Identifier
-
Mp00010:
Binary trees
—to ordered tree: left child = left brother⟶
Ordered trees
St000168: Ordered trees ⟶ ℤ
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
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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.
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.
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