edit this map or download as text // json
Identifier
Mp00139: Ordered trees —Zeilberger's Strahler bijection⟶ Binary trees
Images
=>
Cc0021;cc-rep-0Cc0010;cc-rep-1
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[[[],[[],[[[],[]],[[],[]]]]],[]]=>[[.,[[.,[[[.,.],[.,.]],[.,[[.,.],[.,.]]]]],.]],.] [[[],[[],[[[],[[],[]]],[]]]],[]]=>[[.,[[.,[[[.,[[.,[.,.]],[.,.]]],.],[.,.]]],.]],.] [[[],[[],[[[[],[]],[]],[]]]],[]]=>[[.,[[.,[[[[.,[.,.]],[.,.]],[.,.]],[.,.]]],.]],.] [[[],[[[],[]],[[],[[],[]]]]],[]]=>[[.,[[[.,.],[.,.]],[.,[[.,[[.,.],[.,.]]],.]]]],.] [[[],[[[],[]],[[[],[]],[]]]],[]]=>[[.,[[[.,.],[.,.]],[.,[[[.,.],[.,.]],[.,.]]]]],.] [[[],[[[],[[],[]]],[[],[]]]],[]]=>[[.,[[[.,[[.,.],[.,.]]],.],[.,[[.,.],[.,.]]]]],.] [[[],[[[[],[]],[]],[[],[]]]],[]]=>[[.,[[[[.,.],[.,.]],[.,.]],[.,[[.,.],[.,.]]]]],.] [[[],[[[],[[],[[],[]]]],[]]],[]]=>[[.,[[[.,[[.,[[.,[.,.]],[.,.]]],.]],.],[.,.]]],.] [[[],[[[],[[[],[]],[]]],[]]],[]]=>[[.,[[[.,[[[.,[.,.]],[.,.]],[.,.]]],.],[.,.]]],.] [[[],[[[[],[]],[[],[]]],[]]],[]]=>[[.,[[[[.,.],[.,.]],[.,[[.,.],[.,.]]]],[.,.]]],.] [[[],[[[[],[[],[]]],[]],[]]],[]]=>[[.,[[[[.,[[.,[.,.]],[.,.]]],.],[.,.]],[.,.]]],.] [[[],[[[[[],[]],[]],[]],[]]],[]]=>[[.,[[[[[.,[.,.]],[.,.]],[.,.]],[.,.]],[.,.]]],.] [[[[],[]],[[],[[],[[],[]]]]],[]]=>[[[.,.],[.,.]],[.,[[.,[[.,[[.,.],[.,.]]],.]],.]]] [[[[],[]],[[],[[[],[]],[]]]],[]]=>[[[.,.],[.,.]],[.,[[.,[[[.,.],[.,.]],[.,.]]],.]]] [[[[],[]],[[[],[]],[[],[]]]],[]]=>[[[.,[.,.]],[[[.,.],[.,.]],[.,[[.,.],[.,.]]]]],.] [[[[],[]],[[[],[[],[]]],[]]],[]]=>[[[.,.],[.,.]],[.,[[[.,[[.,.],[.,.]]],.],[.,.]]]] [[[[],[]],[[[[],[]],[]],[]]],[]]=>[[[.,.],[.,.]],[.,[[[[.,.],[.,.]],[.,.]],[.,.]]]] [[[[],[[],[]]],[[],[[],[]]]],[]]=>[[[.,[[.,.],[.,.]]],.],[.,[[.,[[.,.],[.,.]]],.]]] [[[[],[[],[]]],[[[],[]],[]]],[]]=>[[[.,[[.,.],[.,.]]],.],[.,[[[.,.],[.,.]],[.,.]]]] [[[[[],[]],[]],[[],[[],[]]]],[]]=>[[[[.,.],[.,.]],[.,.]],[.,[[.,[[.,.],[.,.]]],.]]] [[[[[],[]],[]],[[[],[]],[]]],[]]=>[[[[.,.],[.,.]],[.,.]],[.,[[[.,.],[.,.]],[.,.]]]] [[[[],[[],[[],[]]]],[[],[]]],[]]=>[[[.,[[.,[[.,.],[.,.]]],.]],.],[.,[[.,.],[.,.]]]] [[[[],[[[],[]],[]]],[[],[]]],[]]=>[[[.,[[[.,.],[.,.]],[.,.]]],.],[.,[[.,.],[.,.]]]] [[[[[],[]],[[],[]]],[[],[]]],[]]=>[[[[.,.],[.,.]],[.,[[.,.],[.,.]]]],[[.,.],[.,.]]] [[[[[],[[],[]]],[]],[[],[]]],[]]=>[[[[.,[[.,.],[.,.]]],.],[.,.]],[.,[[.,.],[.,.]]]] [[[[[[],[]],[]],[]],[[],[]]],[]]=>[[[[[.,.],[.,.]],[.,.]],[.,.]],[.,[[.,.],[.,.]]]] [[[[],[[],[[],[[],[]]]]],[]],[]]=>[[[.,[[.,[[.,[[.,[.,.]],[.,.]]],.]],.]],.],[.,.]] [[[[],[[],[[[],[]],[]]]],[]],[]]=>[[[.,[[.,[[[.,[.,.]],[.,.]],[.,.]]],.]],.],[.,.]] [[[[],[[[],[]],[[],[]]]],[]],[]]=>[[[.,[[[.,.],[.,.]],[.,[[.,.],[.,.]]]]],.],[.,.]] [[[[],[[[],[[],[]]],[]]],[]],[]]=>[[[.,[[[.,[[.,[.,.]],[.,.]]],.],[.,.]]],.],[.,.]] [[[[],[[[[],[]],[]],[]]],[]],[]]=>[[[.,[[[[.,[.,.]],[.,.]],[.,.]],[.,.]]],.],[.,.]] [[[[[],[]],[[],[[],[]]]],[]],[]]=>[[[[.,.],[.,.]],[.,[[.,[[.,.],[.,.]]],.]]],[.,.]] [[[[[],[]],[[[],[]],[]]],[]],[]]=>[[[[.,.],[.,.]],[.,[[[.,.],[.,.]],[.,.]]]],[.,.]] [[[[[],[[],[]]],[[],[]]],[]],[]]=>[[[[.,[[.,.],[.,.]]],.],[.,[[.,.],[.,.]]]],[.,.]] [[[[[[],[]],[]],[[],[]]],[]],[]]=>[[[[[.,.],[.,.]],[.,.]],[.,[[.,.],[.,.]]]],[.,.]] [[[[[],[[],[[],[]]]],[]],[]],[]]=>[[[[.,[[.,[[.,[.,.]],[.,.]]],.]],.],[.,.]],[.,.]] [[[[[],[[[],[]],[]]],[]],[]],[]]=>[[[[.,[[[.,[.,.]],[.,.]],[.,.]]],.],[.,.]],[.,.]] [[[[[[],[]],[[],[]]],[]],[]],[]]=>[[[[[.,.],[.,.]],[.,[[.,.],[.,.]]]],[.,.]],[.,.]] [[[[[[],[[],[]]],[]],[]],[]],[]]=>[[[[[.,[[.,[.,.]],[.,.]]],.],[.,.]],[.,.]],[.,.]] [[[[[[[],[]],[]],[]],[]],[]],[]]=>[[[[[[.,[.,.]],[.,.]],[.,.]],[.,.]],[.,.]],[.,.]]
click to show experimental identities (only identities of compositions of up to three maps are shown)
Description
Zeilberger's Strahler bijection between ordered and binary trees.
This is a bijection sending the pruning number of the ordered tree to the Strahler number of the binary tree.
References
[1] Zeilberger, D. A bijection from ordered trees to binary trees that sends the pruning order to the Strahler number MathSciNet:1058712
Properties
bijective, graded
Sage code
def mapping(t):
    """
    sage: mapping([])
    .

    sage: mapping([[]])
    [., .]

    sage: [mapping(t) for t in OrderedTrees(3)]
    [[., [., .]], [[., .], .]]
    """
    def prune(t):
        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 pruning_order(t):
        o = 0
        while len(t) > 0:
            t = prune(t)
            o += 1
        return o

    def below(t):
        t1 = t[0]
        t2 = t[1:]

        k = pruning_order(t)
        k1 = pruning_order(t1)
        k2 = pruning_order(t2)
        if (k1 <= k-1 and k2 == k) or (k1 == k-1 and k2 == k-1):
            return (t1, t2)
        elif k1 == k-1 and k2 <= k-2:
            i = 0
            while pruning_order(t1[:i+1]) <= k-2:
                i += 1
            f1 = t1[:i]
            s1 = t1[i]
            j = i+1
            while pruning_order(t1[i+1:j+1]) <= k-2:
                j += 1
            f2 = t1[i+1:j]
            return (f1+[s1]+f2, t2+t1[j:])
        elif k1 == k:
            i = 0
            while pruning_order(t1[:i+1]) <= k-1:
                i += 1
            f1 = t1[:i]
            s1 = t1[i]
            f2 = t1[i+1:]
            return (f1+[s1]+t2, f2)

    if len(t) == 0:
        return BinaryTree()
    else:
        u1, u2 = below(t)
        return BinaryTree([mapping(u1), mapping(u2)])
Weight
26
Created
Jan 29, 2020 at 13:27 by FindStatCrew
Updated
Jan 29, 2020 at 13:27 by Martin Rubey