Identifier
            
            - 
Mp00010:
    Binary trees
    
—to ordered tree: left child = left brother⟶
Ordered trees
		
Mp00046: Ordered trees —to graph⟶ Graphs
St000537: Graphs ⟶ ℤ 
                Values
            
            [.,.] => [[]] => ([(0,1)],2) => 1
[.,[.,.]] => [[[]]] => ([(0,2),(1,2)],3) => 1
[[.,.],.] => [[],[]] => ([(0,2),(1,2)],3) => 1
[.,[.,[.,.]]] => [[[[]]]] => ([(0,3),(1,2),(2,3)],4) => 1
[.,[[.,.],.]] => [[[],[]]] => ([(0,3),(1,3),(2,3)],4) => 2
[[.,.],[.,.]] => [[],[[]]] => ([(0,3),(1,2),(2,3)],4) => 1
[[.,[.,.]],.] => [[[]],[]] => ([(0,3),(1,2),(2,3)],4) => 1
[[[.,.],.],.] => [[],[],[]] => ([(0,3),(1,3),(2,3)],4) => 2
[.,[.,[.,[.,.]]]] => [[[[[]]]]] => ([(0,4),(1,3),(2,3),(2,4)],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),(2,4),(3,4)],5) => 2
[[.,.],[.,[.,.]]] => [[],[[[]]]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[[.,.],[[.,.],.]] => [[],[[],[]]] => ([(0,4),(1,4),(2,3),(3,4)],5) => 2
[[.,[.,.]],[.,.]] => [[[]],[[]]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[[[.,.],.],[.,.]] => [[],[],[[]]] => ([(0,4),(1,4),(2,3),(3,4)],5) => 2
[[.,[.,[.,.]]],.] => [[[[]]],[]] => ([(0,4),(1,3),(2,3),(2,4)],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),(2,4),(3,4)],5) => 2
[.,[.,[.,[.,[.,.]]]]] => [[[[[[]]]]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[.,[.,[.,[[.,.],.]]]] => [[[[[],[]]]]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 2
[.,[.,[[.,.],[.,.]]]] => [[[[],[[]]]]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 2
[.,[.,[[.,[.,.]],.]]] => [[[[[]],[]]]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 2
[.,[.,[[[.,.],.],.]]] => [[[[],[],[]]]] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 2
[.,[[.,.],[.,[.,.]]]] => [[[],[[[]]]]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 2
[.,[[.,.],[[.,.],.]]] => [[[],[[],[]]]] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 2
[.,[[.,[.,.]],[.,.]]] => [[[[]],[[]]]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 2
[.,[[[.,.],.],[.,.]]] => [[[],[],[[]]]] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 2
[.,[[.,[.,[.,.]]],.]] => [[[[[]]],[]]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 2
[.,[[.,[[.,.],.]],.]] => [[[[],[]],[]]] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 2
[.,[[[.,.],[.,.]],.]] => [[[],[[]],[]]] => ([(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),(2,5),(3,5),(4,5)],6) => 3
[[.,.],[.,[.,[.,.]]]] => [[],[[[[]]]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[[.,.],[.,[[.,.],.]]] => [[],[[[],[]]]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 2
[[.,.],[[.,.],[.,.]]] => [[],[[],[[]]]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 2
[[.,.],[[.,[.,.]],.]] => [[],[[[]],[]]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 2
[[.,.],[[[.,.],.],.]] => [[],[[],[],[]]] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 2
[[.,[.,.]],[.,[.,.]]] => [[[]],[[[]]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[[.,[.,.]],[[.,.],.]] => [[[]],[[],[]]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 2
[[[.,.],.],[.,[.,.]]] => [[],[],[[[]]]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 2
[[[.,.],.],[[.,.],.]] => [[],[],[[],[]]] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 2
[[.,[.,[.,.]]],[.,.]] => [[[[]]],[[]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[[.,[[.,.],.]],[.,.]] => [[[],[]],[[]]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 2
[[[.,.],[.,.]],[.,.]] => [[],[[]],[[]]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 2
[[[.,[.,.]],.],[.,.]] => [[[]],[],[[]]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 2
[[[[.,.],.],.],[.,.]] => [[],[],[],[[]]] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 2
[[.,[.,[.,[.,.]]]],.] => [[[[[]]]],[]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[[.,[.,[[.,.],.]]],.] => [[[[],[]]],[]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 2
[[.,[[.,.],[.,.]]],.] => [[[],[[]]],[]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 2
[[.,[[.,[.,.]],.]],.] => [[[[]],[]],[]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 2
[[.,[[[.,.],.],.]],.] => [[[],[],[]],[]] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 2
[[[.,.],[.,[.,.]]],.] => [[],[[[]]],[]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 2
[[[.,.],[[.,.],.]],.] => [[],[[],[]],[]] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 2
[[[.,[.,.]],[.,.]],.] => [[[]],[[]],[]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 2
[[[[.,.],.],[.,.]],.] => [[],[],[[]],[]] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 2
[[[.,[.,[.,.]]],.],.] => [[[[]]],[],[]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 2
[[[.,[[.,.],.]],.],.] => [[[],[]],[],[]] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 2
[[[[.,.],[.,.]],.],.] => [[],[[]],[],[]] => ([(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),(2,5),(3,5),(4,5)],6) => 3
[.,[.,[.,[.,[.,[.,.]]]]]] => [[[[[[[]]]]]]] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7) => 1
[.,[.,[.,[.,[[.,.],.]]]]] => [[[[[[],[]]]]]] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7) => 2
[.,[.,[.,[[.,.],[.,.]]]]] => [[[[[],[[]]]]]] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => 2
[.,[.,[.,[[.,[.,.]],.]]]] => [[[[[[]],[]]]]] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => 2
[.,[.,[.,[[[.,.],.],.]]]] => [[[[[],[],[]]]]] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7) => 2
[.,[.,[[.,.],[.,[.,.]]]]] => [[[[],[[[]]]]]] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => 2
[.,[.,[[.,.],[[.,.],.]]]] => [[[[],[[],[]]]]] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7) => 2
[.,[.,[[.,[.,.]],[.,.]]]] => [[[[[]],[[]]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[.,[.,[[[.,.],.],[.,.]]]] => [[[[],[],[[]]]]] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7) => 2
[.,[.,[[.,[.,[.,.]]],.]]] => [[[[[[]]],[]]]] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => 2
[.,[.,[[.,[[.,.],.]],.]]] => [[[[[],[]],[]]]] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7) => 2
[.,[.,[[[.,.],[.,.]],.]]] => [[[[],[[]],[]]]] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7) => 2
[.,[.,[[[.,[.,.]],.],.]]] => [[[[[]],[],[]]]] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7) => 2
[.,[.,[[[[.,.],.],.],.]]] => [[[[],[],[],[]]]] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7) => 3
[.,[[.,.],[.,[.,[.,.]]]]] => [[[],[[[[]]]]]] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7) => 2
[.,[[.,.],[.,[[.,.],.]]]] => [[[],[[[],[]]]]] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7) => 2
[.,[[.,.],[[.,.],[.,.]]]] => [[[],[[],[[]]]]] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7) => 2
[.,[[.,.],[[.,[.,.]],.]]] => [[[],[[[]],[]]]] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7) => 2
[.,[[.,.],[[[.,.],.],.]]] => [[[],[[],[],[]]]] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7) => 2
[.,[[.,[.,.]],[.,[.,.]]]] => [[[[]],[[[]]]]] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => 2
[.,[[.,[.,.]],[[.,.],.]]] => [[[[]],[[],[]]]] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7) => 2
[.,[[[.,.],.],[.,[.,.]]]] => [[[],[],[[[]]]]] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7) => 2
[.,[[[.,.],.],[[.,.],.]]] => [[[],[],[[],[]]]] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7) => 2
[.,[[.,[.,[.,.]]],[.,.]]] => [[[[[]]],[[]]]] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => 2
[.,[[.,[[.,.],.]],[.,.]]] => [[[[],[]],[[]]]] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7) => 2
[.,[[[.,.],[.,.]],[.,.]]] => [[[],[[]],[[]]]] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7) => 2
[.,[[[.,[.,.]],.],[.,.]]] => [[[[]],[],[[]]]] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7) => 2
[.,[[[[.,.],.],.],[.,.]]] => [[[],[],[],[[]]]] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7) => 3
[.,[[.,[.,[.,[.,.]]]],.]] => [[[[[[]]]],[]]] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7) => 2
[.,[[.,[.,[[.,.],.]]],.]] => [[[[[],[]]],[]]] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7) => 2
[.,[[.,[[.,.],[.,.]]],.]] => [[[[],[[]]],[]]] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7) => 2
[.,[[.,[[.,[.,.]],.]],.]] => [[[[[]],[]],[]]] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(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) => 2
[.,[[[.,.],[[.,.],.]],.]] => [[[],[[],[]],[]]] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7) => 2
[.,[[[.,[.,.]],[.,.]],.]] => [[[[]],[[]],[]]] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7) => 2
[.,[[[[.,.],.],[.,.]],.]] => [[[],[],[[]],[]]] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7) => 3
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                Description
            The cutwidth of a graph.
This is the minimum possible width of a linear ordering of its vertices, where the width of an ordering $\sigma$ is the maximum, among all the prefixes of $\sigma$, of the number of edges that have exactly one vertex in a prefix.
	This is the minimum possible width of a linear ordering of its vertices, where the width of an ordering $\sigma$ is the maximum, among all the prefixes of $\sigma$, of the number of edges that have exactly one vertex in a prefix.
Map
            to graph
	    
	Description
            Return the undirected graph obtained from the tree nodes and edges.
	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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