- FindStat

Identifier

Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
St000196: Binary trees ⟶ ℤ (values match St000201The number of leaf nodes in a binary tree.)

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The number of occurrences of the contiguous pattern [[.,.],[.,.]] in a binary tree.
Equivalently, this is the number of branches in the tree, i.e. the number of nodes with two children. Binary trees avoiding this pattern are counted by $2^{n-2}$.

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 $t$.