- FindStat

Identifier

Mp00019: Binary trees —right rotate⟶ Binary trees
Mp00015: Binary trees —to ordered tree: right child = right brother⟶ Ordered trees
Mp00046: Ordered trees —to graph⟶ Graphs
St001692: Graphs ⟶ ℤ

Values

[.,.] => [.,.] => [[]] => ([(0,1)],2) => 0

[.,[.,.]] => [[.,.],.] => [[[]]] => ([(0,2),(1,2)],3) => 1

[[.,.],.] => [.,[.,.]] => [[],[]] => ([(0,2),(1,2)],3) => 1

[.,[.,[.,.]]] => [[.,[.,.]],.] => [[[],[]]] => ([(0,3),(1,3),(2,3)],4) => 1

[.,[[.,.],.]] => [[[.,.],.],.] => [[[[]]]] => ([(0,3),(1,2),(2,3)],4) => 2

[[.,.],[.,.]] => [.,[.,[.,.]]] => [[],[],[]] => ([(0,3),(1,3),(2,3)],4) => 1

[[.,[.,.]],.] => [.,[[.,.],.]] => [[],[[]]] => ([(0,3),(1,2),(2,3)],4) => 2

[[[.,.],.],.] => [[.,.],[.,.]] => [[[]],[]] => ([(0,3),(1,2),(2,3)],4) => 2

[.,[.,[.,[.,.]]]] => [[.,[.,[.,.]]],.] => [[[],[],[]]] => ([(0,4),(1,4),(2,4),(3,4)],5) => 1

[.,[.,[[.,.],.]]] => [[.,[[.,.],.]],.] => [[[],[[]]]] => ([(0,4),(1,4),(2,3),(3,4)],5) => 2

[.,[[.,.],[.,.]]] => [[[.,.],[.,.]],.] => [[[[]],[]]] => ([(0,4),(1,4),(2,3),(3,4)],5) => 2

[.,[[.,[.,.]],.]] => [[[.,[.,.]],.],.] => [[[[],[]]]] => ([(0,4),(1,4),(2,3),(3,4)],5) => 2

[.,[[[.,.],.],.]] => [[[[.,.],.],.],.] => [[[[[]]]]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 3

[[.,.],[.,[.,.]]] => [.,[.,[.,[.,.]]]] => [[],[],[],[]] => ([(0,4),(1,4),(2,4),(3,4)],5) => 1

[[.,.],[[.,.],.]] => [.,[.,[[.,.],.]]] => [[],[],[[]]] => ([(0,4),(1,4),(2,3),(3,4)],5) => 2

[[.,[.,.]],[.,.]] => [.,[[.,.],[.,.]]] => [[],[[]],[]] => ([(0,4),(1,4),(2,3),(3,4)],5) => 2

[[[.,.],.],[.,.]] => [[.,.],[.,[.,.]]] => [[[]],[],[]] => ([(0,4),(1,4),(2,3),(3,4)],5) => 2

[[.,[.,[.,.]]],.] => [.,[[.,[.,.]],.]] => [[],[[],[]]] => ([(0,4),(1,4),(2,3),(3,4)],5) => 2

[[.,[[.,.],.]],.] => [.,[[[.,.],.],.]] => [[],[[[]]]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 3

[[[.,.],[.,.]],.] => [[.,.],[[.,.],.]] => [[[]],[[]]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 3

[[[.,[.,.]],.],.] => [[.,[.,.]],[.,.]] => [[[],[]],[]] => ([(0,4),(1,4),(2,3),(3,4)],5) => 2

[[[[.,.],.],.],.] => [[[.,.],.],[.,.]] => [[[[]]],[]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 3

[.,[.,[.,[.,[.,.]]]]] => [[.,[.,[.,[.,.]]]],.] => [[[],[],[],[]]] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 1

[.,[.,[.,[[.,.],.]]]] => [[.,[.,[[.,.],.]]],.] => [[[],[],[[]]]] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 2

[.,[.,[[.,.],[.,.]]]] => [[.,[[.,.],[.,.]]],.] => [[[],[[]],[]]] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 2

[.,[.,[[.,[.,.]],.]]] => [[.,[[.,[.,.]],.]],.] => [[[],[[],[]]]] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 2

[.,[.,[[[.,.],.],.]]] => [[.,[[[.,.],.],.]],.] => [[[],[[[]]]]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 3

[.,[[.,.],[.,[.,.]]]] => [[[.,.],[.,[.,.]]],.] => [[[[]],[],[]]] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 2

[.,[[.,.],[[.,.],.]]] => [[[.,.],[[.,.],.]],.] => [[[[]],[[]]]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 3

[.,[[.,[.,.]],[.,.]]] => [[[.,[.,.]],[.,.]],.] => [[[[],[]],[]]] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 2

[.,[[[.,.],.],[.,.]]] => [[[[.,.],.],[.,.]],.] => [[[[[]]],[]]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 3

[.,[[.,[.,[.,.]]],.]] => [[[.,[.,[.,.]]],.],.] => [[[[],[],[]]]] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 2

[.,[[.,[[.,.],.]],.]] => [[[.,[[.,.],.]],.],.] => [[[[],[[]]]]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 3

[.,[[[.,.],[.,.]],.]] => [[[[.,.],[.,.]],.],.] => [[[[[]],[]]]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 3

[.,[[[.,[.,.]],.],.]] => [[[[.,[.,.]],.],.],.] => [[[[[],[]]]]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 3

[.,[[[[.,.],.],.],.]] => [[[[[.,.],.],.],.],.] => [[[[[[]]]]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 4

[[.,.],[.,[.,[.,.]]]] => [.,[.,[.,[.,[.,.]]]]] => [[],[],[],[],[]] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 1

[[.,.],[.,[[.,.],.]]] => [.,[.,[.,[[.,.],.]]]] => [[],[],[],[[]]] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 2

[[.,.],[[.,.],[.,.]]] => [.,[.,[[.,.],[.,.]]]] => [[],[],[[]],[]] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 2

[[.,.],[[.,[.,.]],.]] => [.,[.,[[.,[.,.]],.]]] => [[],[],[[],[]]] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 2

[[.,.],[[[.,.],.],.]] => [.,[.,[[[.,.],.],.]]] => [[],[],[[[]]]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 3

[[.,[.,.]],[.,[.,.]]] => [.,[[.,.],[.,[.,.]]]] => [[],[[]],[],[]] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 2

[[.,[.,.]],[[.,.],.]] => [.,[[.,.],[[.,.],.]]] => [[],[[]],[[]]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 3

[[[.,.],.],[.,[.,.]]] => [[.,.],[.,[.,[.,.]]]] => [[[]],[],[],[]] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 2

[[[.,.],.],[[.,.],.]] => [[.,.],[.,[[.,.],.]]] => [[[]],[],[[]]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 3

[[.,[.,[.,.]]],[.,.]] => [.,[[.,[.,.]],[.,.]]] => [[],[[],[]],[]] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 2

[[.,[[.,.],.]],[.,.]] => [.,[[[.,.],.],[.,.]]] => [[],[[[]]],[]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 3

[[[.,.],[.,.]],[.,.]] => [[.,.],[[.,.],[.,.]]] => [[[]],[[]],[]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 3

[[[.,[.,.]],.],[.,.]] => [[.,[.,.]],[.,[.,.]]] => [[[],[]],[],[]] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 2

[[[[.,.],.],.],[.,.]] => [[[.,.],.],[.,[.,.]]] => [[[[]]],[],[]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 3

[[.,[.,[.,[.,.]]]],.] => [.,[[.,[.,[.,.]]],.]] => [[],[[],[],[]]] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 2

[[.,[.,[[.,.],.]]],.] => [.,[[.,[[.,.],.]],.]] => [[],[[],[[]]]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 3

[[.,[[.,.],[.,.]]],.] => [.,[[[.,.],[.,.]],.]] => [[],[[[]],[]]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 3

[[.,[[.,[.,.]],.]],.] => [.,[[[.,[.,.]],.],.]] => [[],[[[],[]]]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 3

[[.,[[[.,.],.],.]],.] => [.,[[[[.,.],.],.],.]] => [[],[[[[]]]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 4

[[[.,.],[.,[.,.]]],.] => [[.,.],[[.,[.,.]],.]] => [[[]],[[],[]]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 3

[[[.,.],[[.,.],.]],.] => [[.,.],[[[.,.],.],.]] => [[[]],[[[]]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 4

[[[.,[.,.]],[.,.]],.] => [[.,[.,.]],[[.,.],.]] => [[[],[]],[[]]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 3

[[[[.,.],.],[.,.]],.] => [[[.,.],.],[[.,.],.]] => [[[[]]],[[]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 4

[[[.,[.,[.,.]]],.],.] => [[.,[.,[.,.]]],[.,.]] => [[[],[],[]],[]] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 2

[[[.,[[.,.],.]],.],.] => [[.,[[.,.],.]],[.,.]] => [[[],[[]]],[]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 3

[[[[.,.],[.,.]],.],.] => [[[.,.],[.,.]],[.,.]] => [[[[]],[]],[]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 3

[[[[.,[.,.]],.],.],.] => [[[.,[.,.]],.],[.,.]] => [[[[],[]]],[]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 3

[[[[[.,.],.],.],.],.] => [[[[.,.],.],.],[.,.]] => [[[[[]]]],[]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 4

[.,[.,[.,[.,[.,[.,.]]]]]] => [[.,[.,[.,[.,[.,.]]]]],.] => [[[],[],[],[],[]]] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 1

[.,[.,[.,[.,[[.,.],.]]]]] => [[.,[.,[.,[[.,.],.]]]],.] => [[[],[],[],[[]]]] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7) => 2

[.,[.,[.,[[.,.],[.,.]]]]] => [[.,[.,[[.,.],[.,.]]]],.] => [[[],[],[[]],[]]] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7) => 2

[.,[.,[.,[[.,[.,.]],.]]]] => [[.,[.,[[.,[.,.]],.]]],.] => [[[],[],[[],[]]]] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7) => 2

[.,[.,[.,[[[.,.],.],.]]]] => [[.,[.,[[[.,.],.],.]]],.] => [[[],[],[[[]]]]] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7) => 3

[.,[.,[[.,.],[.,[.,.]]]]] => [[.,[[.,.],[.,[.,.]]]],.] => [[[],[[]],[],[]]] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7) => 2

[.,[.,[[.,.],[[.,.],.]]]] => [[.,[[.,.],[[.,.],.]]],.] => [[[],[[]],[[]]]] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7) => 3

[.,[.,[[.,[.,.]],[.,.]]]] => [[.,[[.,[.,.]],[.,.]]],.] => [[[],[[],[]],[]]] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7) => 2

[.,[.,[[[.,.],.],[.,.]]]] => [[.,[[[.,.],.],[.,.]]],.] => [[[],[[[]]],[]]] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7) => 3

[.,[.,[[.,[.,[.,.]]],.]]] => [[.,[[.,[.,[.,.]]],.]],.] => [[[],[[],[],[]]]] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7) => 2

[.,[.,[[.,[[.,.],.]],.]]] => [[.,[[.,[[.,.],.]],.]],.] => [[[],[[],[[]]]]] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7) => 3

[.,[.,[[[.,.],[.,.]],.]]] => [[.,[[[.,.],[.,.]],.]],.] => [[[],[[[]],[]]]] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7) => 3

[.,[.,[[[.,[.,.]],.],.]]] => [[.,[[[.,[.,.]],.],.]],.] => [[[],[[[],[]]]]] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7) => 3

[.,[.,[[[[.,.],.],.],.]]] => [[.,[[[[.,.],.],.],.]],.] => [[[],[[[[]]]]]] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7) => 4

[.,[[.,.],[.,[.,[.,.]]]]] => [[[.,.],[.,[.,[.,.]]]],.] => [[[[]],[],[],[]]] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7) => 2

[.,[[.,.],[.,[[.,.],.]]]] => [[[.,.],[.,[[.,.],.]]],.] => [[[[]],[],[[]]]] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7) => 3

[.,[[.,.],[[.,.],[.,.]]]] => [[[.,.],[[.,.],[.,.]]],.] => [[[[]],[[]],[]]] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7) => 3

[.,[[.,.],[[.,[.,.]],.]]] => [[[.,.],[[.,[.,.]],.]],.] => [[[[]],[[],[]]]] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7) => 3

[.,[[.,.],[[[.,.],.],.]]] => [[[.,.],[[[.,.],.],.]],.] => [[[[]],[[[]]]]] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => 4

[.,[[.,[.,.]],[.,[.,.]]]] => [[[.,[.,.]],[.,[.,.]]],.] => [[[[],[]],[],[]]] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7) => 2

[.,[[.,[.,.]],[[.,.],.]]] => [[[.,[.,.]],[[.,.],.]],.] => [[[[],[]],[[]]]] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7) => 3

[.,[[[.,.],.],[.,[.,.]]]] => [[[[.,.],.],[.,[.,.]]],.] => [[[[[]]],[],[]]] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7) => 3

[.,[[[.,.],.],[[.,.],.]]] => [[[[.,.],.],[[.,.],.]],.] => [[[[[]]],[[]]]] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => 4

[.,[[.,[.,[.,.]]],[.,.]]] => [[[.,[.,[.,.]]],[.,.]],.] => [[[[],[],[]],[]]] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7) => 2

[.,[[.,[[.,.],.]],[.,.]]] => [[[.,[[.,.],.]],[.,.]],.] => [[[[],[[]]],[]]] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7) => 3

[.,[[[.,.],[.,.]],[.,.]]] => [[[[.,.],[.,.]],[.,.]],.] => [[[[[]],[]],[]]] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7) => 3

[.,[[[.,[.,.]],.],[.,.]]] => [[[[.,[.,.]],.],[.,.]],.] => [[[[[],[]]],[]]] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7) => 3

[.,[[[[.,.],.],.],[.,.]]] => [[[[[.,.],.],.],[.,.]],.] => [[[[[[]]]],[]]] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7) => 4

[.,[[.,[.,[.,[.,.]]]],.]] => [[[.,[.,[.,[.,.]]]],.],.] => [[[[],[],[],[]]]] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7) => 2

[.,[[.,[.,[[.,.],.]]],.]] => [[[.,[.,[[.,.],.]]],.],.] => [[[[],[],[[]]]]] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7) => 3

[.,[[.,[[.,.],[.,.]]],.]] => [[[.,[[.,.],[.,.]]],.],.] => [[[[],[[]],[]]]] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7) => 3

[.,[[.,[[.,[.,.]],.]],.]] => [[[.,[[.,[.,.]],.]],.],.] => [[[[],[[],[]]]]] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7) => 3

[.,[[.,[[[.,.],.],.]],.]] => [[[.,[[[.,.],.],.]],.],.] => [[[[],[[[]]]]]] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => 4

[.,[[[.,.],[.,[.,.]]],.]] => [[[[.,.],[.,[.,.]]],.],.] => [[[[[]],[],[]]]] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7) => 3

[.,[[[.,.],[[.,.],.]],.]] => [[[[.,.],[[.,.],.]],.],.] => [[[[[]],[[]]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 4

[.,[[[.,[.,.]],[.,.]],.]] => [[[[.,[.,.]],[.,.]],.],.] => [[[[[],[]],[]]]] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7) => 3

[.,[[[[.,.],.],[.,.]],.]] => [[[[[.,.],.],[.,.]],.],.] => [[[[[[]]],[]]]] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => 4

>>> Load all 196 entries. <<<

[.,[[[.,[.,[.,.]]],.],.]] => [[[[.,[.,[.,.]]],.],.],.] => [[[[[],[],[]]]]] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7) => 3

[.,[[[.,[[.,.],.]],.],.]] => [[[[.,[[.,.],.]],.],.],.] => [[[[[],[[]]]]]] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => 4

[.,[[[[.,.],[.,.]],.],.]] => [[[[[.,.],[.,.]],.],.],.] => [[[[[[]],[]]]]] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => 4

[.,[[[[.,[.,.]],.],.],.]] => [[[[[.,[.,.]],.],.],.],.] => [[[[[[],[]]]]]] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7) => 4

[.,[[[[[.,.],.],.],.],.]] => [[[[[[.,.],.],.],.],.],.] => [[[[[[[]]]]]]] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7) => 5

[[.,.],[.,[.,[.,[.,.]]]]] => [.,[.,[.,[.,[.,[.,.]]]]]] => [[],[],[],[],[],[]] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 1

[[.,.],[.,[.,[[.,.],.]]]] => [.,[.,[.,[.,[[.,.],.]]]]] => [[],[],[],[],[[]]] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7) => 2

[[.,.],[.,[[.,.],[.,.]]]] => [.,[.,[.,[[.,.],[.,.]]]]] => [[],[],[],[[]],[]] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7) => 2

[[.,.],[.,[[.,[.,.]],.]]] => [.,[.,[.,[[.,[.,.]],.]]]] => [[],[],[],[[],[]]] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7) => 2

[[.,.],[.,[[[.,.],.],.]]] => [.,[.,[.,[[[.,.],.],.]]]] => [[],[],[],[[[]]]] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7) => 3

[[.,.],[[.,.],[.,[.,.]]]] => [.,[.,[[.,.],[.,[.,.]]]]] => [[],[],[[]],[],[]] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7) => 2

[[.,.],[[.,.],[[.,.],.]]] => [.,[.,[[.,.],[[.,.],.]]]] => [[],[],[[]],[[]]] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7) => 3

[[.,.],[[.,[.,.]],[.,.]]] => [.,[.,[[.,[.,.]],[.,.]]]] => [[],[],[[],[]],[]] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7) => 2

[[.,.],[[[.,.],.],[.,.]]] => [.,[.,[[[.,.],.],[.,.]]]] => [[],[],[[[]]],[]] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7) => 3

[[.,.],[[.,[.,[.,.]]],.]] => [.,[.,[[.,[.,[.,.]]],.]]] => [[],[],[[],[],[]]] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7) => 2

[[.,.],[[.,[[.,.],.]],.]] => [.,[.,[[.,[[.,.],.]],.]]] => [[],[],[[],[[]]]] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7) => 3

[[.,.],[[[.,.],[.,.]],.]] => [.,[.,[[[.,.],[.,.]],.]]] => [[],[],[[[]],[]]] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7) => 3

[[.,.],[[[.,[.,.]],.],.]] => [.,[.,[[[.,[.,.]],.],.]]] => [[],[],[[[],[]]]] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7) => 3

[[.,.],[[[[.,.],.],.],.]] => [.,[.,[[[[.,.],.],.],.]]] => [[],[],[[[[]]]]] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7) => 4

[[.,[.,.]],[.,[.,[.,.]]]] => [.,[[.,.],[.,[.,[.,.]]]]] => [[],[[]],[],[],[]] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7) => 2

[[.,[.,.]],[.,[[.,.],.]]] => [.,[[.,.],[.,[[.,.],.]]]] => [[],[[]],[],[[]]] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7) => 3

[[.,[.,.]],[[.,.],[.,.]]] => [.,[[.,.],[[.,.],[.,.]]]] => [[],[[]],[[]],[]] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7) => 3

[[.,[.,.]],[[.,[.,.]],.]] => [.,[[.,.],[[.,[.,.]],.]]] => [[],[[]],[[],[]]] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7) => 3

[[.,[.,.]],[[[.,.],.],.]] => [.,[[.,.],[[[.,.],.],.]]] => [[],[[]],[[[]]]] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => 4

[[[.,.],.],[.,[.,[.,.]]]] => [[.,.],[.,[.,[.,[.,.]]]]] => [[[]],[],[],[],[]] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7) => 2

[[[.,.],.],[.,[[.,.],.]]] => [[.,.],[.,[.,[[.,.],.]]]] => [[[]],[],[],[[]]] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7) => 3

[[[.,.],.],[[.,.],[.,.]]] => [[.,.],[.,[[.,.],[.,.]]]] => [[[]],[],[[]],[]] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7) => 3

[[[.,.],.],[[.,[.,.]],.]] => [[.,.],[.,[[.,[.,.]],.]]] => [[[]],[],[[],[]]] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7) => 3

[[[.,.],.],[[[.,.],.],.]] => [[.,.],[.,[[[.,.],.],.]]] => [[[]],[],[[[]]]] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => 4

[[.,[.,[.,.]]],[.,[.,.]]] => [.,[[.,[.,.]],[.,[.,.]]]] => [[],[[],[]],[],[]] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7) => 2

[[.,[.,[.,.]]],[[.,.],.]] => [.,[[.,[.,.]],[[.,.],.]]] => [[],[[],[]],[[]]] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7) => 3

[[.,[[.,.],.]],[.,[.,.]]] => [.,[[[.,.],.],[.,[.,.]]]] => [[],[[[]]],[],[]] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7) => 3

[[.,[[.,.],.]],[[.,.],.]] => [.,[[[.,.],.],[[.,.],.]]] => [[],[[[]]],[[]]] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => 4

[[[.,.],[.,.]],[.,[.,.]]] => [[.,.],[[.,.],[.,[.,.]]]] => [[[]],[[]],[],[]] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7) => 3

[[[.,.],[.,.]],[[.,.],.]] => [[.,.],[[.,.],[[.,.],.]]] => [[[]],[[]],[[]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 4

[[[.,[.,.]],.],[.,[.,.]]] => [[.,[.,.]],[.,[.,[.,.]]]] => [[[],[]],[],[],[]] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7) => 2

[[[.,[.,.]],.],[[.,.],.]] => [[.,[.,.]],[.,[[.,.],.]]] => [[[],[]],[],[[]]] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7) => 3

[[[[.,.],.],.],[.,[.,.]]] => [[[.,.],.],[.,[.,[.,.]]]] => [[[[]]],[],[],[]] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7) => 3

[[[[.,.],.],.],[[.,.],.]] => [[[.,.],.],[.,[[.,.],.]]] => [[[[]]],[],[[]]] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => 4

[[.,[.,[.,[.,.]]]],[.,.]] => [.,[[.,[.,[.,.]]],[.,.]]] => [[],[[],[],[]],[]] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7) => 2

[[.,[.,[[.,.],.]]],[.,.]] => [.,[[.,[[.,.],.]],[.,.]]] => [[],[[],[[]]],[]] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7) => 3

[[.,[[.,.],[.,.]]],[.,.]] => [.,[[[.,.],[.,.]],[.,.]]] => [[],[[[]],[]],[]] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7) => 3

[[.,[[.,[.,.]],.]],[.,.]] => [.,[[[.,[.,.]],.],[.,.]]] => [[],[[[],[]]],[]] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7) => 3

[[.,[[[.,.],.],.]],[.,.]] => [.,[[[[.,.],.],.],[.,.]]] => [[],[[[[]]]],[]] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7) => 4

[[[.,.],[.,[.,.]]],[.,.]] => [[.,.],[[.,[.,.]],[.,.]]] => [[[]],[[],[]],[]] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7) => 3

[[[.,.],[[.,.],.]],[.,.]] => [[.,.],[[[.,.],.],[.,.]]] => [[[]],[[[]]],[]] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => 4

[[[.,[.,.]],[.,.]],[.,.]] => [[.,[.,.]],[[.,.],[.,.]]] => [[[],[]],[[]],[]] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7) => 3

[[[[.,.],.],[.,.]],[.,.]] => [[[.,.],.],[[.,.],[.,.]]] => [[[[]]],[[]],[]] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => 4

[[[.,[.,[.,.]]],.],[.,.]] => [[.,[.,[.,.]]],[.,[.,.]]] => [[[],[],[]],[],[]] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7) => 2

[[[.,[[.,.],.]],.],[.,.]] => [[.,[[.,.],.]],[.,[.,.]]] => [[[],[[]]],[],[]] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7) => 3

[[[[.,.],[.,.]],.],[.,.]] => [[[.,.],[.,.]],[.,[.,.]]] => [[[[]],[]],[],[]] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7) => 3

[[[[.,[.,.]],.],.],[.,.]] => [[[.,[.,.]],.],[.,[.,.]]] => [[[[],[]]],[],[]] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7) => 3

[[[[[.,.],.],.],.],[.,.]] => [[[[.,.],.],.],[.,[.,.]]] => [[[[[]]]],[],[]] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7) => 4

[[.,[.,[.,[.,[.,.]]]]],.] => [.,[[.,[.,[.,[.,.]]]],.]] => [[],[[],[],[],[]]] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7) => 2

[[.,[.,[.,[[.,.],.]]]],.] => [.,[[.,[.,[[.,.],.]]],.]] => [[],[[],[],[[]]]] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7) => 3

[[.,[.,[[.,.],[.,.]]]],.] => [.,[[.,[[.,.],[.,.]]],.]] => [[],[[],[[]],[]]] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7) => 3

[[.,[.,[[.,[.,.]],.]]],.] => [.,[[.,[[.,[.,.]],.]],.]] => [[],[[],[[],[]]]] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7) => 3

[[.,[.,[[[.,.],.],.]]],.] => [.,[[.,[[[.,.],.],.]],.]] => [[],[[],[[[]]]]] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => 4

[[.,[[.,.],[.,[.,.]]]],.] => [.,[[[.,.],[.,[.,.]]],.]] => [[],[[[]],[],[]]] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7) => 3

[[.,[[.,.],[[.,.],.]]],.] => [.,[[[.,.],[[.,.],.]],.]] => [[],[[[]],[[]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 4

[[.,[[.,[.,.]],[.,.]]],.] => [.,[[[.,[.,.]],[.,.]],.]] => [[],[[[],[]],[]]] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7) => 3

[[.,[[[.,.],.],[.,.]]],.] => [.,[[[[.,.],.],[.,.]],.]] => [[],[[[[]]],[]]] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => 4

[[.,[[.,[.,[.,.]]],.]],.] => [.,[[[.,[.,[.,.]]],.],.]] => [[],[[[],[],[]]]] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7) => 3

[[.,[[.,[[.,.],.]],.]],.] => [.,[[[.,[[.,.],.]],.],.]] => [[],[[[],[[]]]]] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => 4

[[.,[[[.,.],[.,.]],.]],.] => [.,[[[[.,.],[.,.]],.],.]] => [[],[[[[]],[]]]] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => 4

[[.,[[[.,[.,.]],.],.]],.] => [.,[[[[.,[.,.]],.],.],.]] => [[],[[[[],[]]]]] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7) => 4

[[.,[[[[.,.],.],.],.]],.] => [.,[[[[[.,.],.],.],.],.]] => [[],[[[[[]]]]]] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7) => 5

[[[.,.],[.,[.,[.,.]]]],.] => [[.,.],[[.,[.,[.,.]]],.]] => [[[]],[[],[],[]]] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7) => 3

[[[.,.],[.,[[.,.],.]]],.] => [[.,.],[[.,[[.,.],.]],.]] => [[[]],[[],[[]]]] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => 4

[[[.,.],[[.,.],[.,.]]],.] => [[.,.],[[[.,.],[.,.]],.]] => [[[]],[[[]],[]]] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => 4

[[[.,.],[[.,[.,.]],.]],.] => [[.,.],[[[.,[.,.]],.],.]] => [[[]],[[[],[]]]] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7) => 4

[[[.,.],[[[.,.],.],.]],.] => [[.,.],[[[[.,.],.],.],.]] => [[[]],[[[[]]]]] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7) => 5

[[[.,[.,.]],[.,[.,.]]],.] => [[.,[.,.]],[[.,[.,.]],.]] => [[[],[]],[[],[]]] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7) => 3

[[[.,[.,.]],[[.,.],.]],.] => [[.,[.,.]],[[[.,.],.],.]] => [[[],[]],[[[]]]] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7) => 4

[[[[.,.],.],[.,[.,.]]],.] => [[[.,.],.],[[.,[.,.]],.]] => [[[[]]],[[],[]]] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7) => 4

[[[[.,.],.],[[.,.],.]],.] => [[[.,.],.],[[[.,.],.],.]] => [[[[]]],[[[]]]] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7) => 5

[[[.,[.,[.,.]]],[.,.]],.] => [[.,[.,[.,.]]],[[.,.],.]] => [[[],[],[]],[[]]] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7) => 3

[[[.,[[.,.],.]],[.,.]],.] => [[.,[[.,.],.]],[[.,.],.]] => [[[],[[]]],[[]]] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => 4

[[[[.,.],[.,.]],[.,.]],.] => [[[.,.],[.,.]],[[.,.],.]] => [[[[]],[]],[[]]] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => 4

[[[[.,[.,.]],.],[.,.]],.] => [[[.,[.,.]],.],[[.,.],.]] => [[[[],[]]],[[]]] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7) => 4

[[[[[.,.],.],.],[.,.]],.] => [[[[.,.],.],.],[[.,.],.]] => [[[[[]]]],[[]]] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7) => 5

[[[.,[.,[.,[.,.]]]],.],.] => [[.,[.,[.,[.,.]]]],[.,.]] => [[[],[],[],[]],[]] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7) => 2

[[[.,[.,[[.,.],.]]],.],.] => [[.,[.,[[.,.],.]]],[.,.]] => [[[],[],[[]]],[]] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7) => 3

[[[.,[[.,.],[.,.]]],.],.] => [[.,[[.,.],[.,.]]],[.,.]] => [[[],[[]],[]],[]] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7) => 3

[[[.,[[.,[.,.]],.]],.],.] => [[.,[[.,[.,.]],.]],[.,.]] => [[[],[[],[]]],[]] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7) => 3

[[[.,[[[.,.],.],.]],.],.] => [[.,[[[.,.],.],.]],[.,.]] => [[[],[[[]]]],[]] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => 4

[[[[.,.],[.,[.,.]]],.],.] => [[[.,.],[.,[.,.]]],[.,.]] => [[[[]],[],[]],[]] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7) => 3

[[[[.,.],[[.,.],.]],.],.] => [[[.,.],[[.,.],.]],[.,.]] => [[[[]],[[]]],[]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 4

[[[[.,[.,.]],[.,.]],.],.] => [[[.,[.,.]],[.,.]],[.,.]] => [[[[],[]],[]],[]] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7) => 3

[[[[[.,.],.],[.,.]],.],.] => [[[[.,.],.],[.,.]],[.,.]] => [[[[[]]],[]],[]] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => 4

[[[[.,[.,[.,.]]],.],.],.] => [[[.,[.,[.,.]]],.],[.,.]] => [[[[],[],[]]],[]] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7) => 3

[[[[.,[[.,.],.]],.],.],.] => [[[.,[[.,.],.]],.],[.,.]] => [[[[],[[]]]],[]] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => 4

[[[[[.,.],[.,.]],.],.],.] => [[[[.,.],[.,.]],.],[.,.]] => [[[[[]],[]]],[]] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => 4

[[[[[.,[.,.]],.],.],.],.] => [[[[.,[.,.]],.],.],[.,.]] => [[[[[],[]]]],[]] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7) => 4

[[[[[[.,.],.],.],.],.],.] => [[[[[.,.],.],.],.],[.,.]] => [[[[[[]]]]],[]] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7) => 5

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

$F_{1} = 1$

$F_{2} = 2\ q$

$F_{3} = 2\ q + 3\ q^{2}$

$F_{4} = 2\ q + 8\ q^{2} + 4\ q^{3}$

$F_{5} = 2\ q + 15\ q^{2} + 20\ q^{3} + 5\ q^{4}$

$F_{6} = 2\ q + 24\ q^{2} + 60\ q^{3} + 40\ q^{4} + 6\ q^{5}$

Description

The number of vertices with higher degree than the average degree in a graph.

Map

to graph

Description

Return the undirected graph obtained from the tree nodes and edges.

Map

to ordered tree: right child = right brother

Description

Return an ordered tree of size

$n+1$ by the following recursive rule:

if $x$ is the right child of $y$ , $x$ becomes the right brother of $y$ ,
if $x$ is the left child of $y$ , $x$ becomes the first child of $y$ .

Map

right rotate

Description

Return the result of right rotation applied to a binary tree.
Right rotation on binary trees is defined as follows: Let

$T$ be a binary tree such that the left child of the root of

$T$ is a node. Let

$C$ be the right child of the root of

$T$ , and let

$A$ and

$B$ be the left and right children of the left child of the root of

$T$ . (Keep in mind that nodes of trees are identified with the subtrees consisting of their descendants.) Then, the right rotation of

$T$ is the binary tree in which the left child of the root is

$A$ , whereas the right child of the root is a node whose left and right children are

$B$ and

$C$ .