- FindStat

Identifier

Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
St000409: Binary trees ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The number of pitchforks in a binary tree.
A pitchfork is a subtree of a complete binary tree with exactly three leaves, see Section 3.2 of [1].

Map

logarithmic height to pruning number

Description

Francon's map from Dyck paths to binary trees.
This bijection sends the logarithmic height of the Dyck path, St000920The logarithmic height of a Dyck path., to the pruning number of the binary tree, St000396The register function (or Horton-Strahler number) of a binary tree.. The implementation is a literal translation of Knuth's [2].