- FindStat

Identifier

Mp00061: Permutations —to increasing tree⟶ Binary trees
St000050: Binary trees ⟶ ℤ

Values

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

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

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

[1,2,3] => [.,[.,[.,.]]] => 3

[1,3,2] => [.,[[.,.],.]] => 3

[2,1,3] => [[.,.],[.,.]] => 2

[2,3,1] => [[.,[.,.]],.] => 3

[3,1,2] => [[.,.],[.,.]] => 2

[3,2,1] => [[[.,.],.],.] => 3

[1,2,3,4] => [.,[.,[.,[.,.]]]] => 4

[1,2,4,3] => [.,[.,[[.,.],.]]] => 4

[1,3,2,4] => [.,[[.,.],[.,.]]] => 3

[1,3,4,2] => [.,[[.,[.,.]],.]] => 4

[1,4,2,3] => [.,[[.,.],[.,.]]] => 3

[1,4,3,2] => [.,[[[.,.],.],.]] => 4

[2,1,3,4] => [[.,.],[.,[.,.]]] => 3

[2,1,4,3] => [[.,.],[[.,.],.]] => 3

[2,3,1,4] => [[.,[.,.]],[.,.]] => 3

[2,3,4,1] => [[.,[.,[.,.]]],.] => 4

[2,4,1,3] => [[.,[.,.]],[.,.]] => 3

[2,4,3,1] => [[.,[[.,.],.]],.] => 4

[3,1,2,4] => [[.,.],[.,[.,.]]] => 3

[3,1,4,2] => [[.,.],[[.,.],.]] => 3

[3,2,1,4] => [[[.,.],.],[.,.]] => 3

[3,2,4,1] => [[[.,.],[.,.]],.] => 3

[3,4,1,2] => [[.,[.,.]],[.,.]] => 3

[3,4,2,1] => [[[.,[.,.]],.],.] => 4

[4,1,2,3] => [[.,.],[.,[.,.]]] => 3

[4,1,3,2] => [[.,.],[[.,.],.]] => 3

[4,2,1,3] => [[[.,.],.],[.,.]] => 3

[4,2,3,1] => [[[.,.],[.,.]],.] => 3

[4,3,1,2] => [[[.,.],.],[.,.]] => 3

[4,3,2,1] => [[[[.,.],.],.],.] => 4

[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]] => 5

[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]] => 5

[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]] => 4

[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]] => 5

[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]] => 4

[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]] => 5

[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]] => 4

[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]] => 4

[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]] => 4

[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]] => 5

[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]] => 4

[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]] => 5

[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]] => 4

[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]] => 4

[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]] => 4

[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]] => 4

[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]] => 4

[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]] => 5

[1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]] => 4

[1,5,2,4,3] => [.,[[.,.],[[.,.],.]]] => 4

[1,5,3,2,4] => [.,[[[.,.],.],[.,.]]] => 4

[1,5,3,4,2] => [.,[[[.,.],[.,.]],.]] => 4

[1,5,4,2,3] => [.,[[[.,.],.],[.,.]]] => 4

[1,5,4,3,2] => [.,[[[[.,.],.],.],.]] => 5

[2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]] => 4

[2,1,3,5,4] => [[.,.],[.,[[.,.],.]]] => 4

[2,1,4,3,5] => [[.,.],[[.,.],[.,.]]] => 3

[2,1,4,5,3] => [[.,.],[[.,[.,.]],.]] => 4

[2,1,5,3,4] => [[.,.],[[.,.],[.,.]]] => 3

[2,1,5,4,3] => [[.,.],[[[.,.],.],.]] => 4

[2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]] => 3

[2,3,1,5,4] => [[.,[.,.]],[[.,.],.]] => 3

[2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]] => 4

[2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.] => 5

[2,3,5,1,4] => [[.,[.,[.,.]]],[.,.]] => 4

[2,3,5,4,1] => [[.,[.,[[.,.],.]]],.] => 5

[2,4,1,3,5] => [[.,[.,.]],[.,[.,.]]] => 3

[2,4,1,5,3] => [[.,[.,.]],[[.,.],.]] => 3

[2,4,3,1,5] => [[.,[[.,.],.]],[.,.]] => 4

[2,4,3,5,1] => [[.,[[.,.],[.,.]]],.] => 4

[2,4,5,1,3] => [[.,[.,[.,.]]],[.,.]] => 4

[2,4,5,3,1] => [[.,[[.,[.,.]],.]],.] => 5

[2,5,1,3,4] => [[.,[.,.]],[.,[.,.]]] => 3

[2,5,1,4,3] => [[.,[.,.]],[[.,.],.]] => 3

[2,5,3,1,4] => [[.,[[.,.],.]],[.,.]] => 4

[2,5,3,4,1] => [[.,[[.,.],[.,.]]],.] => 4

[2,5,4,1,3] => [[.,[[.,.],.]],[.,.]] => 4

[2,5,4,3,1] => [[.,[[[.,.],.],.]],.] => 5

[3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]] => 4

[3,1,2,5,4] => [[.,.],[.,[[.,.],.]]] => 4

[3,1,4,2,5] => [[.,.],[[.,.],[.,.]]] => 3

[3,1,4,5,2] => [[.,.],[[.,[.,.]],.]] => 4

[3,1,5,2,4] => [[.,.],[[.,.],[.,.]]] => 3

[3,1,5,4,2] => [[.,.],[[[.,.],.],.]] => 4

[3,2,1,4,5] => [[[.,.],.],[.,[.,.]]] => 3

[3,2,1,5,4] => [[[.,.],.],[[.,.],.]] => 3

[3,2,4,1,5] => [[[.,.],[.,.]],[.,.]] => 3

[3,2,4,5,1] => [[[.,.],[.,[.,.]]],.] => 4

[3,2,5,1,4] => [[[.,.],[.,.]],[.,.]] => 3

[3,2,5,4,1] => [[[.,.],[[.,.],.]],.] => 4

[3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]] => 3

[3,4,1,5,2] => [[.,[.,.]],[[.,.],.]] => 3

[3,4,2,1,5] => [[[.,[.,.]],.],[.,.]] => 4

[3,4,2,5,1] => [[[.,[.,.]],[.,.]],.] => 4

[3,4,5,1,2] => [[.,[.,[.,.]]],[.,.]] => 4

[3,4,5,2,1] => [[[.,[.,[.,.]]],.],.] => 5

[3,5,1,2,4] => [[.,[.,.]],[.,[.,.]]] => 3

[3,5,1,4,2] => [[.,[.,.]],[[.,.],.]] => 3

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Description

The depth or height of a binary tree.
The depth (or height) of a binary tree is the maximal depth (or height) of one of its vertices.
The height of a vertex is the number of edges on the longest path between that node and a leaf. The depth of a vertex is the number of edges from the vertex to the root.
See [1] and [2] for this terminology.
The depth (or height) of a tree $T$ can be recursively defined: $\operatorname{depth}(T) = 0$ if $T$ is empty and
$$\operatorname{depth}(T) = 1 + max(\operatorname{depth}(L),\operatorname{depth}(R))$$
if $T$ is nonempty with left and right subtrees $L$ and $R$, respectively.
The upper and lower bounds on the depth of a binary tree $T$ of size $n$ are $log_2(n) \leq \operatorname{depth}(T) \leq n$.

Map

to increasing tree

Description

Sends a permutation to its associated increasing tree.
This tree is recursively obtained by sending the unique permutation of length $0$ to the empty tree, and sending a permutation $\sigma$ of length $n \geq 1$ to a root node with two subtrees $L$ and $R$ by splitting $\sigma$ at the index $\sigma^{-1}(1)$, normalizing both sides again to permutations and sending the permutations on the left and on the right of $\sigma^{-1}(1)$ to the trees $L$ and $R$, respectively.