- FindStat

Identifier

Mp00018: Binary trees —left border symmetry⟶ Binary trees
St000414: Binary trees ⟶ ℤ

Values

[.,[.,.]] => [.,[.,.]] => 1

[[.,.],.] => [[.,.],.] => 1

[.,[.,[.,.]]] => [.,[.,[.,.]]] => 2

[.,[[.,.],.]] => [.,[[.,.],.]] => 2

[[.,.],[.,.]] => [[.,[.,.]],.] => 2

[[.,[.,.]],.] => [[.,.],[.,.]] => 0

[[[.,.],.],.] => [[[.,.],.],.] => 2

[.,[.,[.,[.,.]]]] => [.,[.,[.,[.,.]]]] => 3

[.,[.,[[.,.],.]]] => [.,[.,[[.,.],.]]] => 3

[.,[[.,.],[.,.]]] => [.,[[.,[.,.]],.]] => 3

[.,[[.,[.,.]],.]] => [.,[[.,.],[.,.]]] => 1

[.,[[[.,.],.],.]] => [.,[[[.,.],.],.]] => 3

[[.,.],[.,[.,.]]] => [[.,[.,[.,.]]],.] => 3

[[.,.],[[.,.],.]] => [[.,[[.,.],.]],.] => 3

[[.,[.,.]],[.,.]] => [[.,[.,.]],[.,.]] => 2

[[[.,.],.],[.,.]] => [[[.,[.,.]],.],.] => 3

[[.,[.,[.,.]]],.] => [[.,.],[.,[.,.]]] => 2

[[.,[[.,.],.]],.] => [[.,.],[[.,.],.]] => 2

[[[.,.],[.,.]],.] => [[[.,.],[.,.]],.] => 1

[[[.,[.,.]],.],.] => [[[.,.],.],[.,.]] => 2

[[[[.,.],.],.],.] => [[[[.,.],.],.],.] => 3

[.,[.,[.,[.,[.,.]]]]] => [.,[.,[.,[.,[.,.]]]]] => 4

[.,[.,[.,[[.,.],.]]]] => [.,[.,[.,[[.,.],.]]]] => 4

[.,[.,[[.,.],[.,.]]]] => [.,[.,[[.,[.,.]],.]]] => 4

[.,[.,[[.,[.,.]],.]]] => [.,[.,[[.,.],[.,.]]]] => 2

[.,[.,[[[.,.],.],.]]] => [.,[.,[[[.,.],.],.]]] => 4

[.,[[.,.],[.,[.,.]]]] => [.,[[.,[.,[.,.]]],.]] => 4

[.,[[.,.],[[.,.],.]]] => [.,[[.,[[.,.],.]],.]] => 4

[.,[[.,[.,.]],[.,.]]] => [.,[[.,[.,.]],[.,.]]] => 3

[.,[[[.,.],.],[.,.]]] => [.,[[[.,[.,.]],.],.]] => 4

[.,[[.,[.,[.,.]]],.]] => [.,[[.,.],[.,[.,.]]]] => 3

[.,[[.,[[.,.],.]],.]] => [.,[[.,.],[[.,.],.]]] => 3

[.,[[[.,.],[.,.]],.]] => [.,[[[.,.],[.,.]],.]] => 2

[.,[[[.,[.,.]],.],.]] => [.,[[[.,.],.],[.,.]]] => 3

[.,[[[[.,.],.],.],.]] => [.,[[[[.,.],.],.],.]] => 4

[[.,.],[.,[.,[.,.]]]] => [[.,[.,[.,[.,.]]]],.] => 4

[[.,.],[.,[[.,.],.]]] => [[.,[.,[[.,.],.]]],.] => 4

[[.,.],[[.,.],[.,.]]] => [[.,[[.,[.,.]],.]],.] => 4

[[.,.],[[.,[.,.]],.]] => [[.,[[.,.],[.,.]]],.] => 2

[[.,.],[[[.,.],.],.]] => [[.,[[[.,.],.],.]],.] => 4

[[.,[.,.]],[.,[.,.]]] => [[.,[.,[.,.]]],[.,.]] => 3

[[.,[.,.]],[[.,.],.]] => [[.,[[.,.],.]],[.,.]] => 3

[[[.,.],.],[.,[.,.]]] => [[[.,[.,[.,.]]],.],.] => 4

[[[.,.],.],[[.,.],.]] => [[[.,[[.,.],.]],.],.] => 4

[[.,[.,[.,.]]],[.,.]] => [[.,[.,.]],[.,[.,.]]] => 2

[[.,[[.,.],.]],[.,.]] => [[.,[.,.]],[[.,.],.]] => 2

[[[.,.],[.,.]],[.,.]] => [[[.,[.,.]],[.,.]],.] => 3

[[[.,[.,.]],.],[.,.]] => [[[.,[.,.]],.],[.,.]] => 3

[[[[.,.],.],.],[.,.]] => [[[[.,[.,.]],.],.],.] => 4

[[.,[.,[.,[.,.]]]],.] => [[.,.],[.,[.,[.,.]]]] => 3

[[.,[.,[[.,.],.]]],.] => [[.,.],[.,[[.,.],.]]] => 3

[[.,[[.,.],[.,.]]],.] => [[.,.],[[.,[.,.]],.]] => 3

[[.,[[.,[.,.]],.]],.] => [[.,.],[[.,.],[.,.]]] => 1

[[.,[[[.,.],.],.]],.] => [[.,.],[[[.,.],.],.]] => 3

[[[.,.],[.,[.,.]]],.] => [[[.,.],[.,[.,.]]],.] => 3

[[[.,.],[[.,.],.]],.] => [[[.,.],[[.,.],.]],.] => 3

[[[.,[.,.]],[.,.]],.] => [[[.,.],[.,.]],[.,.]] => 1

[[[[.,.],.],[.,.]],.] => [[[[.,.],[.,.]],.],.] => 2

[[[.,[.,[.,.]]],.],.] => [[[.,.],.],[.,[.,.]]] => 2

[[[.,[[.,.],.]],.],.] => [[[.,.],.],[[.,.],.]] => 2

[[[[.,.],[.,.]],.],.] => [[[[.,.],.],[.,.]],.] => 3

[[[[.,[.,.]],.],.],.] => [[[[.,.],.],.],[.,.]] => 3

[[[[[.,.],.],.],.],.] => [[[[[.,.],.],.],.],.] => 4

[.,[.,[.,[.,[.,[.,.]]]]]] => [.,[.,[.,[.,[.,[.,.]]]]]] => 5

[.,[.,[.,[.,[[.,.],.]]]]] => [.,[.,[.,[.,[[.,.],.]]]]] => 5

[.,[.,[.,[[.,.],[.,.]]]]] => [.,[.,[.,[[.,[.,.]],.]]]] => 5

[.,[.,[.,[[.,[.,.]],.]]]] => [.,[.,[.,[[.,.],[.,.]]]]] => 3

[.,[.,[.,[[[.,.],.],.]]]] => [.,[.,[.,[[[.,.],.],.]]]] => 5

[.,[.,[[.,.],[.,[.,.]]]]] => [.,[.,[[.,[.,[.,.]]],.]]] => 5

[.,[.,[[.,.],[[.,.],.]]]] => [.,[.,[[.,[[.,.],.]],.]]] => 5

[.,[.,[[.,[.,.]],[.,.]]]] => [.,[.,[[.,[.,.]],[.,.]]]] => 4

[.,[.,[[[.,.],.],[.,.]]]] => [.,[.,[[[.,[.,.]],.],.]]] => 5

[.,[.,[[.,[.,[.,.]]],.]]] => [.,[.,[[.,.],[.,[.,.]]]]] => 4

[.,[.,[[.,[[.,.],.]],.]]] => [.,[.,[[.,.],[[.,.],.]]]] => 4

[.,[.,[[[.,.],[.,.]],.]]] => [.,[.,[[[.,.],[.,.]],.]]] => 3

[.,[.,[[[.,[.,.]],.],.]]] => [.,[.,[[[.,.],.],[.,.]]]] => 4

[.,[.,[[[[.,.],.],.],.]]] => [.,[.,[[[[.,.],.],.],.]]] => 5

[.,[[.,.],[.,[.,[.,.]]]]] => [.,[[.,[.,[.,[.,.]]]],.]] => 5

[.,[[.,.],[.,[[.,.],.]]]] => [.,[[.,[.,[[.,.],.]]],.]] => 5

[.,[[.,.],[[.,.],[.,.]]]] => [.,[[.,[[.,[.,.]],.]],.]] => 5

[.,[[.,.],[[.,[.,.]],.]]] => [.,[[.,[[.,.],[.,.]]],.]] => 3

[.,[[.,.],[[[.,.],.],.]]] => [.,[[.,[[[.,.],.],.]],.]] => 5

[.,[[.,[.,.]],[.,[.,.]]]] => [.,[[.,[.,[.,.]]],[.,.]]] => 4

[.,[[.,[.,.]],[[.,.],.]]] => [.,[[.,[[.,.],.]],[.,.]]] => 4

[.,[[[.,.],.],[.,[.,.]]]] => [.,[[[.,[.,[.,.]]],.],.]] => 5

[.,[[[.,.],.],[[.,.],.]]] => [.,[[[.,[[.,.],.]],.],.]] => 5

[.,[[.,[.,[.,.]]],[.,.]]] => [.,[[.,[.,.]],[.,[.,.]]]] => 3

[.,[[.,[[.,.],.]],[.,.]]] => [.,[[.,[.,.]],[[.,.],.]]] => 3

[.,[[[.,.],[.,.]],[.,.]]] => [.,[[[.,[.,.]],[.,.]],.]] => 4

[.,[[[.,[.,.]],.],[.,.]]] => [.,[[[.,[.,.]],.],[.,.]]] => 4

[.,[[[[.,.],.],.],[.,.]]] => [.,[[[[.,[.,.]],.],.],.]] => 5

[.,[[.,[.,[.,[.,.]]]],.]] => [.,[[.,.],[.,[.,[.,.]]]]] => 4

[.,[[.,[.,[[.,.],.]]],.]] => [.,[[.,.],[.,[[.,.],.]]]] => 4

[.,[[.,[[.,.],[.,.]]],.]] => [.,[[.,.],[[.,[.,.]],.]]] => 4

[.,[[.,[[.,[.,.]],.]],.]] => [.,[[.,.],[[.,.],[.,.]]]] => 2

[.,[[.,[[[.,.],.],.]],.]] => [.,[[.,.],[[[.,.],.],.]]] => 4

[.,[[[.,.],[.,[.,.]]],.]] => [.,[[[.,.],[.,[.,.]]],.]] => 4

[.,[[[.,.],[[.,.],.]],.]] => [.,[[[.,.],[[.,.],.]],.]] => 4

[.,[[[.,[.,.]],[.,.]],.]] => [.,[[[.,.],[.,.]],[.,.]]] => 2

[.,[[[[.,.],.],[.,.]],.]] => [.,[[[[.,.],[.,.]],.],.]] => 3

[.,[[[.,[.,[.,.]]],.],.]] => [.,[[[.,.],.],[.,[.,.]]]] => 3

>>> Load all 1200 entries. <<<

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Description

The binary logarithm of the number of binary trees with the same underlying unordered 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 $T_1, T_2, T_3$.