- FindStat

Identifier

Mp00016: Binary trees —left-right symmetry⟶ Binary trees
St000203: Binary trees ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[.,[.,[.,[.,[.,[.,.]]]]]] => [[[[[[.,.],.],.],.],.],.] => 6

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 955 entries. <<<

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

$F_{1} = q$

$F_{2} = 2\ q^{2}$

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

$F_{4} = 4\ q^{2} + 6\ q^{3} + 4\ q^{4}$

$F_{5} = 10\ q^{2} + 15\ q^{3} + 12\ q^{4} + 5\ q^{5}$

$F_{6} = 28\ q^{2} + 42\ q^{3} + 36\ q^{4} + 20\ q^{5} + 6\ q^{6}$

$F_{7} = 84\ q^{2} + 126\ q^{3} + 112\ q^{4} + 70\ q^{5} + 30\ q^{6} + 7\ q^{7}$

Description

The number of external nodes of a binary tree.
That is, the number of nodes that can be reached from the root by only left steps or only right steps, plus

$1$ for the root node itself. A counting formula for the number of external node in all binary trees of size

$n$ can be found in [1].

Map

left-right symmetry

Description

Return the left-right symmetrized tree of a binary tree.