Identifier
-
Mp00139:
Ordered trees
—Zeilberger's Strahler bijection⟶
Binary trees
Mp00008: Binary trees —to complete tree⟶ Ordered trees
St000397: Ordered trees ⟶ ℤ
Values
[[]] => [.,.] => [[],[]] => 2
[[],[]] => [.,[.,.]] => [[],[[],[]]] => 2
[[[]]] => [[.,.],.] => [[[],[]],[]] => 2
[[],[],[]] => [.,[.,[.,.]]] => [[],[[],[[],[]]]] => 2
[[],[[]]] => [.,[[.,.],.]] => [[],[[[],[]],[]]] => 2
[[[]],[]] => [[.,[.,.]],.] => [[[],[[],[]]],[]] => 2
[[[],[]]] => [[.,.],[.,.]] => [[[],[]],[[],[]]] => 3
[[[[]]]] => [[[.,.],.],.] => [[[[],[]],[]],[]] => 2
[[],[],[],[]] => [.,[.,[.,[.,.]]]] => [[],[[],[[],[[],[]]]]] => 2
[[],[],[[]]] => [.,[.,[[.,.],.]]] => [[],[[],[[[],[]],[]]]] => 2
[[],[[]],[]] => [.,[[.,[.,.]],.]] => [[],[[[],[[],[]]],[]]] => 2
[[],[[],[]]] => [.,[[.,.],[.,.]]] => [[],[[[],[]],[[],[]]]] => 3
[[],[[[]]]] => [.,[[[.,.],.],.]] => [[],[[[[],[]],[]],[]]] => 2
[[[]],[],[]] => [[.,[.,[.,.]]],.] => [[[],[[],[[],[]]]],[]] => 2
[[[]],[[]]] => [[.,[[.,.],.]],.] => [[[],[[[],[]],[]]],[]] => 2
[[[],[]],[]] => [[.,[.,.]],[.,.]] => [[[],[[],[]]],[[],[]]] => 3
[[[[]]],[]] => [[[.,[.,.]],.],.] => [[[[],[[],[]]],[]],[]] => 2
[[[],[],[]]] => [[.,.],[.,[.,.]]] => [[[],[]],[[],[[],[]]]] => 3
[[[],[[]]]] => [[.,.],[[.,.],.]] => [[[],[]],[[[],[]],[]]] => 3
[[[[]],[]]] => [[[.,.],.],[.,.]] => [[[[],[]],[]],[[],[]]] => 3
[[[[],[]]]] => [[[.,.],[.,.]],.] => [[[[],[]],[[],[]]],[]] => 3
[[[[[]]]]] => [[[[.,.],.],.],.] => [[[[[],[]],[]],[]],[]] => 2
[[],[],[],[],[]] => [.,[.,[.,[.,[.,.]]]]] => [[],[[],[[],[[],[[],[]]]]]] => 2
[[],[],[],[[]]] => [.,[.,[.,[[.,.],.]]]] => [[],[[],[[],[[[],[]],[]]]]] => 2
[[],[],[[]],[]] => [.,[.,[[.,[.,.]],.]]] => [[],[[],[[[],[[],[]]],[]]]] => 2
[[],[],[[],[]]] => [.,[.,[[.,.],[.,.]]]] => [[],[[],[[[],[]],[[],[]]]]] => 3
[[],[],[[[]]]] => [.,[.,[[[.,.],.],.]]] => [[],[[],[[[[],[]],[]],[]]]] => 2
[[],[[]],[],[]] => [.,[[.,[.,[.,.]]],.]] => [[],[[[],[[],[[],[]]]],[]]] => 2
[[],[[]],[[]]] => [.,[[.,[[.,.],.]],.]] => [[],[[[],[[[],[]],[]]],[]]] => 2
[[],[[],[]],[]] => [.,[[.,[.,.]],[.,.]]] => [[],[[[],[[],[]]],[[],[]]]] => 3
[[],[[[]]],[]] => [.,[[[.,[.,.]],.],.]] => [[],[[[[],[[],[]]],[]],[]]] => 2
[[],[[],[],[]]] => [.,[[.,.],[.,[.,.]]]] => [[],[[[],[]],[[],[[],[]]]]] => 3
[[],[[],[[]]]] => [.,[[.,.],[[.,.],.]]] => [[],[[[],[]],[[[],[]],[]]]] => 3
[[],[[[]],[]]] => [.,[[[.,.],.],[.,.]]] => [[],[[[[],[]],[]],[[],[]]]] => 3
[[],[[[],[]]]] => [.,[[[.,.],[.,.]],.]] => [[],[[[[],[]],[[],[]]],[]]] => 3
[[],[[[[]]]]] => [.,[[[[.,.],.],.],.]] => [[],[[[[[],[]],[]],[]],[]]] => 2
[[[]],[],[],[]] => [[.,[.,[.,[.,.]]]],.] => [[[],[[],[[],[[],[]]]]],[]] => 2
[[[]],[],[[]]] => [[.,[.,[[.,.],.]]],.] => [[[],[[],[[[],[]],[]]]],[]] => 2
[[[]],[[]],[]] => [[.,[[.,[.,.]],.]],.] => [[[],[[[],[[],[]]],[]]],[]] => 2
[[[]],[[],[]]] => [[.,.],[[.,.],[.,.]]] => [[[],[]],[[[],[]],[[],[]]]] => 3
[[[]],[[[]]]] => [[.,[[[.,.],.],.]],.] => [[[],[[[[],[]],[]],[]]],[]] => 2
[[[],[]],[],[]] => [[.,[.,.]],[.,[.,.]]] => [[[],[[],[]]],[[],[[],[]]]] => 3
[[[[]]],[],[]] => [[[.,[.,[.,.]]],.],.] => [[[[],[[],[[],[]]]],[]],[]] => 2
[[[],[]],[[]]] => [[.,[.,.]],[[.,.],.]] => [[[],[[],[]]],[[[],[]],[]]] => 3
[[[[]]],[[]]] => [[[.,[[.,.],.]],.],.] => [[[[],[[[],[]],[]]],[]],[]] => 2
[[[],[],[]],[]] => [[.,[.,[.,.]]],[.,.]] => [[[],[[],[[],[]]]],[[],[]]] => 3
[[[],[[]]],[]] => [[.,[[.,.],.]],[.,.]] => [[[],[[[],[]],[]]],[[],[]]] => 3
[[[[]],[]],[]] => [[[.,[.,.]],.],[.,.]] => [[[[],[[],[]]],[]],[[],[]]] => 3
[[[[],[]]],[]] => [[[.,[.,.]],[.,.]],.] => [[[[],[[],[]]],[[],[]]],[]] => 3
[[[[[]]]],[]] => [[[[.,[.,.]],.],.],.] => [[[[[],[[],[]]],[]],[]],[]] => 2
[[[],[],[],[]]] => [[.,.],[.,[.,[.,.]]]] => [[[],[]],[[],[[],[[],[]]]]] => 3
[[[],[],[[]]]] => [[.,.],[.,[[.,.],.]]] => [[[],[]],[[],[[[],[]],[]]]] => 3
[[[],[[]],[]]] => [[.,.],[[.,[.,.]],.]] => [[[],[]],[[[],[[],[]]],[]]] => 3
[[[],[[],[]]]] => [[.,[[.,.],[.,.]]],.] => [[[],[[[],[]],[[],[]]]],[]] => 3
[[[],[[[]]]]] => [[.,.],[[[.,.],.],.]] => [[[],[]],[[[[],[]],[]],[]]] => 3
[[[[]],[],[]]] => [[[.,.],.],[.,[.,.]]] => [[[[],[]],[]],[[],[[],[]]]] => 3
[[[[]],[[]]]] => [[[.,.],.],[[.,.],.]] => [[[[],[]],[]],[[[],[]],[]]] => 3
[[[[],[]],[]]] => [[[.,.],[.,.]],[.,.]] => [[[[],[]],[[],[]]],[[],[]]] => 3
[[[[[]]],[]]] => [[[[.,.],.],.],[.,.]] => [[[[[],[]],[]],[]],[[],[]]] => 3
[[[[],[],[]]]] => [[[.,.],[.,[.,.]]],.] => [[[[],[]],[[],[[],[]]]],[]] => 3
[[[[],[[]]]]] => [[[.,.],[[.,.],.]],.] => [[[[],[]],[[[],[]],[]]],[]] => 3
[[[[[]],[]]]] => [[[[.,.],.],[.,.]],.] => [[[[[],[]],[]],[[],[]]],[]] => 3
[[[[[],[]]]]] => [[[[.,.],[.,.]],.],.] => [[[[[],[]],[[],[]]],[]],[]] => 3
[[[[[[]]]]]] => [[[[[.,.],.],.],.],.] => [[[[[[],[]],[]],[]],[]],[]] => 2
[[],[],[],[],[],[]] => [.,[.,[.,[.,[.,[.,.]]]]]] => [[],[[],[[],[[],[[],[[],[]]]]]]] => 2
[[],[],[],[],[[]]] => [.,[.,[.,[.,[[.,.],.]]]]] => [[],[[],[[],[[],[[[],[]],[]]]]]] => 2
[[],[],[],[[]],[]] => [.,[.,[.,[[.,[.,.]],.]]]] => [[],[[],[[],[[[],[[],[]]],[]]]]] => 2
[[],[],[],[[],[]]] => [.,[.,[.,[[.,.],[.,.]]]]] => [[],[[],[[],[[[],[]],[[],[]]]]]] => 3
[[],[],[],[[[]]]] => [.,[.,[.,[[[.,.],.],.]]]] => [[],[[],[[],[[[[],[]],[]],[]]]]] => 2
[[],[],[[]],[],[]] => [.,[.,[[.,[.,[.,.]]],.]]] => [[],[[],[[[],[[],[[],[]]]],[]]]] => 2
[[],[],[[]],[[]]] => [.,[.,[[.,[[.,.],.]],.]]] => [[],[[],[[[],[[[],[]],[]]],[]]]] => 2
[[],[],[[],[]],[]] => [.,[.,[[.,[.,.]],[.,.]]]] => [[],[[],[[[],[[],[]]],[[],[]]]]] => 3
[[],[],[[[]]],[]] => [.,[.,[[[.,[.,.]],.],.]]] => [[],[[],[[[[],[[],[]]],[]],[]]]] => 2
[[],[],[[],[],[]]] => [.,[.,[[.,.],[.,[.,.]]]]] => [[],[[],[[[],[]],[[],[[],[]]]]]] => 3
[[],[],[[],[[]]]] => [.,[.,[[.,.],[[.,.],.]]]] => [[],[[],[[[],[]],[[[],[]],[]]]]] => 3
[[],[],[[[]],[]]] => [.,[.,[[[.,.],.],[.,.]]]] => [[],[[],[[[[],[]],[]],[[],[]]]]] => 3
[[],[],[[[],[]]]] => [.,[.,[[[.,.],[.,.]],.]]] => [[],[[],[[[[],[]],[[],[]]],[]]]] => 3
[[],[],[[[[]]]]] => [.,[.,[[[[.,.],.],.],.]]] => [[],[[],[[[[[],[]],[]],[]],[]]]] => 2
[[],[[]],[],[],[]] => [.,[[.,[.,[.,[.,.]]]],.]] => [[],[[[],[[],[[],[[],[]]]]],[]]] => 2
[[],[[]],[],[[]]] => [.,[[.,[.,[[.,.],.]]],.]] => [[],[[[],[[],[[[],[]],[]]]],[]]] => 2
[[],[[]],[[]],[]] => [.,[[.,[[.,[.,.]],.]],.]] => [[],[[[],[[[],[[],[]]],[]]],[]]] => 2
[[],[[]],[[],[]]] => [.,[[.,.],[[.,.],[.,.]]]] => [[],[[[],[]],[[[],[]],[[],[]]]]] => 3
[[],[[]],[[[]]]] => [.,[[.,[[[.,.],.],.]],.]] => [[],[[[],[[[[],[]],[]],[]]],[]]] => 2
[[],[[],[]],[],[]] => [.,[[.,[.,.]],[.,[.,.]]]] => [[],[[[],[[],[]]],[[],[[],[]]]]] => 3
[[],[[[]]],[],[]] => [.,[[[.,[.,[.,.]]],.],.]] => [[],[[[[],[[],[[],[]]]],[]],[]]] => 2
[[],[[],[]],[[]]] => [.,[[.,[.,.]],[[.,.],.]]] => [[],[[[],[[],[]]],[[[],[]],[]]]] => 3
[[],[[[]]],[[]]] => [.,[[[.,[[.,.],.]],.],.]] => [[],[[[[],[[[],[]],[]]],[]],[]]] => 2
[[],[[],[],[]],[]] => [.,[[.,[.,[.,.]]],[.,.]]] => [[],[[[],[[],[[],[]]]],[[],[]]]] => 3
[[],[[],[[]]],[]] => [.,[[.,[[.,.],.]],[.,.]]] => [[],[[[],[[[],[]],[]]],[[],[]]]] => 3
[[],[[[]],[]],[]] => [.,[[[.,[.,.]],.],[.,.]]] => [[],[[[[],[[],[]]],[]],[[],[]]]] => 3
[[],[[[],[]]],[]] => [.,[[[.,[.,.]],[.,.]],.]] => [[],[[[[],[[],[]]],[[],[]]],[]]] => 3
[[],[[[[]]]],[]] => [.,[[[[.,[.,.]],.],.],.]] => [[],[[[[[],[[],[]]],[]],[]],[]]] => 2
[[],[[],[],[],[]]] => [.,[[.,.],[.,[.,[.,.]]]]] => [[],[[[],[]],[[],[[],[[],[]]]]]] => 3
[[],[[],[],[[]]]] => [.,[[.,.],[.,[[.,.],.]]]] => [[],[[[],[]],[[],[[[],[]],[]]]]] => 3
[[],[[],[[]],[]]] => [.,[[.,.],[[.,[.,.]],.]]] => [[],[[[],[]],[[[],[[],[]]],[]]]] => 3
[[],[[],[[],[]]]] => [.,[[.,[[.,.],[.,.]]],.]] => [[],[[[],[[[],[]],[[],[]]]],[]]] => 3
[[],[[],[[[]]]]] => [.,[[.,.],[[[.,.],.],.]]] => [[],[[[],[]],[[[[],[]],[]],[]]]] => 3
[[],[[[]],[],[]]] => [.,[[[.,.],.],[.,[.,.]]]] => [[],[[[[],[]],[]],[[],[[],[]]]]] => 3
[[],[[[]],[[]]]] => [.,[[[.,.],.],[[.,.],.]]] => [[],[[[[],[]],[]],[[[],[]],[]]]] => 3
[[],[[[],[]],[]]] => [.,[[[.,.],[.,.]],[.,.]]] => [[],[[[[],[]],[[],[]]],[[],[]]]] => 3
[[],[[[[]]],[]]] => [.,[[[[.,.],.],.],[.,.]]] => [[],[[[[[],[]],[]],[]],[[],[]]]] => 3
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Description
The Strahler number of a rooted tree.
Map
to complete tree
Description
Return the same tree seen as an ordered tree. By default, leaves are transformed into actual nodes.
Map
Zeilberger's Strahler bijection
Description
Zeilberger's Strahler bijection between ordered and binary trees.
This is a bijection sending the pruning number of the ordered tree to the Strahler number of the binary tree.
This is a bijection sending the pruning number of the ordered tree to the Strahler number of the binary tree.
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