- FindStat

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Identifier

St000701: Binary trees ⟶ ℤ

Values

[.,.] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 1884 entries. <<<

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$F_{1} = q$

$F_{2} = 2\ q$

$F_{3} = 4\ q + q^{2}$

$F_{4} = 10\ q + 4\ q^{2}$

$F_{5} = 28\ q + 14\ q^{2}$

$F_{6} = 84\ q + 48\ q^{2}$

$F_{7} = 264\ q + 164\ q^{2} + q^{3}$

Description

The protection number of a binary tree.
This is the minimal distance from the root to a leaf.

References

[1] Heuberger, C., Prodinger, H. Protection Number in Plane Trees arXiv:1702.06816

Code

def statistic(T):
    if len(T) == 0:
        return 0
    else:
        return 1+min(statistic(S) for S in T)

Created

Feb 23, 2017 at 20:19 by Martin Rubey

Updated

Jul 23, 2019 at 08:47 by Martin Rubey