- FindStat

Identifier

Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
St001554: Binary trees ⟶ ℤ

Values

[1,0] => [.,.] => 1

[1,0,1,0] => [[.,.],.] => 2

[1,1,0,0] => [.,[.,.]] => 2

[1,0,1,0,1,0] => [[[.,.],.],.] => 3

[1,0,1,1,0,0] => [[.,.],[.,.]] => 2

[1,1,0,0,1,0] => [[.,[.,.]],.] => 3

[1,1,0,1,0,0] => [.,[[.,.],.]] => 3

[1,1,1,0,0,0] => [.,[.,[.,.]]] => 3

[1,0,1,0,1,0,1,0] => [[[[.,.],.],.],.] => 4

[1,0,1,0,1,1,0,0] => [[[.,.],.],[.,.]] => 3

[1,0,1,1,0,0,1,0] => [[[.,.],[.,.]],.] => 3

[1,0,1,1,0,1,0,0] => [[.,.],[[.,.],.]] => 3

[1,0,1,1,1,0,0,0] => [[.,.],[.,[.,.]]] => 3

[1,1,0,0,1,0,1,0] => [[[.,[.,.]],.],.] => 4

[1,1,0,0,1,1,0,0] => [[.,[.,.]],[.,.]] => 3

[1,1,0,1,0,0,1,0] => [[.,[[.,.],.]],.] => 4

[1,1,0,1,0,1,0,0] => [.,[[[.,.],.],.]] => 4

[1,1,0,1,1,0,0,0] => [.,[[.,.],[.,.]]] => 3

[1,1,1,0,0,0,1,0] => [[.,[.,[.,.]]],.] => 4

[1,1,1,0,0,1,0,0] => [.,[[.,[.,.]],.]] => 4

[1,1,1,0,1,0,0,0] => [.,[.,[[.,.],.]]] => 4

[1,1,1,1,0,0,0,0] => [.,[.,[.,[.,.]]]] => 4

[1,0,1,0,1,0,1,0,1,0] => [[[[[.,.],.],.],.],.] => 5

[1,0,1,0,1,0,1,1,0,0] => [[[[.,.],.],.],[.,.]] => 4

[1,0,1,0,1,1,0,0,1,0] => [[[[.,.],.],[.,.]],.] => 4

[1,0,1,0,1,1,0,1,0,0] => [[[.,.],.],[[.,.],.]] => 3

[1,0,1,0,1,1,1,0,0,0] => [[[.,.],.],[.,[.,.]]] => 4

[1,0,1,1,0,0,1,0,1,0] => [[[[.,.],[.,.]],.],.] => 4

[1,0,1,1,0,0,1,1,0,0] => [[[.,.],[.,.]],[.,.]] => 3

[1,0,1,1,0,1,0,0,1,0] => [[[.,.],[[.,.],.]],.] => 4

[1,0,1,1,0,1,0,1,0,0] => [[.,.],[[[.,.],.],.]] => 4

[1,0,1,1,0,1,1,0,0,0] => [[.,.],[[.,.],[.,.]]] => 3

[1,0,1,1,1,0,0,0,1,0] => [[[.,.],[.,[.,.]]],.] => 4

[1,0,1,1,1,0,0,1,0,0] => [[.,.],[[.,[.,.]],.]] => 4

[1,0,1,1,1,0,1,0,0,0] => [[.,.],[.,[[.,.],.]]] => 4

[1,0,1,1,1,1,0,0,0,0] => [[.,.],[.,[.,[.,.]]]] => 4

[1,1,0,0,1,0,1,0,1,0] => [[[[.,[.,.]],.],.],.] => 5

[1,1,0,0,1,0,1,1,0,0] => [[[.,[.,.]],.],[.,.]] => 4

[1,1,0,0,1,1,0,0,1,0] => [[[.,[.,.]],[.,.]],.] => 4

[1,1,0,0,1,1,0,1,0,0] => [[.,[.,.]],[[.,.],.]] => 4

[1,1,0,0,1,1,1,0,0,0] => [[.,[.,.]],[.,[.,.]]] => 3

[1,1,0,1,0,0,1,0,1,0] => [[[.,[[.,.],.]],.],.] => 5

[1,1,0,1,0,0,1,1,0,0] => [[.,[[.,.],.]],[.,.]] => 4

[1,1,0,1,0,1,0,0,1,0] => [[.,[[[.,.],.],.]],.] => 5

[1,1,0,1,0,1,0,1,0,0] => [.,[[[[.,.],.],.],.]] => 5

[1,1,0,1,0,1,1,0,0,0] => [.,[[[.,.],.],[.,.]]] => 4

[1,1,0,1,1,0,0,0,1,0] => [[.,[[.,.],[.,.]]],.] => 4

[1,1,0,1,1,0,0,1,0,0] => [.,[[[.,.],[.,.]],.]] => 4

[1,1,0,1,1,0,1,0,0,0] => [.,[[.,.],[[.,.],.]]] => 4

[1,1,0,1,1,1,0,0,0,0] => [.,[[.,.],[.,[.,.]]]] => 4

[1,1,1,0,0,0,1,0,1,0] => [[[.,[.,[.,.]]],.],.] => 5

[1,1,1,0,0,0,1,1,0,0] => [[.,[.,[.,.]]],[.,.]] => 4

[1,1,1,0,0,1,0,0,1,0] => [[.,[[.,[.,.]],.]],.] => 5

[1,1,1,0,0,1,0,1,0,0] => [.,[[[.,[.,.]],.],.]] => 5

[1,1,1,0,0,1,1,0,0,0] => [.,[[.,[.,.]],[.,.]]] => 4

[1,1,1,0,1,0,0,0,1,0] => [[.,[.,[[.,.],.]]],.] => 5

[1,1,1,0,1,0,0,1,0,0] => [.,[[.,[[.,.],.]],.]] => 5

[1,1,1,0,1,0,1,0,0,0] => [.,[.,[[[.,.],.],.]]] => 5

[1,1,1,0,1,1,0,0,0,0] => [.,[.,[[.,.],[.,.]]]] => 4

[1,1,1,1,0,0,0,0,1,0] => [[.,[.,[.,[.,.]]]],.] => 5

[1,1,1,1,0,0,0,1,0,0] => [.,[[.,[.,[.,.]]],.]] => 5

[1,1,1,1,0,0,1,0,0,0] => [.,[.,[[.,[.,.]],.]]] => 5

[1,1,1,1,0,1,0,0,0,0] => [.,[.,[.,[[.,.],.]]]] => 5

[1,1,1,1,1,0,0,0,0,0] => [.,[.,[.,[.,[.,.]]]]] => 5

[1,0,1,0,1,0,1,0,1,0,1,0] => [[[[[[.,.],.],.],.],.],.] => 6

[1,0,1,0,1,0,1,0,1,1,0,0] => [[[[[.,.],.],.],.],[.,.]] => 5

[1,0,1,0,1,0,1,1,0,0,1,0] => [[[[[.,.],.],.],[.,.]],.] => 5

[1,0,1,0,1,0,1,1,0,1,0,0] => [[[[.,.],.],.],[[.,.],.]] => 4

[1,0,1,0,1,0,1,1,1,0,0,0] => [[[[.,.],.],.],[.,[.,.]]] => 5

[1,0,1,0,1,1,0,0,1,0,1,0] => [[[[[.,.],.],[.,.]],.],.] => 5

[1,0,1,0,1,1,0,0,1,1,0,0] => [[[[.,.],.],[.,.]],[.,.]] => 4

[1,0,1,0,1,1,0,1,0,0,1,0] => [[[[.,.],.],[[.,.],.]],.] => 4

[1,0,1,0,1,1,0,1,0,1,0,0] => [[[.,.],.],[[[.,.],.],.]] => 4

[1,0,1,0,1,1,0,1,1,0,0,0] => [[[.,.],.],[[.,.],[.,.]]] => 4

[1,0,1,0,1,1,1,0,0,0,1,0] => [[[[.,.],.],[.,[.,.]]],.] => 5

[1,0,1,0,1,1,1,0,0,1,0,0] => [[[.,.],.],[[.,[.,.]],.]] => 5

[1,0,1,0,1,1,1,0,1,0,0,0] => [[[.,.],.],[.,[[.,.],.]]] => 4

[1,0,1,0,1,1,1,1,0,0,0,0] => [[[.,.],.],[.,[.,[.,.]]]] => 5

[1,0,1,1,0,0,1,0,1,0,1,0] => [[[[[.,.],[.,.]],.],.],.] => 5

[1,0,1,1,0,0,1,0,1,1,0,0] => [[[[.,.],[.,.]],.],[.,.]] => 4

[1,0,1,1,0,0,1,1,0,0,1,0] => [[[[.,.],[.,.]],[.,.]],.] => 4

[1,0,1,1,0,0,1,1,0,1,0,0] => [[[.,.],[.,.]],[[.,.],.]] => 4

[1,0,1,1,0,0,1,1,1,0,0,0] => [[[.,.],[.,.]],[.,[.,.]]] => 4

[1,0,1,1,0,1,0,0,1,0,1,0] => [[[[.,.],[[.,.],.]],.],.] => 5

[1,0,1,1,0,1,0,0,1,1,0,0] => [[[.,.],[[.,.],.]],[.,.]] => 4

[1,0,1,1,0,1,0,1,0,0,1,0] => [[[.,.],[[[.,.],.],.]],.] => 5

[1,0,1,1,0,1,0,1,0,1,0,0] => [[.,.],[[[[.,.],.],.],.]] => 5

[1,0,1,1,0,1,0,1,1,0,0,0] => [[.,.],[[[.,.],.],[.,.]]] => 4

[1,0,1,1,0,1,1,0,0,0,1,0] => [[[.,.],[[.,.],[.,.]]],.] => 4

[1,0,1,1,0,1,1,0,0,1,0,0] => [[.,.],[[[.,.],[.,.]],.]] => 4

[1,0,1,1,0,1,1,0,1,0,0,0] => [[.,.],[[.,.],[[.,.],.]]] => 4

[1,0,1,1,0,1,1,1,0,0,0,0] => [[.,.],[[.,.],[.,[.,.]]]] => 4

[1,0,1,1,1,0,0,0,1,0,1,0] => [[[[.,.],[.,[.,.]]],.],.] => 5

[1,0,1,1,1,0,0,0,1,1,0,0] => [[[.,.],[.,[.,.]]],[.,.]] => 4

[1,0,1,1,1,0,0,1,0,0,1,0] => [[[.,.],[[.,[.,.]],.]],.] => 5

[1,0,1,1,1,0,0,1,0,1,0,0] => [[.,.],[[[.,[.,.]],.],.]] => 5

[1,0,1,1,1,0,0,1,1,0,0,0] => [[.,.],[[.,[.,.]],[.,.]]] => 4

[1,0,1,1,1,0,1,0,0,0,1,0] => [[[.,.],[.,[[.,.],.]]],.] => 5

[1,0,1,1,1,0,1,0,0,1,0,0] => [[.,.],[[.,[[.,.],.]],.]] => 5

[1,0,1,1,1,0,1,0,1,0,0,0] => [[.,.],[.,[[[.,.],.],.]]] => 5

[1,0,1,1,1,0,1,1,0,0,0,0] => [[.,.],[.,[[.,.],[.,.]]]] => 4

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Description

The number of distinct nonempty subtrees of a binary tree.

Map

to binary tree: left tree, up step, right tree, down step

Description

Return the binary tree corresponding to the Dyck path under the transformation left tree - up step - right tree - down step.
A Dyck path $D$ of semilength $n$ with $n > 1$ may be uniquely decomposed into $L 1 R 0$ for Dyck paths $L,R$ of respective semilengths $n_1,n_2$ with $n_1+n_2 = n-1$.
This map sends $D$ to the binary tree $T$ consisting of a root node with a left child according to $L$ and a right child according to $R$ and then recursively proceeds.
The base case of the unique Dyck path of semilength $1$ is sent to a single node.
This map may also be described as the unique map sending the Tamari orders on Dyck paths to the Tamari order on binary trees.