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Identifier

St000679: Ordered trees ⟶ ℤ

Values

[[]] => 1

[[],[]] => 1

[[[]]] => 1

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

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

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

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

[[[[]]]] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

[[[[[]]]]] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[[[[[]]]]]] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The pruning number of an ordered tree.
A hanging branch of an ordered tree is a proper factor of the form $[^r]^r$ for some $r\geq 1$. A hanging branch is a maximal hanging branch if it is not a proper factor of another hanging branch.
A pruning of an ordered tree is the act of deleting all its maximal hanging branches. The pruning order of an ordered tree is the number of prunings required to reduce it to $[]$.

References

[1] Zeilberger, D. A bijection from ordered trees to binary trees that sends the pruning order to the Strahler number MathSciNet:1058712

Code

def prune(t):
    """
    sage: prune([[]])
    []

    sage: prune([[[],[[],[]]],[[],[],[[]]]])
    [[[]], []]
    """
    def is_branch(b):
        n = len(b)
        return n == 0 or (n == 1 and is_branch(b[0]))
    if len(t) == 0:
        return []
    else:
        return [prune(b) for b in t if not is_branch(b)]

def statistic(t):
    """
    sage: statistic([[[],[[],[]]],[[],[],[[]]]])
    2
    """
    o = 0
    while len(t) > 0:
        t = prune(t)
        o += 1
    return o

Created

Jan 03, 2017 at 00:09 by Martin Rubey

Updated

Jan 03, 2017 at 00:09 by Martin Rubey