- FindStat

Identifier

Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
St000201: Binary trees ⟶ ℤ (values match St000196The number of occurrences of the contiguous pattern [[.,.],[.,.)

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The number of leaf nodes in a binary tree.
Equivalently, the number of cherries [1] in the complete binary tree.
The number of binary trees of size $n$, at least $1$, with exactly one leaf node for is $2^{n-1}$, see [2].
The number of binary tree of size $n$, at least $3$, with exactly two leaf nodes is $n(n+1)2^{n-2}$, see [3].

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 $t$.