- FindStat

Identifier

Mp00008: Binary trees —to complete tree⟶ Ordered trees
Mp00048: Ordered trees —left-right symmetry⟶ Ordered trees
Mp00246: Ordered trees —rotate⟶ Ordered trees
St000700: Ordered trees ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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>>> Load all 196 entries. <<<

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search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

$F_{1} = q$

$F_{2} = 2\ q$

$F_{3} = 4\ q + q^{2}$

$F_{4} = 10\ q + 4\ q^{2}$

$F_{5} = 28\ q + 14\ q^{2}$

$F_{6} = 84\ q + 48\ q^{2}$

Description

The protection number of an ordered tree.
This is the minimal distance from the root to a leaf.

Map

to complete tree

Description

Return the same tree seen as an ordered tree. By default, leaves are transformed into actual nodes.

Map

left-right symmetry

Description

Return the left-right symmetrized tree of an ordered tree.

Map

rotate

Description

The rotation of an ordered tree.
Let

$T$ be a rooted ordered tree with root

$r$ and left most leaf

$\ell$ . It's rotation is obtained by attaching a leaf to

$r$ , removing the leaf

$\ell$ and letting the vertex attached to

$\ell$ be the new root.
By duality, this operation corresponds to rotation of polygon dissections.