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Identifier

St000569: Binary trees ⟶ ℤ

Values

[.,[.,.]] => 3

[[.,.],.] => 3

[.,[.,[.,.]]] => 6

[.,[[.,.],.]] => 6

[[.,.],[.,.]] => 4

[[.,[.,.]],.] => 6

[[[.,.],.],.] => 6

[.,[.,[.,[.,.]]]] => 10

[.,[.,[[.,.],.]]] => 10

[.,[[.,.],[.,.]]] => 7

[.,[[.,[.,.]],.]] => 10

[.,[[[.,.],.],.]] => 10

[[.,.],[.,[.,.]]] => 7

[[.,.],[[.,.],.]] => 7

[[.,[.,.]],[.,.]] => 7

[[[.,.],.],[.,.]] => 7

[[.,[.,[.,.]]],.] => 10

[[.,[[.,.],.]],.] => 10

[[[.,.],[.,.]],.] => 7

[[[.,[.,.]],.],.] => 10

[[[[.,.],.],.],.] => 10

[.,[.,[.,[.,[.,.]]]]] => 15

[.,[.,[.,[[.,.],.]]]] => 15

[.,[.,[[.,.],[.,.]]]] => 11

[.,[.,[[.,[.,.]],.]]] => 15

[.,[.,[[[.,.],.],.]]] => 15

[.,[[.,.],[.,[.,.]]]] => 11

[.,[[.,.],[[.,.],.]]] => 11

[.,[[.,[.,.]],[.,.]]] => 11

[.,[[[.,.],.],[.,.]]] => 11

[.,[[.,[.,[.,.]]],.]] => 15

[.,[[.,[[.,.],.]],.]] => 15

[.,[[[.,.],[.,.]],.]] => 11

[.,[[[.,[.,.]],.],.]] => 15

[.,[[[[.,.],.],.],.]] => 15

[[.,.],[.,[.,[.,.]]]] => 11

[[.,.],[.,[[.,.],.]]] => 11

[[.,.],[[.,.],[.,.]]] => 8

[[.,.],[[.,[.,.]],.]] => 11

[[.,.],[[[.,.],.],.]] => 11

[[.,[.,.]],[.,[.,.]]] => 9

[[.,[.,.]],[[.,.],.]] => 9

[[[.,.],.],[.,[.,.]]] => 9

[[[.,.],.],[[.,.],.]] => 9

[[.,[.,[.,.]]],[.,.]] => 11

[[.,[[.,.],.]],[.,.]] => 11

[[[.,.],[.,.]],[.,.]] => 8

[[[.,[.,.]],.],[.,.]] => 11

[[[[.,.],.],.],[.,.]] => 11

[[.,[.,[.,[.,.]]]],.] => 15

[[.,[.,[[.,.],.]]],.] => 15

[[.,[[.,.],[.,.]]],.] => 11

[[.,[[.,[.,.]],.]],.] => 15

[[.,[[[.,.],.],.]],.] => 15

[[[.,.],[.,[.,.]]],.] => 11

[[[.,.],[[.,.],.]],.] => 11

[[[.,[.,.]],[.,.]],.] => 11

[[[[.,.],.],[.,.]],.] => 11

[[[.,[.,[.,.]]],.],.] => 15

[[[.,[[.,.],.]],.],.] => 15

[[[[.,.],[.,.]],.],.] => 11

[[[[.,[.,.]],.],.],.] => 15

[[[[[.,.],.],.],.],.] => 15

[.,[.,[.,[.,[.,[.,.]]]]]] => 21

[.,[.,[.,[.,[[.,.],.]]]]] => 21

[.,[.,[.,[[.,.],[.,.]]]]] => 16

[.,[.,[.,[[.,[.,.]],.]]]] => 21

[.,[.,[.,[[[.,.],.],.]]]] => 21

[.,[.,[[.,.],[.,[.,.]]]]] => 16

[.,[.,[[.,.],[[.,.],.]]]] => 16

[.,[.,[[.,[.,.]],[.,.]]]] => 16

[.,[.,[[[.,.],.],[.,.]]]] => 16

[.,[.,[[.,[.,[.,.]]],.]]] => 21

[.,[.,[[.,[[.,.],.]],.]]] => 21

[.,[.,[[[.,.],[.,.]],.]]] => 16

[.,[.,[[[.,[.,.]],.],.]]] => 21

[.,[.,[[[[.,.],.],.],.]]] => 21

[.,[[.,.],[.,[.,[.,.]]]]] => 16

[.,[[.,.],[.,[[.,.],.]]]] => 16

[.,[[.,.],[[.,.],[.,.]]]] => 12

[.,[[.,.],[[.,[.,.]],.]]] => 16

[.,[[.,.],[[[.,.],.],.]]] => 16

[.,[[.,[.,.]],[.,[.,.]]]] => 13

[.,[[.,[.,.]],[[.,.],.]]] => 13

[.,[[[.,.],.],[.,[.,.]]]] => 13

[.,[[[.,.],.],[[.,.],.]]] => 13

[.,[[.,[.,[.,.]]],[.,.]]] => 16

[.,[[.,[[.,.],.]],[.,.]]] => 16

[.,[[[.,.],[.,.]],[.,.]]] => 12

[.,[[[.,[.,.]],.],[.,.]]] => 16

[.,[[[[.,.],.],.],[.,.]]] => 16

[.,[[.,[.,[.,[.,.]]]],.]] => 21

[.,[[.,[.,[[.,.],.]]],.]] => 21

[.,[[.,[[.,.],[.,.]]],.]] => 16

[.,[[.,[[.,[.,.]],.]],.]] => 21

[.,[[.,[[[.,.],.],.]],.]] => 21

[.,[[[.,.],[.,[.,.]]],.]] => 16

[.,[[[.,.],[[.,.],.]],.]] => 16

[.,[[[.,[.,.]],[.,.]],.]] => 16

[.,[[[[.,.],.],[.,.]],.]] => 16

[.,[[[.,[.,[.,.]]],.],.]] => 21

>>> Load all 1200 entries. <<<

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$F_{2} = 2\ q^{3}$

$F_{3} = q^{4} + 4\ q^{6}$

$F_{4} = 6\ q^{7} + 8\ q^{10}$

$F_{5} = 2\ q^{8} + 4\ q^{9} + 20\ q^{11} + 16\ q^{15}$

$F_{6} = 4\ q^{10} + 16\ q^{12} + 24\ q^{13} + 56\ q^{16} + 32\ q^{21}$

$F_{7} = q^{11} + 4\ q^{13} + 48\ q^{14} + 16\ q^{16} + 72\ q^{17} + 80\ q^{18} + 144\ q^{22} + 64\ q^{28}$

Description

The sum of the heights of the vertices of a binary tree.
The height of a vertex is the number of edges to a leaf, see [3].

References

[1] wikipedia:Tree (data_structure)
[2] The depth or height of a binary tree. St000050
[3] The sum of the depths of the vertices (or total internal path length) of a binary tree. St000398
[4] The external path length of a binary tree. St000399
[5] The path length of an ordered tree. St000400

Code

def height(T):
    if not T:
        return 0
    else:
        return 1 + max(height(T[0]),height(T[1]))

def statistic(T):
    if not T:
        return 0
    else:
        return height(T) + statistic(T[0]) + statistic(T[1])

Created

Aug 09, 2016 at 11:27 by Christian Stump

Updated

Aug 10, 2016 at 11:38 by Christian Stump