- FindStat

Identifier

Mp00010: Binary trees —to ordered tree: left child = left brother⟶ Ordered trees
St000522: Ordered trees ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The number of 1-protected nodes of a rooted tree.
This is the number of nodes with minimal distance one to a leaf.

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 $y$.