Identifier
-
Mp00016:
Binary trees
—left-right symmetry⟶
Binary trees
Mp00018: Binary trees —left border symmetry⟶ Binary trees
Mp00008: Binary trees —to complete tree⟶ Ordered trees
St000397: Ordered trees ⟶ ℤ
Values
[.,.] => [.,.] => [.,.] => [[],[]] => 2
[.,[.,.]] => [[.,.],.] => [[.,.],.] => [[[],[]],[]] => 2
[[.,.],.] => [.,[.,.]] => [.,[.,.]] => [[],[[],[]]] => 2
[.,[.,[.,.]]] => [[[.,.],.],.] => [[[.,.],.],.] => [[[[],[]],[]],[]] => 2
[.,[[.,.],.]] => [[.,[.,.]],.] => [[.,.],[.,.]] => [[[],[]],[[],[]]] => 3
[[.,.],[.,.]] => [[.,.],[.,.]] => [[.,[.,.]],.] => [[[],[[],[]]],[]] => 2
[[.,[.,.]],.] => [.,[[.,.],.]] => [.,[[.,.],.]] => [[],[[[],[]],[]]] => 2
[[[.,.],.],.] => [.,[.,[.,.]]] => [.,[.,[.,.]]] => [[],[[],[[],[]]]] => 2
[.,[.,[.,[.,.]]]] => [[[[.,.],.],.],.] => [[[[.,.],.],.],.] => [[[[[],[]],[]],[]],[]] => 2
[.,[.,[[.,.],.]]] => [[[.,[.,.]],.],.] => [[[.,.],.],[.,.]] => [[[[],[]],[]],[[],[]]] => 3
[.,[[.,.],[.,.]]] => [[[.,.],[.,.]],.] => [[[.,.],[.,.]],.] => [[[[],[]],[[],[]]],[]] => 3
[.,[[.,[.,.]],.]] => [[.,[[.,.],.]],.] => [[.,.],[[.,.],.]] => [[[],[]],[[[],[]],[]]] => 3
[.,[[[.,.],.],.]] => [[.,[.,[.,.]]],.] => [[.,.],[.,[.,.]]] => [[[],[]],[[],[[],[]]]] => 3
[[.,.],[.,[.,.]]] => [[[.,.],.],[.,.]] => [[[.,[.,.]],.],.] => [[[[],[[],[]]],[]],[]] => 2
[[.,.],[[.,.],.]] => [[.,[.,.]],[.,.]] => [[.,[.,.]],[.,.]] => [[[],[[],[]]],[[],[]]] => 3
[[.,[.,.]],[.,.]] => [[.,.],[[.,.],.]] => [[.,[[.,.],.]],.] => [[[],[[[],[]],[]]],[]] => 2
[[[.,.],.],[.,.]] => [[.,.],[.,[.,.]]] => [[.,[.,[.,.]]],.] => [[[],[[],[[],[]]]],[]] => 2
[[.,[.,[.,.]]],.] => [.,[[[.,.],.],.]] => [.,[[[.,.],.],.]] => [[],[[[[],[]],[]],[]]] => 2
[[.,[[.,.],.]],.] => [.,[[.,[.,.]],.]] => [.,[[.,.],[.,.]]] => [[],[[[],[]],[[],[]]]] => 3
[[[.,.],[.,.]],.] => [.,[[.,.],[.,.]]] => [.,[[.,[.,.]],.]] => [[],[[[],[[],[]]],[]]] => 2
[[[.,[.,.]],.],.] => [.,[.,[[.,.],.]]] => [.,[.,[[.,.],.]]] => [[],[[],[[[],[]],[]]]] => 2
[[[[.,.],.],.],.] => [.,[.,[.,[.,.]]]] => [.,[.,[.,[.,.]]]] => [[],[[],[[],[[],[]]]]] => 2
[.,[.,[.,[.,[.,.]]]]] => [[[[[.,.],.],.],.],.] => [[[[[.,.],.],.],.],.] => [[[[[[],[]],[]],[]],[]],[]] => 2
[.,[.,[.,[[.,.],.]]]] => [[[[.,[.,.]],.],.],.] => [[[[.,.],.],.],[.,.]] => [[[[[],[]],[]],[]],[[],[]]] => 3
[.,[.,[[.,.],[.,.]]]] => [[[[.,.],[.,.]],.],.] => [[[[.,.],.],[.,.]],.] => [[[[[],[]],[]],[[],[]]],[]] => 3
[.,[.,[[.,[.,.]],.]]] => [[[.,[[.,.],.]],.],.] => [[[.,.],.],[[.,.],.]] => [[[[],[]],[]],[[[],[]],[]]] => 3
[.,[.,[[[.,.],.],.]]] => [[[.,[.,[.,.]]],.],.] => [[[.,.],.],[.,[.,.]]] => [[[[],[]],[]],[[],[[],[]]]] => 3
[.,[[.,.],[.,[.,.]]]] => [[[[.,.],.],[.,.]],.] => [[[[.,.],[.,.]],.],.] => [[[[[],[]],[[],[]]],[]],[]] => 3
[.,[[.,.],[[.,.],.]]] => [[[.,[.,.]],[.,.]],.] => [[[.,.],[.,.]],[.,.]] => [[[[],[]],[[],[]]],[[],[]]] => 3
[.,[[.,[.,.]],[.,.]]] => [[[.,.],[[.,.],.]],.] => [[[.,.],[[.,.],.]],.] => [[[[],[]],[[[],[]],[]]],[]] => 3
[.,[[[.,.],.],[.,.]]] => [[[.,.],[.,[.,.]]],.] => [[[.,.],[.,[.,.]]],.] => [[[[],[]],[[],[[],[]]]],[]] => 3
[.,[[.,[.,[.,.]]],.]] => [[.,[[[.,.],.],.]],.] => [[.,.],[[[.,.],.],.]] => [[[],[]],[[[[],[]],[]],[]]] => 3
[.,[[.,[[.,.],.]],.]] => [[.,[[.,[.,.]],.]],.] => [[.,.],[[.,.],[.,.]]] => [[[],[]],[[[],[]],[[],[]]]] => 3
[.,[[[.,.],[.,.]],.]] => [[.,[[.,.],[.,.]]],.] => [[.,.],[[.,[.,.]],.]] => [[[],[]],[[[],[[],[]]],[]]] => 3
[.,[[[.,[.,.]],.],.]] => [[.,[.,[[.,.],.]]],.] => [[.,.],[.,[[.,.],.]]] => [[[],[]],[[],[[[],[]],[]]]] => 3
[.,[[[[.,.],.],.],.]] => [[.,[.,[.,[.,.]]]],.] => [[.,.],[.,[.,[.,.]]]] => [[[],[]],[[],[[],[[],[]]]]] => 3
[[.,.],[.,[.,[.,.]]]] => [[[[.,.],.],.],[.,.]] => [[[[.,[.,.]],.],.],.] => [[[[[],[[],[]]],[]],[]],[]] => 2
[[.,.],[.,[[.,.],.]]] => [[[.,[.,.]],.],[.,.]] => [[[.,[.,.]],.],[.,.]] => [[[[],[[],[]]],[]],[[],[]]] => 3
[[.,.],[[.,.],[.,.]]] => [[[.,.],[.,.]],[.,.]] => [[[.,[.,.]],[.,.]],.] => [[[[],[[],[]]],[[],[]]],[]] => 3
[[.,.],[[.,[.,.]],.]] => [[.,[[.,.],.]],[.,.]] => [[.,[.,.]],[[.,.],.]] => [[[],[[],[]]],[[[],[]],[]]] => 3
[[.,.],[[[.,.],.],.]] => [[.,[.,[.,.]]],[.,.]] => [[.,[.,.]],[.,[.,.]]] => [[[],[[],[]]],[[],[[],[]]]] => 3
[[.,[.,.]],[.,[.,.]]] => [[[.,.],.],[[.,.],.]] => [[[.,[[.,.],.]],.],.] => [[[[],[[[],[]],[]]],[]],[]] => 2
[[.,[.,.]],[[.,.],.]] => [[.,[.,.]],[[.,.],.]] => [[.,[[.,.],.]],[.,.]] => [[[],[[[],[]],[]]],[[],[]]] => 3
[[[.,.],.],[.,[.,.]]] => [[[.,.],.],[.,[.,.]]] => [[[.,[.,[.,.]]],.],.] => [[[[],[[],[[],[]]]],[]],[]] => 2
[[[.,.],.],[[.,.],.]] => [[.,[.,.]],[.,[.,.]]] => [[.,[.,[.,.]]],[.,.]] => [[[],[[],[[],[]]]],[[],[]]] => 3
[[.,[.,[.,.]]],[.,.]] => [[.,.],[[[.,.],.],.]] => [[.,[[[.,.],.],.]],.] => [[[],[[[[],[]],[]],[]]],[]] => 2
[[.,[[.,.],.]],[.,.]] => [[.,.],[[.,[.,.]],.]] => [[.,[[.,.],[.,.]]],.] => [[[],[[[],[]],[[],[]]]],[]] => 3
[[[.,.],[.,.]],[.,.]] => [[.,.],[[.,.],[.,.]]] => [[.,[[.,[.,.]],.]],.] => [[[],[[[],[[],[]]],[]]],[]] => 2
[[[.,[.,.]],.],[.,.]] => [[.,.],[.,[[.,.],.]]] => [[.,[.,[[.,.],.]]],.] => [[[],[[],[[[],[]],[]]]],[]] => 2
[[[[.,.],.],.],[.,.]] => [[.,.],[.,[.,[.,.]]]] => [[.,[.,[.,[.,.]]]],.] => [[[],[[],[[],[[],[]]]]],[]] => 2
[[.,[.,[.,[.,.]]]],.] => [.,[[[[.,.],.],.],.]] => [.,[[[[.,.],.],.],.]] => [[],[[[[[],[]],[]],[]],[]]] => 2
[[.,[.,[[.,.],.]]],.] => [.,[[[.,[.,.]],.],.]] => [.,[[[.,.],.],[.,.]]] => [[],[[[[],[]],[]],[[],[]]]] => 3
[[.,[[.,.],[.,.]]],.] => [.,[[[.,.],[.,.]],.]] => [.,[[[.,.],[.,.]],.]] => [[],[[[[],[]],[[],[]]],[]]] => 3
[[.,[[.,[.,.]],.]],.] => [.,[[.,[[.,.],.]],.]] => [.,[[.,.],[[.,.],.]]] => [[],[[[],[]],[[[],[]],[]]]] => 3
[[.,[[[.,.],.],.]],.] => [.,[[.,[.,[.,.]]],.]] => [.,[[.,.],[.,[.,.]]]] => [[],[[[],[]],[[],[[],[]]]]] => 3
[[[.,.],[.,[.,.]]],.] => [.,[[[.,.],.],[.,.]]] => [.,[[[.,[.,.]],.],.]] => [[],[[[[],[[],[]]],[]],[]]] => 2
[[[.,.],[[.,.],.]],.] => [.,[[.,[.,.]],[.,.]]] => [.,[[.,[.,.]],[.,.]]] => [[],[[[],[[],[]]],[[],[]]]] => 3
[[[.,[.,.]],[.,.]],.] => [.,[[.,.],[[.,.],.]]] => [.,[[.,[[.,.],.]],.]] => [[],[[[],[[[],[]],[]]],[]]] => 2
[[[[.,.],.],[.,.]],.] => [.,[[.,.],[.,[.,.]]]] => [.,[[.,[.,[.,.]]],.]] => [[],[[[],[[],[[],[]]]],[]]] => 2
[[[.,[.,[.,.]]],.],.] => [.,[.,[[[.,.],.],.]]] => [.,[.,[[[.,.],.],.]]] => [[],[[],[[[[],[]],[]],[]]]] => 2
[[[.,[[.,.],.]],.],.] => [.,[.,[[.,[.,.]],.]]] => [.,[.,[[.,.],[.,.]]]] => [[],[[],[[[],[]],[[],[]]]]] => 3
[[[[.,.],[.,.]],.],.] => [.,[.,[[.,.],[.,.]]]] => [.,[.,[[.,[.,.]],.]]] => [[],[[],[[[],[[],[]]],[]]]] => 2
[[[[.,[.,.]],.],.],.] => [.,[.,[.,[[.,.],.]]]] => [.,[.,[.,[[.,.],.]]]] => [[],[[],[[],[[[],[]],[]]]]] => 2
[[[[[.,.],.],.],.],.] => [.,[.,[.,[.,[.,.]]]]] => [.,[.,[.,[.,[.,.]]]]] => [[],[[],[[],[[],[[],[]]]]]] => 2
[.,[.,[.,[.,[.,[.,.]]]]]] => [[[[[[.,.],.],.],.],.],.] => [[[[[[.,.],.],.],.],.],.] => [[[[[[[],[]],[]],[]],[]],[]],[]] => 2
[.,[.,[.,[.,[[.,.],.]]]]] => [[[[[.,[.,.]],.],.],.],.] => [[[[[.,.],.],.],.],[.,.]] => [[[[[[],[]],[]],[]],[]],[[],[]]] => 3
[.,[.,[.,[[.,.],[.,.]]]]] => [[[[[.,.],[.,.]],.],.],.] => [[[[[.,.],.],.],[.,.]],.] => [[[[[[],[]],[]],[]],[[],[]]],[]] => 3
[.,[.,[.,[[.,[.,.]],.]]]] => [[[[.,[[.,.],.]],.],.],.] => [[[[.,.],.],.],[[.,.],.]] => [[[[[],[]],[]],[]],[[[],[]],[]]] => 3
[.,[.,[.,[[[.,.],.],.]]]] => [[[[.,[.,[.,.]]],.],.],.] => [[[[.,.],.],.],[.,[.,.]]] => [[[[[],[]],[]],[]],[[],[[],[]]]] => 3
[.,[.,[[.,.],[.,[.,.]]]]] => [[[[[.,.],.],[.,.]],.],.] => [[[[[.,.],.],[.,.]],.],.] => [[[[[[],[]],[]],[[],[]]],[]],[]] => 3
[.,[.,[[.,.],[[.,.],.]]]] => [[[[.,[.,.]],[.,.]],.],.] => [[[[.,.],.],[.,.]],[.,.]] => [[[[[],[]],[]],[[],[]]],[[],[]]] => 3
[.,[.,[[.,[.,.]],[.,.]]]] => [[[[.,.],[[.,.],.]],.],.] => [[[[.,.],.],[[.,.],.]],.] => [[[[[],[]],[]],[[[],[]],[]]],[]] => 3
[.,[.,[[[.,.],.],[.,.]]]] => [[[[.,.],[.,[.,.]]],.],.] => [[[[.,.],.],[.,[.,.]]],.] => [[[[[],[]],[]],[[],[[],[]]]],[]] => 3
[.,[.,[[.,[.,[.,.]]],.]]] => [[[.,[[[.,.],.],.]],.],.] => [[[.,.],.],[[[.,.],.],.]] => [[[[],[]],[]],[[[[],[]],[]],[]]] => 3
[.,[.,[[.,[[.,.],.]],.]]] => [[[.,[[.,[.,.]],.]],.],.] => [[[.,.],.],[[.,.],[.,.]]] => [[[[],[]],[]],[[[],[]],[[],[]]]] => 3
[.,[.,[[[.,.],[.,.]],.]]] => [[[.,[[.,.],[.,.]]],.],.] => [[[.,.],.],[[.,[.,.]],.]] => [[[[],[]],[]],[[[],[[],[]]],[]]] => 3
[.,[.,[[[.,[.,.]],.],.]]] => [[[.,[.,[[.,.],.]]],.],.] => [[[.,.],.],[.,[[.,.],.]]] => [[[[],[]],[]],[[],[[[],[]],[]]]] => 3
[.,[.,[[[[.,.],.],.],.]]] => [[[.,[.,[.,[.,.]]]],.],.] => [[[.,.],.],[.,[.,[.,.]]]] => [[[[],[]],[]],[[],[[],[[],[]]]]] => 3
[.,[[.,.],[.,[.,[.,.]]]]] => [[[[[.,.],.],.],[.,.]],.] => [[[[[.,.],[.,.]],.],.],.] => [[[[[[],[]],[[],[]]],[]],[]],[]] => 3
[.,[[.,.],[.,[[.,.],.]]]] => [[[[.,[.,.]],.],[.,.]],.] => [[[[.,.],[.,.]],.],[.,.]] => [[[[[],[]],[[],[]]],[]],[[],[]]] => 3
[.,[[.,.],[[.,.],[.,.]]]] => [[[[.,.],[.,.]],[.,.]],.] => [[[[.,.],[.,.]],[.,.]],.] => [[[[[],[]],[[],[]]],[[],[]]],[]] => 3
[.,[[.,.],[[.,[.,.]],.]]] => [[[.,[[.,.],.]],[.,.]],.] => [[[.,.],[.,.]],[[.,.],.]] => [[[[],[]],[[],[]]],[[[],[]],[]]] => 3
[.,[[.,.],[[[.,.],.],.]]] => [[[.,[.,[.,.]]],[.,.]],.] => [[[.,.],[.,.]],[.,[.,.]]] => [[[[],[]],[[],[]]],[[],[[],[]]]] => 3
[.,[[.,[.,.]],[.,[.,.]]]] => [[[[.,.],.],[[.,.],.]],.] => [[[[.,.],[[.,.],.]],.],.] => [[[[[],[]],[[[],[]],[]]],[]],[]] => 3
[.,[[.,[.,.]],[[.,.],.]]] => [[[.,[.,.]],[[.,.],.]],.] => [[[.,.],[[.,.],.]],[.,.]] => [[[[],[]],[[[],[]],[]]],[[],[]]] => 3
[.,[[[.,.],.],[.,[.,.]]]] => [[[[.,.],.],[.,[.,.]]],.] => [[[[.,.],[.,[.,.]]],.],.] => [[[[[],[]],[[],[[],[]]]],[]],[]] => 3
[.,[[[.,.],.],[[.,.],.]]] => [[[.,[.,.]],[.,[.,.]]],.] => [[[.,.],[.,[.,.]]],[.,.]] => [[[[],[]],[[],[[],[]]]],[[],[]]] => 3
[.,[[.,[.,[.,.]]],[.,.]]] => [[[.,.],[[[.,.],.],.]],.] => [[[.,.],[[[.,.],.],.]],.] => [[[[],[]],[[[[],[]],[]],[]]],[]] => 3
[.,[[.,[[.,.],.]],[.,.]]] => [[[.,.],[[.,[.,.]],.]],.] => [[[.,.],[[.,.],[.,.]]],.] => [[[[],[]],[[[],[]],[[],[]]]],[]] => 3
[.,[[[.,.],[.,.]],[.,.]]] => [[[.,.],[[.,.],[.,.]]],.] => [[[.,.],[[.,[.,.]],.]],.] => [[[[],[]],[[[],[[],[]]],[]]],[]] => 3
[.,[[[.,[.,.]],.],[.,.]]] => [[[.,.],[.,[[.,.],.]]],.] => [[[.,.],[.,[[.,.],.]]],.] => [[[[],[]],[[],[[[],[]],[]]]],[]] => 3
[.,[[[[.,.],.],.],[.,.]]] => [[[.,.],[.,[.,[.,.]]]],.] => [[[.,.],[.,[.,[.,.]]]],.] => [[[[],[]],[[],[[],[[],[]]]]],[]] => 3
[.,[[.,[.,[.,[.,.]]]],.]] => [[.,[[[[.,.],.],.],.]],.] => [[.,.],[[[[.,.],.],.],.]] => [[[],[]],[[[[[],[]],[]],[]],[]]] => 3
[.,[[.,[.,[[.,.],.]]],.]] => [[.,[[[.,[.,.]],.],.]],.] => [[.,.],[[[.,.],.],[.,.]]] => [[[],[]],[[[[],[]],[]],[[],[]]]] => 3
[.,[[.,[[.,.],[.,.]]],.]] => [[.,[[[.,.],[.,.]],.]],.] => [[.,.],[[[.,.],[.,.]],.]] => [[[],[]],[[[[],[]],[[],[]]],[]]] => 3
[.,[[.,[[.,[.,.]],.]],.]] => [[.,[[.,[[.,.],.]],.]],.] => [[.,.],[[.,.],[[.,.],.]]] => [[[],[]],[[[],[]],[[[],[]],[]]]] => 3
[.,[[.,[[[.,.],.],.]],.]] => [[.,[[.,[.,[.,.]]],.]],.] => [[.,.],[[.,.],[.,[.,.]]]] => [[[],[]],[[[],[]],[[],[[],[]]]]] => 3
[.,[[[.,.],[.,[.,.]]],.]] => [[.,[[[.,.],.],[.,.]]],.] => [[.,.],[[[.,[.,.]],.],.]] => [[[],[]],[[[[],[[],[]]],[]],[]]] => 3
[.,[[[.,.],[[.,.],.]],.]] => [[.,[[.,[.,.]],[.,.]]],.] => [[.,.],[[.,[.,.]],[.,.]]] => [[[],[]],[[[],[[],[]]],[[],[]]]] => 3
[.,[[[.,[.,.]],[.,.]],.]] => [[.,[[.,.],[[.,.],.]]],.] => [[.,.],[[.,[[.,.],.]],.]] => [[[],[]],[[[],[[[],[]],[]]],[]]] => 3
[.,[[[[.,.],.],[.,.]],.]] => [[.,[[.,.],[.,[.,.]]]],.] => [[.,.],[[.,[.,[.,.]]],.]] => [[[],[]],[[[],[[],[[],[]]]],[]]] => 3
>>> Load all 196 entries. <<<
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searching the database for statistics with the same generating function
Description
The Strahler number of a rooted tree.
Map
left-right symmetry
Description
Return the left-right symmetrized tree of a binary tree.
Map
left border symmetry
Description
Return the tree where a symmetry has been applied recursively on all left borders. If a tree is made of three trees $T_1, T_2, T_3$ on its left border, it becomes $T_3, T_2, T_1$ where same symmetry has been applied to $T_1, T_2, T_3$.
Map
to complete tree
Description
Return the same tree seen as an ordered tree. By default, leaves are transformed into actual nodes.
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