]]],.],.] => 13
[[[[.,.],[[.,[.,.]],.]],.],.] => 16
[[[[.,.],[[[.,.],.],.]],.],.] => 16
[[[[.,[.,.]],[.,[.,.]]],.],.] => 13
[[[[.,[.,.]],[[.,.],.]],.],.] => 13
[[[[[.,.],.],[.,[.,.]]],.],.] => 13
[[[[[.,.],.],[[.,.],.]],.],.] => 13
[[[[.,[.,[.,.]]],[.,.]],.],.] => 16
[[[[.,[[.,.],.]],[.,.]],.],.] => 16
[[[[[.,.],[.,.]],[.,.]],.],.] => 13
[[[[[.,[.,.]],.],[.,.]],.],.] => 16
[[[[[[.,.],.],.],[.,.]],.],.] => 16
[[[[.,[.,[.,[.,.]]]],.],.],.] => 21
[[[[.,[.,[[.,.],.]]],.],.],.] => 21
[[[[.,[[.,.],[.,.]]],.],.],.] => 18
[[[[.,[[.,[.,.]],.]],.],.],.] => 21
[[[[.,[[[.,.],.],.]],.],.],.] => 21
[[[[[.,.],[.,[.,.]]],.],.],.] => 17
[[[[[.,.],[[.,.],.]],.],.],.] => 17
[[[[[.,[.,.]],[.,.]],.],.],.] => 17
[[[[[[.,.],.],[.,.]],.],.],.] => 17
[[[[[.,[.,[.,.]]],.],.],.],.] => 21
[[[[[.,[[.,.],.]],.],.],.],.] => 21
[[[[[[.,.],[.,.]],.],.],.],.] => 18
[[[[[[.,[.,.]],.],.],.],.],.] => 21
[[[[[[[.,.],.],.],.],.],.],.] => 21
[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]] => 28
[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]] => 28
[.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]] => 25
[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]] => 28
[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]] => 28
[.,[.,[.,[.,[[.,.],[.,[.,.]]]]]]] => 24
[.,[.,[.,[.,[[.,.],[[.,.],.]]]]]] => 24
[.,[.,[.,[.,[[.,[.,.]],[.,.]]]]]] => 24
[.,[.,[.,[.,[[[.,.],.],[.,.]]]]]] => 24
[.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]] => 28
[.,[.,[.,[.,[[.,[[.,.],.]],.]]]]] => 28
[.,[.,[.,[.,[[[.,.],[.,.]],.]]]]] => 25
[.,[.,[.,[.,[[[.,[.,.]],.],.]]]]] => 28
[.,[.,[.,[.,[[[[.,.],.],.],.]]]]] => 28
[.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]] => 23
[.,[.,[.,[[.,.],[.,[[.,.],.]]]]]] => 23
[.,[.,[.,[[.,.],[[.,.],[.,.]]]]]] => 20
[.,[.,[.,[[.,.],[[.,[.,.]],.]]]]] => 23
[.,[.,[.,[[.,.],[[[.,.],.],.]]]]] => 23
[.,[.,[.,[[.,[.,.]],[.,[.,.]]]]]] => 20
[.,[.,[.,[[.,[.,.]],[[.,.],.]]]]] => 20
[.,[.,[.,[[[.,.],.],[.,[.,.]]]]]] => 20
[.,[.,[.,[[[.,.],.],[[.,.],.]]]]] => 20
[.,[.,[.,[[.,[.,[.,.]]],[.,.]]]]] => 23
[.,[.,[.,[[.,[[.,.],.]],[.,.]]]]] => 23
[.,[.,[.,[[[.,.],[.,.]],[.,.]]]]] => 20
[.,[.,[.,[[[.,[.,.]],.],[.,.]]]]] => 23
[.,[.,[.,[[[[.,.],.],.],[.,.]]]]] => 23
[.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]] => 28
[.,[.,[.,[[.,[.,[[.,.],.]]],.]]]] => 28
[.,[.,[.,[[.,[[.,.],[.,.]]],.]]]] => 25
[.,[.,[.,[[.,[[.,[.,.]],.]],.]]]] => 28
[.,[.,[.,[[.,[[[.,.],.],.]],.]]]] => 28
[.,[.,[.,[[[.,.],[.,[.,.]]],.]]]] => 24
[.,[.,[.,[[[.,.],[[.,.],.]],.]]]] => 24
[.,[.,[.,[[[.,[.,.]],[.,.]],.]]]] => 24
[.,[.,[.,[[[[.,.],.],[.,.]],.]]]] => 24
[.,[.,[.,[[[.,[.,[.,.]]],.],.]]]] => 28
[.,[.,[.,[[[.,[[.,.],.]],.],.]]]] => 28
[.,[.,[.,[[[[.,.],[.,.]],.],.]]]] => 25
[.,[.,[.,[[[[.,[.,.]],.],.],.]]]] => 28
[.,[.,[.,[[[[[.,.],.],.],.],.]]]] => 28
[.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]] => 22
[.,[.,[[.,.],[.,[.,[[.,.],.]]]]]] => 22
[.,[.,[[.,.],[.,[[.,.],[.,.]]]]]] => 19
[.,[.,[[.,.],[.,[[.,[.,.]],.]]]]] => 22
[.,[.,[[.,.],[.,[[[.,.],.],.]]]]] => 22
[.,[.,[[.,.],[[.,.],[.,[.,.]]]]]] => 18
[.,[.,[[.,.],[[.,.],[[.,.],.]]]]] => 18
[.,[.,[[.,.],[[.,[.,.]],[.,.]]]]] => 18
[.,[.,[[.,.],[[[.,.],.],[.,.]]]]] => 18
[.,[.,[[.,.],[[.,[.,[.,.]]],.]]]] => 22
[.,[.,[[.,.],[[.,[[.,.],.]],.]]]] => 22
[.,[.,[[.,.],[[[.,.],[.,.]],.]]]] => 19
[.,[.,[[.,.],[[[.,[.,.]],.],.]]]] => 22
[.,[.,[[.,.],[[[[.,.],.],.],.]]]] => 22
[.,[.,[[.,[.,.]],[.,[.,[.,.]]]]]] => 18
[.,[.,[[.,[.,.]],[.,[[.,.],.]]]]] => 18
[.,[.,[[.,[.,.]],[[.,.],[.,.]]]]] => 15
[.,[.,[[.,[.,.]],[[.,[.,.]],.]]]] => 18
[.,[.,[[.,[.,.]],[[[.,.],.],.]]]] => 18
[.,[.,[[[.,.],.],[.,[.,[.,.]]]]]] => 18
[.,[.,[[[.,.],.],[.,[[.,.],.]]]]] => 18
[.,[.,[[[.,.],.],[[.,.],[.,.]]]]] => 15
[.,[.,[[[.,.],.],[[.,[.,.]],.]]]] => 18
[.,[.,[[[.,.],.],[[[.,.],.],.]]]] => 18
[.,[.,[[.,[.,[.,.]]],[.,[.,.]]]]] => 18
[.,[.,[[.,[.,[.,.]]],[[.,.],.]]]] => 18
[.,[.,[[.,[[.,.],.]],[.,[.,.]]]]] => 18
[.,[.,[[.,[[.,.],.]],[[.,.],.]]]] => 18
[.,[.,[[[.,.],[.,.]],[.,[.,.]]]]] => 15
[.,[.,[[[.,.],[.,.]],[[.,.],.]]]] => 15
[.,[.,[[[.,[.,.]],.],[.,[.,.]]]]] => 18
[.,[.,[[[.,[.,.]],.],[[.,.],.]]]] => 18
[.,[.,[[[[.,.],.],.],[.,[.,.]]]]] => 18
[.,[.,[[[[.,.],.],.],[[.,.],.]]]] => 18
[.,[.,[[.,[.,[.,[.,.]]]],[.,.]]]] => 22
[.,[.,[[.,[.,[[.,.],.]]],[.,.]]]] => 22
[.,[.,[[.,[[.,.],[.,.]]],[.,.]]]] => 19
[.,[.,[[.,[[.,[.,.]],.]],[.,.]]]] => 22
[.,[.,[[.,[[[.,.],.],.]],[.,.]]]] => 22
[.,[.,[[[.,.],[.,[.,.]]],[.,.]]]] => 18
[.,[.,[[[.,.],[[.,.],.]],[.,.]]]] => 18
[.,[.,[[[.,[.,.]],[.,.]],[.,.]]]] => 18
[.,[.,[[[[.,.],.],[.,.]],[.,.]]]] => 18
[.,[.,[[[.,[.,[.,.]]],.],[.,.]]]] => 22
[.,[.,[[[.,[[.,.],.]],.],[.,.]]]] => 22
[.,[.,[[[[.,.],[.,.]],.],[.,.]]]] => 19
[.,[.,[[[[.,[.,.]],.],.],[.,.]]]] => 22
[.,[.,[[[[[.,.],.],.],.],[.,.]]]] => 22
[.,[.,[[.,[.,[.,[.,[.,.]]]]],.]]] => 28
[.,[.,[[.,[.,[.,[[.,.],.]]]],.]]] => 28
[.,[.,[[.,[.,[[.,.],[.,.]]]],.]]] => 25
[.,[.,[[.,[.,[[.,[.,.]],.]]],.]]] => 28
[.,[.,[[.,[.,[[[.,.],.],.]]],.]]] => 28
[.,[.,[[.,[[.,.],[.,[.,.]]]],.]]] => 24
[.,[.,[[.,[[.,.],[[.,.],.]]],.]]] => 24
[.,[.,[[.,[[.,[.,.]],[.,.]]],.]]] => 24
[.,[.,[[.,[[[.,.],.],[.,.]]],.]]] => 24
[.,[.,[[.,[[.,[.,[.,.]]],.]],.]]] => 28
[.,[.,[[.,[[.,[[.,.],.]],.]],.]]] => 28
[.,[.,[[.,[[[.,.],[.,.]],.]],.]]] => 25
[.,[.,[[.,[[[.,[.,.]],.],.]],.]]] => 28
[.,[.,[[.,[[[[.,.],.],.],.]],.]]] => 28
[.,[.,[[[.,.],[.,[.,[.,.]]]],.]]] => 23
[.,[.,[[[.,.],[.,[[.,.],.]]],.]]] => 23
[.,[.,[[[.,.],[[.,.],[.,.]]],.]]] => 20
[.,[.,[[[.,.],[[.,[.,.]],.]],.]]] => 23
[.,[.,[[[.,.],[[[.,.],.],.]],.]]] => 23
[.,[.,[[[.,[.,.]],[.,[.,.]]],.]]] => 20
[.,[.,[[[.,[.,.]],[[.,.],.]],.]]] => 20
[.,[.,[[[[.,.],.],[.,[.,.]]],.]]] => 20
[.,[.,[[[[.,.],.],[[.,.],.]],.]]] => 20
[.,[.,[[[.,[.,[.,.]]],[.,.]],.]]] => 23
[.,[.,[[[.,[[.,.],.]],[.,.]],.]]] => 23
[.,[.,[[[[.,.],[.,.]],[.,.]],.]]] => 20
[.,[.,[[[[.,[.,.]],.],[.,.]],.]]] => 23
[.,[.,[[[[[.,.],.],.],[.,.]],.]]] => 23
[.,[.,[[[.,[.,[.,[.,.]]]],.],.]]] => 28
[.,[.,[[[.,[.,[[.,.],.]]],.],.]]] => 28
[.,[.,[[[.,[[.,.],[.,.]]],.],.]]] => 25
[.,[.,[[[.,[[.,[.,.]],.]],.],.]]] => 28
[.,[.,[[[.,[[[.,.],.],.]],.],.]]] => 28
[.,[.,[[[[.,.],[.,[.,.]]],.],.]]] => 24
[.,[.,[[[[.,.],[[.,.],.]],.],.]]] => 24
[.,[.,[[[[.,[.,.]],[.,.]],.],.]]] => 24
[.,[.,[[[[[.,.],.],[.,.]],.],.]]] => 24
[.,[.,[[[[.,[.,[.,.]]],.],.],.]]] => 28
[.,[.,[[[[.,[[.,.],.]],.],.],.]]] => 28
[.,[.,[[[[[.,.],[.,.]],.],.],.]]] => 25
[.,[.,[[[[[.,[.,.]],.],.],.],.]]] => 28
[.,[.,[[[[[[.,.],.],.],.],.],.]]] => 28
[.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]] => 21
[.,[[.,.],[.,[.,[.,[[.,.],.]]]]]] => 21
[.,[[.,.],[.,[.,[[.,.],[.,.]]]]]] => 18
[.,[[.,.],[.,[.,[[.,[.,.]],.]]]]] => 21
[.,[[.,.],[.,[.,[[[.,.],.],.]]]]] => 21
[.,[[.,.],[.,[[.,.],[.,[.,.]]]]]] => 17
[.,[[.,.],[.,[[.,.],[[.,.],.]]]]] => 17
[.,[[.,.],[.,[[.,[.,.]],[.,.]]]]] => 17
[.,[[.,.],[.,[[[.,.],.],[.,.]]]]] => 17
[.,[[.,.],[.,[[.,[.,[.,.]]],.]]]] => 21
[.,[[.,.],[.,[[.,[[.,.],.]],.]]]] => 21
[.,[[.,.],[.,[[[.,.],[.,.]],.]]]] => 18
[.,[[.,.],[.,[[[.,[.,.]],.],.]]]] => 21
[.,[[.,.],[.,[[[[.,.],.],.],.]]]] => 21
[.,[[.,.],[[.,.],[.,[.,[.,.]]]]]] => 16
[.,[[.,.],[[.,.],[.,[[.,.],.]]]]] => 16
[.,[[.,.],[[.,.],[[.,.],[.,.]]]]] => 13
[.,[[.,.],[[.,.],[[.,[.,.]],.]]]] => 16
[.,[[.,.],[[.,.],[[[.,.],.],.]]]] => 16
[.,[[.,.],[[.,[.,.]],[.,[.,.]]]]] => 13
[.,[[.,.],[[.,[.,.]],[[.,.],.]]]] => 13
[.,[[.,.],[[[.,.],.],[.,[.,.]]]]] => 13
[.,[[.,.],[[[.,.],.],[[.,.],.]]]] => 13
[.,[[.,.],[[.,[.,[.,.]]],[.,.]]]] => 16
[.,[[.,.],[[.,[[.,.],.]],[.,.]]]] => 16
[.,[[.,.],[[[.,.],[.,.]],[.,.]]]] => 13
[.,[[.,.],[[[.,[.,.]],.],[.,.]]]] => 16
[.,[[.,.],[[[[.,.],.],.],[.,.]]]] => 16
[.,[[.,.],[[.,[.,[.,[.,.]]]],.]]] => 21
[.,[[.,.],[[.,[.,[[.,.],.]]],.]]] => 21
[.,[[.,.],[[.,[[.,.],[.,.]]],.]]] => 18
[.,[[.,.],[[.,[[.,[.,.]],.]],.]]] => 21
[.,[[.,.],[[.,[[[.,.],.],.]],.]]] => 21
[.,[[.,.],[[[.,.],[.,[.,.]]],.]]] => 17
[.,[[.,.],[[[.,.],[[.,.],.]],.]]] => 17
[.,[[.,.],[[[.,[.,.]],[.,.]],.]]] => 17
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[[.,.],[[.,[.,.]],[.,[[.,.],.]]]] => 10
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[[.,.],[[[.,.],.],[[[.,.],.],.]]] => 10
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[[.,.],[[.,[[.,.],.]],[[.,.],.]]] => 10
[[.,.],[[[.,.],[.,.]],[.,[.,.]]]] => 7
[[.,.],[[[.,.],[.,.]],[[.,.],.]]] => 7
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[[.,.],[[[.,[.,.]],.],[[.,.],.]]] => 10
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[[.,.],[[.,[.,[.,[.,.]]]],[.,.]]] => 14
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[[.,.],[[.,[[.,.],[.,.]]],[.,.]]] => 11
[[.,.],[[.,[[.,[.,.]],.]],[.,.]]] => 14
[[.,.],[[.,[[[.,.],.],.]],[.,.]]] => 14
[[.,.],[[[.,.],[.,[.,.]]],[.,.]]] => 10
[[.,.],[[[.,.],[[.,.],.]],[.,.]]] => 10
[[.,.],[[[.,[.,.]],[.,.]],[.,.]]] => 10
[[.,.],[[[[.,.],.],[.,.]],[.,.]]] => 10
[[.,.],[[[.,[.,[.,.]]],.],[.,.]]] => 14
[[.,.],[[[.,[[.,.],.]],.],[.,.]]] => 14
[[.,.],[[[[.,.],[.,.]],.],[.,.]]] => 11
[[.,.],[[[[.,[.,.]],.],.],[.,.]]] => 14
[[.,.],[[[[[.,.],.],.],.],[.,.]]] => 14
[[.,.],[[.,[.,[.,[.,[.,.]]]]],.]] => 20
[[.,.],[[.,[.,[.,[[.,.],.]]]],.]] => 20
[[.,.],[[.,[.,[[.,.],[.,.]]]],.]] => 17
[[.,.],[[.,[.,[[.,[.,.]],.]]],.]] => 20
[[.,.],[[.,[.,[[[.,.],.],.]]],.]] => 20
[[.,.],[[.,[[.,.],[.,[.,.]]]],.]] => 16
[[.,.],[[.,[[.,.],[[.,.],.]]],.]] => 16
[[.,.],[[.,[[.,[.,.]],[.,.]]],.]] => 16
[[.,.],[[.,[[[.,.],.],[.,.]]],.]] => 16
[[.,.],[[.,[[.,[.,[.,.]]],.]],.]] => 20
[[.,.],[[.,[[.,[[.,.],.]],.]],.]] => 20
[[.,.],[[.,[[[.,.],[.,.]],.]],.]] => 17
[[.,.],[[.,[[[.,[.,.]],.],.]],.]] => 20
[[.,.],[[.,[[[[.,.],.],.],.]],.]] => 20
[[.,.],[[[.,.],[.,[.,[.,.]]]],.]] => 15
[[.,.],[[[.,.],[.,[[.,.],.]]],.]] => 15
[[.,.],[[[.,.],[[.,.],[.,.]]],.]] => 12
[[.,.],[[[.,.],[[.,[.,.]],.]],.]] => 15
[[.,.],[[[.,.],[[[.,.],.],.]],.]] => 15
[[.,.],[[[.,[.,.]],[.,[.,.]]],.]] => 12
[[.,.],[[[.,[.,.]],[[.,.],.]],.]] => 12
[[.,.],[[[[.,.],.],[.,[.,.]]],.]] => 12
[[.,.],[[[[.,.],.],[[.,.],.]],.]] => 12
[[.,.],[[[.,[.,[.,.]]],[.,.]],.]] => 15
[[.,.],[[[.,[[.,.],.]],[.,.]],.]] => 15
[[.,.],[[[[.,.],[.,.]],[.,.]],.]] => 12
[[.,.],[[[[.,[.,.]],.],[.,.]],.]] => 15
[[.,.],[[[[[.,.],.],.],[.,.]],.]] => 15
[[.,.],[[[.,[.,[.,[.,.]]]],.],.]] => 20
[[.,.],[[[.,[.,[[.,.],.]]],.],.]] => 20
[[.,.],[[[.,[[.,.],[.,.]]],.],.]] => 17
[[.,.],[[[.,[[.,[.,.]],.]],.],.]] => 20
[[.,.],[[[.,[[[.,.],.],.]],.],.]] => 20
[[.,.],[[[[.,.],[.,[.,.]]],.],.]] => 16
[[.,.],[[[[.,.],[[.,.],.]],.],.]] => 16
[[.,.],[[[[.,[.,.]],[.,.]],.],.]] => 16
[[.,.],[[[[[.,.],.],[.,.]],.],.]] => 16
[[.,.],[[[[.,[.,[.,.]]],.],.],.]] => 20
[[.,.],[[[[.,[[.,.],.]],.],.],.]] => 20
[[.,.],[[[[[.,.],[.,.]],.],.],.]] => 17
[[.,.],[[[[[.,[.,.]],.],.],.],.]] => 20
[[.,.],[[[[[[.,.],.],.],.],.],.]] => 20
[[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]] => 14
[[.,[.,.]],[.,[.,[.,[[.,.],.]]]]] => 14
[[.,[.,.]],[.,[.,[[.,.],[.,.]]]]] => 11
[[.,[.,.]],[.,[.,[[.,[.,.]],.]]]] => 14
[[.,[.,.]],[.,[.,[[[.,.],.],.]]]] => 14
[[.,[.,.]],[.,[[.,.],[.,[.,.]]]]] => 10
[[.,[.,.]],[.,[[.,.],[[.,.],.]]]] => 10
[[.,[.,.]],[.,[[.,[.,.]],[.,.]]]] => 10
[[.,[.,.]],[.,[[[.,.],.],[.,.]]]] => 10
[[.,[.,.]],[.,[[.,[.,[.,.]]],.]]] => 14
[[.,[.,.]],[.,[[.,[[.,.],.]],.]]] => 14
[[.,[.,.]],[.,[[[.,.],[.,.]],.]]] => 11
[[.,[.,.]],[.,[[[.,[.,.]],.],.]]] => 14
[[.,[.,.]],[.,[[[[.,.],.],.],.]]] => 14
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Generating function
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Search the OEIS for these generating functions
Search the Online Encyclopedia of Integer
Sequences for the coefficients of a few of the
first generating functions, in the case at hand:
1,0,0,4 4,2,0,0,8 6,0,0,8,8,4,0,0,16 4,0,0,24,4,12,0,16,16,16,8,0,0,32 1,0,0,8,0,16,36,0,8,48,16,56,8,24,32,32,32,32,16,0,0,64
$F_{1} = 1$
$F_{2} = 2\ q$
$F_{3} = 1 + 4\ q^{3}$
$F_{4} = 4\ q^{2} + 2\ q^{3} + 8\ q^{6}$
$F_{5} = 6\ q^{2} + 8\ q^{5} + 8\ q^{6} + 4\ q^{7} + 16\ q^{10}$
$F_{6} = 4\ q^{2} + 24\ q^{5} + 4\ q^{6} + 12\ q^{7} + 16\ q^{9} + 16\ q^{10} + 16\ q^{11} + 8\ q^{12} + 32\ q^{15}$
$F_{7} = 1 + 8\ q^{3} + 16\ q^{5} + 36\ q^{6} + 8\ q^{8} + 48\ q^{9} + 16\ q^{10} + 56\ q^{11} + 8\ q^{12} + 24\ q^{13} + 32\ q^{14} + 32\ q^{15} + 32\ q^{16} + 32\ q^{17} + 16\ q^{18} + 64\ q^{21}$
Description
The Colless index of a binary tree.
The Colless index is the sum of the absolute values of the differences of the number of leaves in the left and the right subtree over all internal nodes.
The Colless index is the sum of the absolute values of the differences of the number of leaves in the left and the right subtree over all internal nodes.
References
[1] Coronado, T. M., Rosselló, F. The minimum value of the Colless index arXiv:1903.11670
Code
def statistic(B):
return sum(abs(t[0].node_number()-t[1].node_number()) for t in B.subtrees())
Created
Mar 29, 2019 at 10:03 by Martin Rubey
Updated
Mar 29, 2019 at 10:03 by Martin Rubey
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