- FindStat

Identifier

Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
St000919: Binary trees ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The number of maximal left branches of a binary tree.
A maximal left branch of a binary tree is an inclusion wise maximal path which consists of left edges only. This statistic records the number of distinct maximal left branches in the 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.