- FindStat

Identifier

Mp00018: Binary trees —left border symmetry⟶ Binary trees
St000385: Binary trees ⟶ ℤ

Values

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

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

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

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

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

[[.,[.,.]],.] => [[.,.],[.,.]] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[.,[[.,[.,.]],.]],.] => [[.,.],[[.,.],[.,.]]] => 0

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

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

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

[[[.,[.,.]],[.,.]],.] => [[[.,.],[.,.]],[.,.]] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The number of vertices with out-degree 1 in a binary tree.
See the references for several connections of this statistic.
In particular, the number $T(n,k)$ of binary trees with $n$ vertices and $k$ out-degree $1$ vertices is given by $T(n,k) = 0$ for $n-k$ odd and
$$T(n,k)=\frac{2^k}{n+1}\binom{n+1}{k}\binom{n+1-k}{(n-k)/2}$$
for $n-k$ is even.

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 $T_1, T_2, T_3$.