- FindStat

Identifier

Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
St000568: Binary trees ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The hook number of a binary tree.
A hook of a binary tree is a vertex together with is left- and its right-most branch. Then there is a unique decomposition of the tree into hooks and the hook number is the number of hooks in this decomposition.

Map

to binary tree: left brother = left child

Description

Return a binary tree of size $n-1$ (where $n$ is the size of $t$, and where $t$ is an ordered tree) by the following recursive rule:
- if $x$ is the left brother of $y$ in $t$, then $x$ becomes the left child of $y$;
- if $x$ is the last child of $y$ in $t$, then $x$ becomes the right child of $y$,
and removing the root of $t$.