Identifier
-
Mp00029:
Dyck paths
—to binary tree: left tree, up step, right tree, down step⟶
Binary trees
Mp00008: Binary trees —to complete tree⟶ Ordered trees
St000397: Ordered trees ⟶ ℤ
Values
[1,0] => [.,.] => [[],[]] => 2
[1,0,1,0] => [[.,.],.] => [[[],[]],[]] => 2
[1,1,0,0] => [.,[.,.]] => [[],[[],[]]] => 2
[1,0,1,0,1,0] => [[[.,.],.],.] => [[[[],[]],[]],[]] => 2
[1,0,1,1,0,0] => [[.,.],[.,.]] => [[[],[]],[[],[]]] => 3
[1,1,0,0,1,0] => [[.,[.,.]],.] => [[[],[[],[]]],[]] => 2
[1,1,0,1,0,0] => [.,[[.,.],.]] => [[],[[[],[]],[]]] => 2
[1,1,1,0,0,0] => [.,[.,[.,.]]] => [[],[[],[[],[]]]] => 2
[1,0,1,0,1,0,1,0] => [[[[.,.],.],.],.] => [[[[[],[]],[]],[]],[]] => 2
[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] => [[[.,[.,.]],.],.] => [[[[],[[],[]]],[]],[]] => 2
[1,1,0,0,1,1,0,0] => [[.,[.,.]],[.,.]] => [[[],[[],[]]],[[],[]]] => 3
[1,1,0,1,0,0,1,0] => [[.,[[.,.],.]],.] => [[[],[[[],[]],[]]],[]] => 2
[1,1,0,1,0,1,0,0] => [.,[[[.,.],.],.]] => [[],[[[[],[]],[]],[]]] => 2
[1,1,0,1,1,0,0,0] => [.,[[.,.],[.,.]]] => [[],[[[],[]],[[],[]]]] => 3
[1,1,1,0,0,0,1,0] => [[.,[.,[.,.]]],.] => [[[],[[],[[],[]]]],[]] => 2
[1,1,1,0,0,1,0,0] => [.,[[.,[.,.]],.]] => [[],[[[],[[],[]]],[]]] => 2
[1,1,1,0,1,0,0,0] => [.,[.,[[.,.],.]]] => [[],[[],[[[],[]],[]]]] => 2
[1,1,1,1,0,0,0,0] => [.,[.,[.,[.,.]]]] => [[],[[],[[],[[],[]]]]] => 2
[1,0,1,0,1,0,1,0,1,0] => [[[[[.,.],.],.],.],.] => [[[[[[],[]],[]],[]],[]],[]] => 2
[1,0,1,0,1,0,1,1,0,0] => [[[[.,.],.],.],[.,.]] => [[[[[],[]],[]],[]],[[],[]]] => 3
[1,0,1,0,1,1,0,0,1,0] => [[[[.,.],.],[.,.]],.] => [[[[[],[]],[]],[[],[]]],[]] => 3
[1,0,1,0,1,1,0,1,0,0] => [[[.,.],.],[[.,.],.]] => [[[[],[]],[]],[[[],[]],[]]] => 3
[1,0,1,0,1,1,1,0,0,0] => [[[.,.],.],[.,[.,.]]] => [[[[],[]],[]],[[],[[],[]]]] => 3
[1,0,1,1,0,0,1,0,1,0] => [[[[.,.],[.,.]],.],.] => [[[[[],[]],[[],[]]],[]],[]] => 3
[1,0,1,1,0,0,1,1,0,0] => [[[.,.],[.,.]],[.,.]] => [[[[],[]],[[],[]]],[[],[]]] => 3
[1,0,1,1,0,1,0,0,1,0] => [[[.,.],[[.,.],.]],.] => [[[[],[]],[[[],[]],[]]],[]] => 3
[1,0,1,1,0,1,0,1,0,0] => [[.,.],[[[.,.],.],.]] => [[[],[]],[[[[],[]],[]],[]]] => 3
[1,0,1,1,0,1,1,0,0,0] => [[.,.],[[.,.],[.,.]]] => [[[],[]],[[[],[]],[[],[]]]] => 3
[1,0,1,1,1,0,0,0,1,0] => [[[.,.],[.,[.,.]]],.] => [[[[],[]],[[],[[],[]]]],[]] => 3
[1,0,1,1,1,0,0,1,0,0] => [[.,.],[[.,[.,.]],.]] => [[[],[]],[[[],[[],[]]],[]]] => 3
[1,0,1,1,1,0,1,0,0,0] => [[.,.],[.,[[.,.],.]]] => [[[],[]],[[],[[[],[]],[]]]] => 3
[1,0,1,1,1,1,0,0,0,0] => [[.,.],[.,[.,[.,.]]]] => [[[],[]],[[],[[],[[],[]]]]] => 3
[1,1,0,0,1,0,1,0,1,0] => [[[[.,[.,.]],.],.],.] => [[[[[],[[],[]]],[]],[]],[]] => 2
[1,1,0,0,1,0,1,1,0,0] => [[[.,[.,.]],.],[.,.]] => [[[[],[[],[]]],[]],[[],[]]] => 3
[1,1,0,0,1,1,0,0,1,0] => [[[.,[.,.]],[.,.]],.] => [[[[],[[],[]]],[[],[]]],[]] => 3
[1,1,0,0,1,1,0,1,0,0] => [[.,[.,.]],[[.,.],.]] => [[[],[[],[]]],[[[],[]],[]]] => 3
[1,1,0,0,1,1,1,0,0,0] => [[.,[.,.]],[.,[.,.]]] => [[[],[[],[]]],[[],[[],[]]]] => 3
[1,1,0,1,0,0,1,0,1,0] => [[[.,[[.,.],.]],.],.] => [[[[],[[[],[]],[]]],[]],[]] => 2
[1,1,0,1,0,0,1,1,0,0] => [[.,[[.,.],.]],[.,.]] => [[[],[[[],[]],[]]],[[],[]]] => 3
[1,1,0,1,0,1,0,0,1,0] => [[.,[[[.,.],.],.]],.] => [[[],[[[[],[]],[]],[]]],[]] => 2
[1,1,0,1,0,1,0,1,0,0] => [.,[[[[.,.],.],.],.]] => [[],[[[[[],[]],[]],[]],[]]] => 2
[1,1,0,1,0,1,1,0,0,0] => [.,[[[.,.],.],[.,.]]] => [[],[[[[],[]],[]],[[],[]]]] => 3
[1,1,0,1,1,0,0,0,1,0] => [[.,[[.,.],[.,.]]],.] => [[[],[[[],[]],[[],[]]]],[]] => 3
[1,1,0,1,1,0,0,1,0,0] => [.,[[[.,.],[.,.]],.]] => [[],[[[[],[]],[[],[]]],[]]] => 3
[1,1,0,1,1,0,1,0,0,0] => [.,[[.,.],[[.,.],.]]] => [[],[[[],[]],[[[],[]],[]]]] => 3
[1,1,0,1,1,1,0,0,0,0] => [.,[[.,.],[.,[.,.]]]] => [[],[[[],[]],[[],[[],[]]]]] => 3
[1,1,1,0,0,0,1,0,1,0] => [[[.,[.,[.,.]]],.],.] => [[[[],[[],[[],[]]]],[]],[]] => 2
[1,1,1,0,0,0,1,1,0,0] => [[.,[.,[.,.]]],[.,.]] => [[[],[[],[[],[]]]],[[],[]]] => 3
[1,1,1,0,0,1,0,0,1,0] => [[.,[[.,[.,.]],.]],.] => [[[],[[[],[[],[]]],[]]],[]] => 2
[1,1,1,0,0,1,0,1,0,0] => [.,[[[.,[.,.]],.],.]] => [[],[[[[],[[],[]]],[]],[]]] => 2
[1,1,1,0,0,1,1,0,0,0] => [.,[[.,[.,.]],[.,.]]] => [[],[[[],[[],[]]],[[],[]]]] => 3
[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] => [.,[.,[[[.,.],.],.]]] => [[],[[],[[[[],[]],[]],[]]]] => 2
[1,1,1,0,1,1,0,0,0,0] => [.,[.,[[.,.],[.,.]]]] => [[],[[],[[[],[]],[[],[]]]]] => 3
[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] => [.,[.,[[.,[.,.]],.]]] => [[],[[],[[[],[[],[]]],[]]]] => 2
[1,1,1,1,0,1,0,0,0,0] => [.,[.,[.,[[.,.],.]]]] => [[],[[],[[],[[[],[]],[]]]]] => 2
[1,1,1,1,1,0,0,0,0,0] => [.,[.,[.,[.,[.,.]]]]] => [[],[[],[[],[[],[[],[]]]]]] => 2
[1,0,1,0,1,0,1,0,1,0,1,0] => [[[[[[.,.],.],.],.],.],.] => [[[[[[[],[]],[]],[]],[]],[]],[]] => 2
[1,0,1,0,1,0,1,0,1,1,0,0] => [[[[[.,.],.],.],.],[.,.]] => [[[[[[],[]],[]],[]],[]],[[],[]]] => 3
[1,0,1,0,1,0,1,1,0,0,1,0] => [[[[[.,.],.],.],[.,.]],.] => [[[[[[],[]],[]],[]],[[],[]]],[]] => 3
[1,0,1,0,1,0,1,1,0,1,0,0] => [[[[.,.],.],.],[[.,.],.]] => [[[[[],[]],[]],[]],[[[],[]],[]]] => 3
[1,0,1,0,1,0,1,1,1,0,0,0] => [[[[.,.],.],.],[.,[.,.]]] => [[[[[],[]],[]],[]],[[],[[],[]]]] => 3
[1,0,1,0,1,1,0,0,1,0,1,0] => [[[[[.,.],.],[.,.]],.],.] => [[[[[[],[]],[]],[[],[]]],[]],[]] => 3
[1,0,1,0,1,1,0,0,1,1,0,0] => [[[[.,.],.],[.,.]],[.,.]] => [[[[[],[]],[]],[[],[]]],[[],[]]] => 3
[1,0,1,0,1,1,0,1,0,0,1,0] => [[[[.,.],.],[[.,.],.]],.] => [[[[[],[]],[]],[[[],[]],[]]],[]] => 3
[1,0,1,0,1,1,0,1,0,1,0,0] => [[[.,.],.],[[[.,.],.],.]] => [[[[],[]],[]],[[[[],[]],[]],[]]] => 3
[1,0,1,0,1,1,0,1,1,0,0,0] => [[[.,.],.],[[.,.],[.,.]]] => [[[[],[]],[]],[[[],[]],[[],[]]]] => 3
[1,0,1,0,1,1,1,0,0,0,1,0] => [[[[.,.],.],[.,[.,.]]],.] => [[[[[],[]],[]],[[],[[],[]]]],[]] => 3
[1,0,1,0,1,1,1,0,0,1,0,0] => [[[.,.],.],[[.,[.,.]],.]] => [[[[],[]],[]],[[[],[[],[]]],[]]] => 3
[1,0,1,0,1,1,1,0,1,0,0,0] => [[[.,.],.],[.,[[.,.],.]]] => [[[[],[]],[]],[[],[[[],[]],[]]]] => 3
[1,0,1,0,1,1,1,1,0,0,0,0] => [[[.,.],.],[.,[.,[.,.]]]] => [[[[],[]],[]],[[],[[],[[],[]]]]] => 3
[1,0,1,1,0,0,1,0,1,0,1,0] => [[[[[.,.],[.,.]],.],.],.] => [[[[[[],[]],[[],[]]],[]],[]],[]] => 3
[1,0,1,1,0,0,1,0,1,1,0,0] => [[[[.,.],[.,.]],.],[.,.]] => [[[[[],[]],[[],[]]],[]],[[],[]]] => 3
[1,0,1,1,0,0,1,1,0,0,1,0] => [[[[.,.],[.,.]],[.,.]],.] => [[[[[],[]],[[],[]]],[[],[]]],[]] => 3
[1,0,1,1,0,0,1,1,0,1,0,0] => [[[.,.],[.,.]],[[.,.],.]] => [[[[],[]],[[],[]]],[[[],[]],[]]] => 3
[1,0,1,1,0,0,1,1,1,0,0,0] => [[[.,.],[.,.]],[.,[.,.]]] => [[[[],[]],[[],[]]],[[],[[],[]]]] => 3
[1,0,1,1,0,1,0,0,1,0,1,0] => [[[[.,.],[[.,.],.]],.],.] => [[[[[],[]],[[[],[]],[]]],[]],[]] => 3
[1,0,1,1,0,1,0,0,1,1,0,0] => [[[.,.],[[.,.],.]],[.,.]] => [[[[],[]],[[[],[]],[]]],[[],[]]] => 3
[1,0,1,1,0,1,0,1,0,0,1,0] => [[[.,.],[[[.,.],.],.]],.] => [[[[],[]],[[[[],[]],[]],[]]],[]] => 3
[1,0,1,1,0,1,0,1,0,1,0,0] => [[.,.],[[[[.,.],.],.],.]] => [[[],[]],[[[[[],[]],[]],[]],[]]] => 3
[1,0,1,1,0,1,0,1,1,0,0,0] => [[.,.],[[[.,.],.],[.,.]]] => [[[],[]],[[[[],[]],[]],[[],[]]]] => 3
[1,0,1,1,0,1,1,0,0,0,1,0] => [[[.,.],[[.,.],[.,.]]],.] => [[[[],[]],[[[],[]],[[],[]]]],[]] => 3
[1,0,1,1,0,1,1,0,0,1,0,0] => [[.,.],[[[.,.],[.,.]],.]] => [[[],[]],[[[[],[]],[[],[]]],[]]] => 3
[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] => [[.,.],[[.,.],[.,[.,.]]]] => [[[],[]],[[[],[]],[[],[[],[]]]]] => 3
[1,0,1,1,1,0,0,0,1,0,1,0] => [[[[.,.],[.,[.,.]]],.],.] => [[[[[],[]],[[],[[],[]]]],[]],[]] => 3
[1,0,1,1,1,0,0,0,1,1,0,0] => [[[.,.],[.,[.,.]]],[.,.]] => [[[[],[]],[[],[[],[]]]],[[],[]]] => 3
[1,0,1,1,1,0,0,1,0,0,1,0] => [[[.,.],[[.,[.,.]],.]],.] => [[[[],[]],[[[],[[],[]]],[]]],[]] => 3
[1,0,1,1,1,0,0,1,0,1,0,0] => [[.,.],[[[.,[.,.]],.],.]] => [[[],[]],[[[[],[[],[]]],[]],[]]] => 3
[1,0,1,1,1,0,0,1,1,0,0,0] => [[.,.],[[.,[.,.]],[.,.]]] => [[[],[]],[[[],[[],[]]],[[],[]]]] => 3
[1,0,1,1,1,0,1,0,0,0,1,0] => [[[.,.],[.,[[.,.],.]]],.] => [[[[],[]],[[],[[[],[]],[]]]],[]] => 3
[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] => [[.,.],[.,[[[.,.],.],.]]] => [[[],[]],[[],[[[[],[]],[]],[]]]] => 3
[1,0,1,1,1,0,1,1,0,0,0,0] => [[.,.],[.,[[.,.],[.,.]]]] => [[[],[]],[[],[[[],[]],[[],[]]]]] => 3
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Description
The Strahler number of a rooted 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.
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.
Map
to complete tree
Description
Return the same tree seen as an ordered tree. By default, leaves are transformed into actual nodes.
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