- FindStat

Identifier

Mp00018: Binary trees —left border symmetry⟶ Binary trees
St000396: Binary trees ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The register function (or Horton-Strahler number) of a binary tree.
This is different from the dimension of the associated poset for the tree $[[[.,.],[.,.]],[[.,.],[.,.]]]$: its register function is 3, whereas the dimension of the associated poset is 2.

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 $T_1, T_2, T_3$.