Identifier
Values
[.,.] => [[]] => 0
[.,[.,.]] => [[[]]] => 1
[[.,.],.] => [[],[]] => 0
[.,[.,[.,.]]] => [[[[]]]] => 2
[.,[[.,.],.]] => [[[],[]]] => 1
[[.,.],[.,.]] => [[],[[]]] => 1
[[.,[.,.]],.] => [[[]],[]] => 1
[[[.,.],.],.] => [[],[],[]] => 0
[.,[.,[.,[.,.]]]] => [[[[[]]]]] => 3
[.,[.,[[.,.],.]]] => [[[[],[]]]] => 2
[.,[[.,.],[.,.]]] => [[[],[[]]]] => 2
[.,[[.,[.,.]],.]] => [[[[]],[]]] => 2
[.,[[[.,.],.],.]] => [[[],[],[]]] => 1
[[.,.],[.,[.,.]]] => [[],[[[]]]] => 2
[[.,.],[[.,.],.]] => [[],[[],[]]] => 1
[[.,[.,.]],[.,.]] => [[[]],[[]]] => 2
[[[.,.],.],[.,.]] => [[],[],[[]]] => 1
[[.,[.,[.,.]]],.] => [[[[]]],[]] => 2
[[.,[[.,.],.]],.] => [[[],[]],[]] => 1
[[[.,.],[.,.]],.] => [[],[[]],[]] => 1
[[[.,[.,.]],.],.] => [[[]],[],[]] => 1
[[[[.,.],.],.],.] => [[],[],[],[]] => 0
[.,[.,[.,[.,[.,.]]]]] => [[[[[[]]]]]] => 4
[.,[.,[.,[[.,.],.]]]] => [[[[[],[]]]]] => 3
[.,[.,[[.,.],[.,.]]]] => [[[[],[[]]]]] => 3
[.,[.,[[.,[.,.]],.]]] => [[[[[]],[]]]] => 3
[.,[.,[[[.,.],.],.]]] => [[[[],[],[]]]] => 2
[.,[[.,.],[.,[.,.]]]] => [[[],[[[]]]]] => 3
[.,[[.,.],[[.,.],.]]] => [[[],[[],[]]]] => 2
[.,[[.,[.,.]],[.,.]]] => [[[[]],[[]]]] => 3
[.,[[[.,.],.],[.,.]]] => [[[],[],[[]]]] => 2
[.,[[.,[.,[.,.]]],.]] => [[[[[]]],[]]] => 3
[.,[[.,[[.,.],.]],.]] => [[[[],[]],[]]] => 2
[.,[[[.,.],[.,.]],.]] => [[[],[[]],[]]] => 2
[.,[[[.,[.,.]],.],.]] => [[[[]],[],[]]] => 2
[.,[[[[.,.],.],.],.]] => [[[],[],[],[]]] => 1
[[.,.],[.,[.,[.,.]]]] => [[],[[[[]]]]] => 3
[[.,.],[.,[[.,.],.]]] => [[],[[[],[]]]] => 2
[[.,.],[[.,.],[.,.]]] => [[],[[],[[]]]] => 2
[[.,.],[[.,[.,.]],.]] => [[],[[[]],[]]] => 2
[[.,.],[[[.,.],.],.]] => [[],[[],[],[]]] => 1
[[.,[.,.]],[.,[.,.]]] => [[[]],[[[]]]] => 3
[[.,[.,.]],[[.,.],.]] => [[[]],[[],[]]] => 2
[[[.,.],.],[.,[.,.]]] => [[],[],[[[]]]] => 2
[[[.,.],.],[[.,.],.]] => [[],[],[[],[]]] => 1
[[.,[.,[.,.]]],[.,.]] => [[[[]]],[[]]] => 3
[[.,[[.,.],.]],[.,.]] => [[[],[]],[[]]] => 2
[[[.,.],[.,.]],[.,.]] => [[],[[]],[[]]] => 2
[[[.,[.,.]],.],[.,.]] => [[[]],[],[[]]] => 2
[[[[.,.],.],.],[.,.]] => [[],[],[],[[]]] => 1
[[.,[.,[.,[.,.]]]],.] => [[[[[]]]],[]] => 3
[[.,[.,[[.,.],.]]],.] => [[[[],[]]],[]] => 2
[[.,[[.,.],[.,.]]],.] => [[[],[[]]],[]] => 2
[[.,[[.,[.,.]],.]],.] => [[[[]],[]],[]] => 2
[[.,[[[.,.],.],.]],.] => [[[],[],[]],[]] => 1
[[[.,.],[.,[.,.]]],.] => [[],[[[]]],[]] => 2
[[[.,.],[[.,.],.]],.] => [[],[[],[]],[]] => 1
[[[.,[.,.]],[.,.]],.] => [[[]],[[]],[]] => 2
[[[[.,.],.],[.,.]],.] => [[],[],[[]],[]] => 1
[[[.,[.,[.,.]]],.],.] => [[[[]]],[],[]] => 2
[[[.,[[.,.],.]],.],.] => [[[],[]],[],[]] => 1
[[[[.,.],[.,.]],.],.] => [[],[[]],[],[]] => 1
[[[[.,[.,.]],.],.],.] => [[[]],[],[],[]] => 1
[[[[[.,.],.],.],.],.] => [[],[],[],[],[]] => 0
[.,[.,[.,[.,[.,[.,.]]]]]] => [[[[[[[]]]]]]] => 5
[.,[.,[.,[.,[[.,.],.]]]]] => [[[[[[],[]]]]]] => 4
[.,[.,[.,[[.,.],[.,.]]]]] => [[[[[],[[]]]]]] => 4
[.,[.,[.,[[.,[.,.]],.]]]] => [[[[[[]],[]]]]] => 4
[.,[.,[.,[[[.,.],.],.]]]] => [[[[[],[],[]]]]] => 3
[.,[.,[[.,.],[.,[.,.]]]]] => [[[[],[[[]]]]]] => 4
[.,[.,[[.,.],[[.,.],.]]]] => [[[[],[[],[]]]]] => 3
[.,[.,[[.,[.,.]],[.,.]]]] => [[[[[]],[[]]]]] => 4
[.,[.,[[[.,.],.],[.,.]]]] => [[[[],[],[[]]]]] => 3
[.,[.,[[.,[.,[.,.]]],.]]] => [[[[[[]]],[]]]] => 4
[.,[.,[[.,[[.,.],.]],.]]] => [[[[[],[]],[]]]] => 3
[.,[.,[[[.,.],[.,.]],.]]] => [[[[],[[]],[]]]] => 3
[.,[.,[[[.,[.,.]],.],.]]] => [[[[[]],[],[]]]] => 3
[.,[.,[[[[.,.],.],.],.]]] => [[[[],[],[],[]]]] => 2
[.,[[.,.],[.,[.,[.,.]]]]] => [[[],[[[[]]]]]] => 4
[.,[[.,.],[.,[[.,.],.]]]] => [[[],[[[],[]]]]] => 3
[.,[[.,.],[[.,.],[.,.]]]] => [[[],[[],[[]]]]] => 3
[.,[[.,.],[[.,[.,.]],.]]] => [[[],[[[]],[]]]] => 3
[.,[[.,.],[[[.,.],.],.]]] => [[[],[[],[],[]]]] => 2
[.,[[.,[.,.]],[.,[.,.]]]] => [[[[]],[[[]]]]] => 4
[.,[[.,[.,.]],[[.,.],.]]] => [[[[]],[[],[]]]] => 3
[.,[[[.,.],.],[.,[.,.]]]] => [[[],[],[[[]]]]] => 3
[.,[[[.,.],.],[[.,.],.]]] => [[[],[],[[],[]]]] => 2
[.,[[.,[.,[.,.]]],[.,.]]] => [[[[[]]],[[]]]] => 4
[.,[[.,[[.,.],.]],[.,.]]] => [[[[],[]],[[]]]] => 3
[.,[[[.,.],[.,.]],[.,.]]] => [[[],[[]],[[]]]] => 3
[.,[[[.,[.,.]],.],[.,.]]] => [[[[]],[],[[]]]] => 3
[.,[[[[.,.],.],.],[.,.]]] => [[[],[],[],[[]]]] => 2
[.,[[.,[.,[.,[.,.]]]],.]] => [[[[[[]]]],[]]] => 4
[.,[[.,[.,[[.,.],.]]],.]] => [[[[[],[]]],[]]] => 3
[.,[[.,[[.,.],[.,.]]],.]] => [[[[],[[]]],[]]] => 3
[.,[[.,[[.,[.,.]],.]],.]] => [[[[[]],[]],[]]] => 3
[.,[[.,[[[.,.],.],.]],.]] => [[[[],[],[]],[]]] => 2
[.,[[[.,.],[.,[.,.]]],.]] => [[[],[[[]]],[]]] => 3
[.,[[[.,.],[[.,.],.]],.]] => [[[],[[],[]],[]]] => 2
[.,[[[.,[.,.]],[.,.]],.]] => [[[[]],[[]],[]]] => 3
[.,[[[[.,.],.],[.,.]],.]] => [[[],[],[[]],[]]] => 2
>>> Load all 196 entries. <<<
[.,[[[.,[.,[.,.]]],.],.]] => [[[[[]]],[],[]]] => 3
[.,[[[.,[[.,.],.]],.],.]] => [[[[],[]],[],[]]] => 2
[.,[[[[.,.],[.,.]],.],.]] => [[[],[[]],[],[]]] => 2
[.,[[[[.,[.,.]],.],.],.]] => [[[[]],[],[],[]]] => 2
[.,[[[[[.,.],.],.],.],.]] => [[[],[],[],[],[]]] => 1
[[.,.],[.,[.,[.,[.,.]]]]] => [[],[[[[[]]]]]] => 4
[[.,.],[.,[.,[[.,.],.]]]] => [[],[[[[],[]]]]] => 3
[[.,.],[.,[[.,.],[.,.]]]] => [[],[[[],[[]]]]] => 3
[[.,.],[.,[[.,[.,.]],.]]] => [[],[[[[]],[]]]] => 3
[[.,.],[.,[[[.,.],.],.]]] => [[],[[[],[],[]]]] => 2
[[.,.],[[.,.],[.,[.,.]]]] => [[],[[],[[[]]]]] => 3
[[.,.],[[.,.],[[.,.],.]]] => [[],[[],[[],[]]]] => 2
[[.,.],[[.,[.,.]],[.,.]]] => [[],[[[]],[[]]]] => 3
[[.,.],[[[.,.],.],[.,.]]] => [[],[[],[],[[]]]] => 2
[[.,.],[[.,[.,[.,.]]],.]] => [[],[[[[]]],[]]] => 3
[[.,.],[[.,[[.,.],.]],.]] => [[],[[[],[]],[]]] => 2
[[.,.],[[[.,.],[.,.]],.]] => [[],[[],[[]],[]]] => 2
[[.,.],[[[.,[.,.]],.],.]] => [[],[[[]],[],[]]] => 2
[[.,.],[[[[.,.],.],.],.]] => [[],[[],[],[],[]]] => 1
[[.,[.,.]],[.,[.,[.,.]]]] => [[[]],[[[[]]]]] => 4
[[.,[.,.]],[.,[[.,.],.]]] => [[[]],[[[],[]]]] => 3
[[.,[.,.]],[[.,.],[.,.]]] => [[[]],[[],[[]]]] => 3
[[.,[.,.]],[[.,[.,.]],.]] => [[[]],[[[]],[]]] => 3
[[.,[.,.]],[[[.,.],.],.]] => [[[]],[[],[],[]]] => 2
[[[.,.],.],[.,[.,[.,.]]]] => [[],[],[[[[]]]]] => 3
[[[.,.],.],[.,[[.,.],.]]] => [[],[],[[[],[]]]] => 2
[[[.,.],.],[[.,.],[.,.]]] => [[],[],[[],[[]]]] => 2
[[[.,.],.],[[.,[.,.]],.]] => [[],[],[[[]],[]]] => 2
[[[.,.],.],[[[.,.],.],.]] => [[],[],[[],[],[]]] => 1
[[.,[.,[.,.]]],[.,[.,.]]] => [[[[]]],[[[]]]] => 4
[[.,[.,[.,.]]],[[.,.],.]] => [[[[]]],[[],[]]] => 3
[[.,[[.,.],.]],[.,[.,.]]] => [[[],[]],[[[]]]] => 3
[[.,[[.,.],.]],[[.,.],.]] => [[[],[]],[[],[]]] => 2
[[[.,.],[.,.]],[.,[.,.]]] => [[],[[]],[[[]]]] => 3
[[[.,.],[.,.]],[[.,.],.]] => [[],[[]],[[],[]]] => 2
[[[.,[.,.]],.],[.,[.,.]]] => [[[]],[],[[[]]]] => 3
[[[.,[.,.]],.],[[.,.],.]] => [[[]],[],[[],[]]] => 2
[[[[.,.],.],.],[.,[.,.]]] => [[],[],[],[[[]]]] => 2
[[[[.,.],.],.],[[.,.],.]] => [[],[],[],[[],[]]] => 1
[[.,[.,[.,[.,.]]]],[.,.]] => [[[[[]]]],[[]]] => 4
[[.,[.,[[.,.],.]]],[.,.]] => [[[[],[]]],[[]]] => 3
[[.,[[.,.],[.,.]]],[.,.]] => [[[],[[]]],[[]]] => 3
[[.,[[.,[.,.]],.]],[.,.]] => [[[[]],[]],[[]]] => 3
[[.,[[[.,.],.],.]],[.,.]] => [[[],[],[]],[[]]] => 2
[[[.,.],[.,[.,.]]],[.,.]] => [[],[[[]]],[[]]] => 3
[[[.,.],[[.,.],.]],[.,.]] => [[],[[],[]],[[]]] => 2
[[[.,[.,.]],[.,.]],[.,.]] => [[[]],[[]],[[]]] => 3
[[[[.,.],.],[.,.]],[.,.]] => [[],[],[[]],[[]]] => 2
[[[.,[.,[.,.]]],.],[.,.]] => [[[[]]],[],[[]]] => 3
[[[.,[[.,.],.]],.],[.,.]] => [[[],[]],[],[[]]] => 2
[[[[.,.],[.,.]],.],[.,.]] => [[],[[]],[],[[]]] => 2
[[[[.,[.,.]],.],.],[.,.]] => [[[]],[],[],[[]]] => 2
[[[[[.,.],.],.],.],[.,.]] => [[],[],[],[],[[]]] => 1
[[.,[.,[.,[.,[.,.]]]]],.] => [[[[[[]]]]],[]] => 4
[[.,[.,[.,[[.,.],.]]]],.] => [[[[[],[]]]],[]] => 3
[[.,[.,[[.,.],[.,.]]]],.] => [[[[],[[]]]],[]] => 3
[[.,[.,[[.,[.,.]],.]]],.] => [[[[[]],[]]],[]] => 3
[[.,[.,[[[.,.],.],.]]],.] => [[[[],[],[]]],[]] => 2
[[.,[[.,.],[.,[.,.]]]],.] => [[[],[[[]]]],[]] => 3
[[.,[[.,.],[[.,.],.]]],.] => [[[],[[],[]]],[]] => 2
[[.,[[.,[.,.]],[.,.]]],.] => [[[[]],[[]]],[]] => 3
[[.,[[[.,.],.],[.,.]]],.] => [[[],[],[[]]],[]] => 2
[[.,[[.,[.,[.,.]]],.]],.] => [[[[[]]],[]],[]] => 3
[[.,[[.,[[.,.],.]],.]],.] => [[[[],[]],[]],[]] => 2
[[.,[[[.,.],[.,.]],.]],.] => [[[],[[]],[]],[]] => 2
[[.,[[[.,[.,.]],.],.]],.] => [[[[]],[],[]],[]] => 2
[[.,[[[[.,.],.],.],.]],.] => [[[],[],[],[]],[]] => 1
[[[.,.],[.,[.,[.,.]]]],.] => [[],[[[[]]]],[]] => 3
[[[.,.],[.,[[.,.],.]]],.] => [[],[[[],[]]],[]] => 2
[[[.,.],[[.,.],[.,.]]],.] => [[],[[],[[]]],[]] => 2
[[[.,.],[[.,[.,.]],.]],.] => [[],[[[]],[]],[]] => 2
[[[.,.],[[[.,.],.],.]],.] => [[],[[],[],[]],[]] => 1
[[[.,[.,.]],[.,[.,.]]],.] => [[[]],[[[]]],[]] => 3
[[[.,[.,.]],[[.,.],.]],.] => [[[]],[[],[]],[]] => 2
[[[[.,.],.],[.,[.,.]]],.] => [[],[],[[[]]],[]] => 2
[[[[.,.],.],[[.,.],.]],.] => [[],[],[[],[]],[]] => 1
[[[.,[.,[.,.]]],[.,.]],.] => [[[[]]],[[]],[]] => 3
[[[.,[[.,.],.]],[.,.]],.] => [[[],[]],[[]],[]] => 2
[[[[.,.],[.,.]],[.,.]],.] => [[],[[]],[[]],[]] => 2
[[[[.,[.,.]],.],[.,.]],.] => [[[]],[],[[]],[]] => 2
[[[[[.,.],.],.],[.,.]],.] => [[],[],[],[[]],[]] => 1
[[[.,[.,[.,[.,.]]]],.],.] => [[[[[]]]],[],[]] => 3
[[[.,[.,[[.,.],.]]],.],.] => [[[[],[]]],[],[]] => 2
[[[.,[[.,.],[.,.]]],.],.] => [[[],[[]]],[],[]] => 2
[[[.,[[.,[.,.]],.]],.],.] => [[[[]],[]],[],[]] => 2
[[[.,[[[.,.],.],.]],.],.] => [[[],[],[]],[],[]] => 1
[[[[.,.],[.,[.,.]]],.],.] => [[],[[[]]],[],[]] => 2
[[[[.,.],[[.,.],.]],.],.] => [[],[[],[]],[],[]] => 1
[[[[.,[.,.]],[.,.]],.],.] => [[[]],[[]],[],[]] => 2
[[[[[.,.],.],[.,.]],.],.] => [[],[],[[]],[],[]] => 1
[[[[.,[.,[.,.]]],.],.],.] => [[[[]]],[],[],[]] => 2
[[[[.,[[.,.],.]],.],.],.] => [[[],[]],[],[],[]] => 1
[[[[[.,.],[.,.]],.],.],.] => [[],[[]],[],[],[]] => 1
[[[[[.,[.,.]],.],.],.],.] => [[[]],[],[],[],[]] => 1
[[[[[[.,.],.],.],.],.],.] => [[],[],[],[],[],[]] => 0
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Description
The number of internal nodes of an ordered tree.
A node is internal if it is neither the root nor a leaf.
Map
to ordered tree: left child = left brother
Description
Return an ordered tree of size $n+1$ by the following recursive rule:
  • if $x$ is the left child of $y$, $x$ becomes the left brother of $y$,
  • if $x$ is the right child of $y$, $x$ becomes the last child of $y$.