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Identifier

St000396: Binary trees ⟶ ℤ

Values

[.,.] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The register function (or Horton-Strahler number) of a binary tree.
This is different from the dimension of the associated poset for the tree $[[[.,.],[.,.]],[[.,.],[.,.]]]$: its register function is 3, whereas the dimension of the associated poset is 2.

References

[1] Hackl, B., Heuberger, C., Prodinger, H. The Register Function and Reductions of Binary Trees and Lattice Paths arXiv:1602.06200

Code

def statistic(B):
    if B.is_empty():
        return 0
    else:
        r1 = statistic(B[0])
        r2 = statistic(B[1])
        if r1 == r2:
            return r1+1
        else:
            return max(r1, r2)

Created

Feb 22, 2016 at 08:21 by Martin Rubey

Updated

Feb 22, 2016 at 08:57 by Martin Rubey