- FindStat

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Identifier

Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees

Images

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              Cc0005;cc-rep-0Cc0010;cc-rep-1
            

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[1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,0]=>[[[[.,.],[.,.]],[[.,.],[[[.,.],.],.]]],.]
[1,1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]=>[[[[[.,.],[[[[.,.],.],.],.]],[.,.]],.],.]
[1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]=>[[[[.,.],[[[[[[.,.],.],.],.],.],.]],.],.]
[1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[[.,[[[[[[[[.,.],.],.],.],.],.],.],.]],.]
[1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0]=>[[[[[[.,.],.],.],.],[.,.]],[[.,.],[.,.]]]
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0]=>[[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]],[.,.]]
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0]=>[[[[[[[.,.],[[[.,.],.],.]],.],.],.],.],.]
[1,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0,0]=>[[[[[[[[.,.],.],[[.,.],.]],.],.],.],.],.]
[1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]=>[[[[[[.,[[[[.,.],.],.],.]],.],.],.],.],.]
[1,1,0,1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0,0]=>[[[[[[.,.],.],.],.],[[[[.,.],.],.],.]],.]
[1,1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0,1,0,0]=>[[[[[[.,.],.],.],[[[[.,.],.],.],.]],.],.]
[1,1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0,0]=>[[[[[[.,.],.],[[[[.,.],.],.],.]],.],.],.]
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]=>[.,[.,[.,[.,[.,[.,[[[.,[.,.]],.],.]]]]]]]
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]=>[.,[[[[[[[.,[.,[.,.]]],.],.],.],.],.],.]]
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]=>[.,[.,[[[[[[[.,[.,.]],.],.],.],.],.],.]]]
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,1,0,0]=>[.,[.,[[[.,[.,[.,[.,[[.,.],.]]]]],.],.]]]
[1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,1,0,0]=>[.,[[[[.,[.,[.,[.,[[.,.],.]]]]],.],.],.]]
[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,1,0,0]=>[[[[[.,[.,[.,[.,[[.,.],.]]]]],.],.],.],.]
[1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]=>[[[.,.],[.,.]],[[[[[[.,.],.],.],.],.],.]]
[1,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0]=>[[[.,[.,[.,[.,.]]]],[.,.]],[[.,.],[.,.]]]
[1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0,0,0]=>[[[.,[.,[.,.]]],[.,.]],[[.,[.,.]],[.,.]]]
[1,1,1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]=>[[.,[[.,[.,[.,.]]],[.,.]]],[[.,.],[.,.]]]
[1,1,1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0,0,0]=>[[[[[[[.,.],[.,.]],.],.],.],.],[[.,.],.]]
[1,1,1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0,0,0]=>[[[[[[.,.],[.,.]],.],.],.],[[.,.],[.,.]]]
[1,1,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0,0]=>[[.,[.,[.,[[.,.],[.,.]]]]],[[.,.],[.,.]]]
[1,1,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0]=>[[.,[.,[[[.,.],.],[.,.]]]],[[.,.],[.,.]]]
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0,0]=>[[[[.,.],.],[.,.]],[[.,.],[[.,.],[.,.]]]]
[1,1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0,0]=>[[[.,.],[[[.,.],[.,.]],[[.,.],.]]],[.,.]]
[1,1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[[[.,.],.],[[[.,.],[.,.]],[[.,.],[.,.]]]]
[1,1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]=>[[.,.],[[[[.,.],.],[.,.]],[[.,.],[.,.]]]]
[1,1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0]=>[[[[[.,.],.],[[.,.],.]],[[.,.],[.,.]]],.]
[1,1,1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0]=>[[.,[[[.,.],.],[.,[.,.]]]],[[.,.],[.,.]]]
[1,1,1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0]=>[[[[[.,.],.],[.,.]],.],[[[.,.],.],[.,.]]]
[1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0,0,0]=>[[[[.,.],.],[.,.]],[.,[[.,[.,.]],[.,.]]]]
[1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0,0]=>[[[[.,.],.],[.,.]],[[[.,.],[[.,.],.]],.]]
[1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,0,1,0]=>[[.,[.,[.,[.,.]]]],[[[[.,[.,.]],.],.],.]]
[1,1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0,1,0]=>[[[.,[.,[.,.]]],[[[[.,[.,.]],.],.],.]],.]
[1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0]=>[[[[.,[.,.]],[[[[.,[.,.]],.],.],.]],.],.]
[1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0,1,0]=>[[[[[.,.],[[[[.,[.,.]],.],.],.]],.],.],.]
[1,1,1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]=>[[[[.,.],[.,.]],.],[.,[.,[.,[.,[.,.]]]]]]
[1,1,1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0,0,0]=>[[[.,.],[.,.]],[[[[[.,[.,.]],.],.],.],.]]
[1,1,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]=>[[.,[[.,.],[.,.]]],[.,[.,[.,[.,[.,.]]]]]]
[1,1,1,1,1,1,0,1,0,1,0,0,1,0,1,0,0,0,0,0]=>[[[.,.],[.,.]],[[[[.,[.,[.,.]]],.],.],.]]
[1,1,1,1,1,1,0,1,0,0,1,0,1,0,1,0,0,0,0,0]=>[[[.,.],[.,.]],[[[.,[.,[.,[.,.]]]],.],.]]
[1,1,1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0,0,0]=>[[[[.,[.,.]],.],[[.,.],.]],[[.,.],[.,.]]]
[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]=>[[.,[.,[.,[.,[.,[[[.,[.,.]],.],.]]]]]],.]
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,1,0,1,0]=>[[.,[.,[.,[.,[.,[.,.]]]]]],[.,[.,[.,.]]]]
[1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0,1,0,1,0]=>[[.,[.,[.,[.,[.,.]]]]],[.,[.,[.,[.,.]]]]]
[1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0,1,0,1,0]=>[[.,[.,[.,[.,.]]]],[.,[.,[.,[.,[.,.]]]]]]
[1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0,1,0,1,0]=>[[.,[.,[.,.]]],[.,[.,[.,[.,[.,[.,.]]]]]]]
[1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[[.,[.,.]],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]=>[[[[[[[[.,[[.,.],.]],.],.],.],.],.],.],.]
[1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0,1,0,1,0]=>[[[.,.],[.,[.,[.,[.,[.,.]]]]]],[.,[.,.]]]
[1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0,1,0,1,0]=>[[[.,.],[.,[.,[.,[.,.]]]]],[[[.,.],.],.]]
[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,1,0,1,0]=>[[[[[[[.,.],.],.],.],.],.],[.,[.,[.,.]]]]
[1,1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,0,0]=>[[[[[[[.,[.,[.,.]]],.],.],.],.],.],[.,.]]
[1,1,1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0,0,0]=>[[[[[[[.,.],[.,.]],.],.],.],.],[.,[.,.]]]
[1,1,1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0,0,0]=>[[[[[[.,.],.],[.,.]],.],.],[[.,.],[.,.]]]
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,1,0,0,1,0]=>[[.,[[.,[[.,[[[.,[.,.]],.],.]],.]],.]],.]
[1,1,1,0,1,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[[.,[.,.]],[.,[[.,[[.,[[.,.],.]],.]],.]]]
[1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0,0,0]=>[[[[.,.],.],[.,.]],[.,[.,[.,[.,[.,.]]]]]]
[1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0,0,0]=>[[[[.,.],.],[.,.]],[[[[[.,.],.],.],.],.]]
[1,1,1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]=>[[[[.,.],[.,.]],.],[[[[[.,.],.],.],.],.]]
[1,0,1,1,1,1,0,1,0,1,1,1,1,0,0,0,0,0,0,0]=>[.,[[[[[.,.],[.,.]],.],.],[[.,.],[.,.]]]]
[1,0,1,1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0,0]=>[.,[[.,[.,[[.,.],[.,.]]]],[[.,.],[.,.]]]]
[1,0,1,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[.,[[[[[.,.],[.,.]],[[.,.],[.,.]]],.],.]]
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]=>[.,[.,[.,[.,[[[.,[.,[.,[.,.]]]],.],.]]]]]
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]=>[.,[.,[.,[[[.,[.,[.,[.,[.,.]]]]],.],.]]]]
[1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[.,[[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.],.]]
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[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]=>[.,[[[[[[[[.,[.,.]],.],.],.],.],.],.],.]]
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[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]=>[.,[.,[.,[.,[.,[.,[[[.,.],[.,.]],.]]]]]]]
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[1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,1,0,0]=>[[[.,.],[[.,.],.]],[.,[.,[.,[.,[.,.]]]]]]
[1,1,1,1,1,0,1,0,1,1,0,1,0,1,0,0,0,0,0,0]=>[[[.,.],[.,.]],[.,[.,[[[[.,.],.],.],.]]]]
[1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0,0,0]=>[[[.,[.,.]],[.,.]],[.,[.,[.,[.,[.,.]]]]]]
[1,1,1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0,0,0]=>[[[.,.],[.,.]],[.,[[[[[.,.],.],.],.],.]]]
[1,1,0,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]=>[[.,[[[.,[.,[.,[.,[.,[.,.]]]]]],.],.]],.]
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[1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[[.,.],[[.,[[.,[[.,[[.,.],.]],.]],.]],.]]
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[1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]=>[[.,.],[[.,.],[[.,.],[[.,.],[[.,.],.]]]]]
[1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0]=>[[.,.],[[.,.],[[.,.],[[.,.],[.,[.,.]]]]]]
[1,1,1,0,0,1,1,0,0,1,1,0,0,0,1,1,0,0,1,0]=>[[.,.],[[.,.],[[.,.],[.,[[.,[.,.]],.]]]]]
[1,1,1,0,0,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]=>[[.,.],[[.,.],[.,[[.,[[.,[.,.]],.]],.]]]]
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]=>[[.,.],[[[[[[[.,[.,.]],.],.],.],.],.],.]]
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[1,1,0,1,0,1,0,0,1,0,1,1,0,1,0,1,0,0,1,0]=>[[[[.,[.,[[[[.,[.,.]],.],.],.]]],.],.],.]
[1,1,0,1,0,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]=>[[[.,[.,[[[[[.,[.,.]],.],.],.],.]]],.],.]
[1,1,0,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]=>[[.,[.,[[[[[[.,[.,.]],.],.],.],.],.]]],.]
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0,1,0]=>[[[.,[.,.]],[.,.]],[[.,.],[.,[.,[.,.]]]]]
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0,1,0,0]=>[[[.,[.,.]],[.,.]],[[.,.],[[[.,.],.],.]]]
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0,0,0]=>[[[.,[.,.]],[.,.]],[[.,[.,[.,.]]],[.,.]]]
[1,1,1,1,1,1,1,0,1,0,0,0,0,1,0,1,0,0,0,0]=>[[[.,[.,.]],[.,.]],[[[[.,.],.],.],[.,.]]]
[1,1,1,1,1,1,1,0,1,0,0,0,1,0,1,0,0,0,0,0]=>[[[.,[.,.]],[.,[.,[.,.]]]],[[.,.],[.,.]]]
[1,1,1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0,0,0]=>[[[.,[.,.]],[[[.,.],.],.]],[[.,.],[.,.]]]
[1,1,1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0,0,0]=>[[[.,.],[[[.,.],.],[.,.]]],[[.,.],[.,.]]]
[1,0,1,1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0,0]=>[.,[[[.,.],[[[.,.],.],[.,.]]],[[.,.],.]]]
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0,0]=>[.,[[[.,.],[.,.]],[[.,[[.,.],.]],[.,.]]]]
[1,1,1,1,1,0,0,1,1,0,1,0,1,0,1,0,0,0,0,0]=>[[[[.,.],[.,.]],[.,[.,[.,[.,.]]]]],[.,.]]
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]=>[.,[.,[.,[.,[.,[[.,.],[[.,[.,.]],.]]]]]]]
[1,1,0,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]=>[[.,[[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]],.]
[1,1,1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0,0,0]=>[[[[.,.],[.,.]],[[.,.],[.,.]]],[[.,.],.]]
[1,1,1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0,0,0]=>[[[[[[[.,.],.],.],.],.],[.,.]],[.,[.,.]]]
[1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0,0]=>[[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]],[.,.]]
[1,1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0,0]=>[[[[.,[.,[.,[.,[.,[.,.]]]]]],.],.],[.,.]]
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]=>[[[[[[[[.,.],[[.,.],.]],.],.],.],.],.],.]
[1,1,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0]=>[[[.,[[[.,.],.],[.,.]]],.],[[.,.],[.,.]]]
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,1,1,0,0,0,0]=>[[[.,.],.],[.,[[[.,[[.,.],.]],.],[.,.]]]]
[1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,1,0,0,0]=>[[[.,.],[.,[[[.,[.,.]],.],[.,.]]]],[.,.]]
[1,1,1,1,1,0,0,0,0,0,1,1,1,0,1,1,0,0,0,0]=>[[[.,.],[.,[[.,[[.,.],.]],[.,.]]]],[.,.]]
[1,1,1,0,0,0,1,1,1,1,1,0,0,0,1,1,0,0,0,0]=>[[.,.],[.,[[[.,[[.,.],.]],[.,.]],[.,.]]]]
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]=>[.,[[[[[[[[.,.],[.,.]],.],.],.],.],.],.]]
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0,0,0]=>[[[[[.,.],.],.],[.,.]],[[.,[.,.]],[.,.]]]
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0,0,0]=>[[[[[.,.],.],.],[.,.]],[[[.,.],.],[.,.]]]
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0,0,0]=>[[[[[.,.],.],.],[.,[.,.]]],[[.,.],[.,.]]]
[1,1,1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0,0,0]=>[[[[[.,.],.],.],[[.,.],.]],[[.,.],[.,.]]]
[1,1,1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0,0,0]=>[[[[[.,.],.],.],[.,.]],[[[.,.],[.,.]],.]]
[1,1,1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0,0,0]=>[[[[[.,.],.],.],[.,.]],[.,[[.,.],[.,.]]]]
[1,1,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]=>[[.,[[[[.,.],.],.],[.,.]]],[[.,.],[.,.]]]
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,0]=>[[[[.,.],.],[.,.]],[[.,.],[[[.,.],.],.]]]
[1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0,0]=>[[[.,.],[.,.]],[[.,.],[[[[.,.],.],.],.]]]
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]=>[[.,[[.,[[.,[[.,.],[[.,.],.]]],.]],.]],.]
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]=>[[.,[[.,[[.,.],[[.,.],[[.,.],.]]]],.]],.]
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]=>[[.,[[.,.],[[.,.],[[.,.],[[.,.],.]]]]],.]
[1,0,1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]=>[.,[[[.,.],[[.,.],[[.,.],[[.,.],.]]]],.]]
[1,0,1,1,0,0,1,1,0,1,1,0,0,1,1,0,0,1,0,0]=>[.,[[.,[[[.,.],[[.,.],[[.,.],.]]],.]],.]]
[1,0,1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,1,0,0]=>[.,[[.,[[.,[[[.,.],[[.,.],.]],.]],.]],.]]
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,1,0,0]=>[.,[[.,[[.,[[.,[[[.,.],.],.]],.]],.]],.]]
[1,0,1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0,1,0]=>[.,[[[[.,.],.],.],[.,[.,[.,[.,[.,.]]]]]]]
[1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]=>[.,[.,[[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]]]
[1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[[[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.]
[1,1,1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0,0,0]=>[[[[[[[.,.],.],.],.],.],[.,.]],[[.,.],.]]
[1,1,1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,0,0,0]=>[[[[[[[[.,.],.],.],.],.],[.,.]],.],[.,.]]
[1,1,1,1,0,1,0,0,1,1,0,1,0,1,0,1,0,0,0,0]=>[[[[.,[[[[[.,.],.],.],.],.]],.],.],[.,.]]
[1,1,1,1,0,1,0,0,0,1,1,1,0,1,0,1,0,0,0,0]=>[[[[.,.],.],.],[[[[[.,.],.],.],.],[.,.]]]
[1,1,1,1,1,1,0,1,0,1,1,0,1,0,0,0,0,0,0,0]=>[[[.,[.,.]],[.,.]],[[[[.,.],[.,.]],.],.]]
[1,1,1,0,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]=>[[.,.],[[[.,[.,.]],[.,.]],[[.,.],[.,.]]]]
[1,1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0,0]=>[[[[[[.,.],.],[[.,.],[[.,.],.]]],.],.],.]
[1,1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0,0]=>[[[.,.],[[[.,.],.],[[[[.,.],.],.],.]]],.]
[1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]=>[[.,[[[[[[[.,[.,.]],.],.],.],.],.],.]],.]
[1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[[.,.],[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]]
[1,1,0,0,1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]=>[[.,[[[.,[[[.,.],[.,.]],.]],[.,.]],.]],.]
[1,1,0,1,1,0,0,0,1,1,0,1,1,0,1,1,0,0,0,0]=>[[[.,.],[.,[[[.,[[.,.],.]],[.,.]],.]]],.]
[1,1,0,1,1,0,0,1,1,0,0,0,1,1,0,1,1,0,0,0]=>[[[.,.],[[.,.],[.,[[[.,.],[.,.]],.]]]],.]
[1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0,1,0]=>[[[.,.],[[.,.],[[.,.],[.,[.,[.,.]]]]]],.]
[1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0,0]=>[[[.,.],[[.,.],[[.,.],[[[.,.],.],.]]]],.]
[1,1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,1,0,0,0]=>[[[.,.],[[.,.],[[[.,[.,.]],[.,.]],.]]],.]
[1,1,0,1,1,0,0,1,0,1,1,0,1,1,0,1,0,0,0,0]=>[[[.,.],[[[.,[[[.,.],.],.]],[.,.]],.]],.]
[1,1,0,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]=>[[[[.,[[[.,.],[.,.]],.]],[.,[.,.]]],.],.]
[1,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0,0]=>[[.,[[[.,.],[.,.]],.]],[.,[[[.,.],.],.]]]
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,1,0]=>[[[[[[[[.,.],[.,[.,.]]],.],.],.],.],.],.]
[1,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0,0,1,0]=>[[[.,[.,[.,[.,[.,[.,.]]]]]],.],[.,[.,.]]]
[1,1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0,0]=>[[[[.,[.,.]],.],[.,.]],[[[.,[.,.]],.],.]]
[1,1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0,0]=>[[.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]],[.,.]]
[1,1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,0]=>[[[[.,.],.],[[[[[[.,.],.],.],.],.],.]],.]
[1,1,0,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]=>[[.,[[[.,.],[.,[.,.]]],[[[.,.],.],.]]],.]
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,1,0,0]=>[.,[[[.,.],[.,[[.,.],.]]],[[[.,.],.],.]]]
[1,1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0,0]=>[[.,[.,[.,[.,[[[[.,.],.],.],.]]]]],[.,.]]
[1,1,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]=>[[.,[[[[[[[.,.],.],.],.],.],.],.]],[.,.]]
[1,1,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0,0]=>[[.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]],[.,.]]
[1,1,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0,0]=>[[.,[.,[.,[.,[[.,.],[.,.]]]]]],[.,[.,.]]]
[1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0]=>[.,[[.,.],[[.,.],[[.,.],[[.,.],[.,.]]]]]]
[1,0,1,1,0,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]=>[.,[[.,[[.,.],[[.,.],[[.,.],[.,.]]]]],.]]
[1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,1,1,0,0,0]=>[.,[[.,[[.,[[.,.],[[.,.],[.,.]]]],.]],.]]
[1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]=>[[.,[.,[.,[.,[[[[[.,.],.],.],.],.]]]]],.]
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,1,0,0]=>[.,[.,[.,[.,[[[.,.],[.,[[.,.],.]]],.]]]]]
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,1,0,0]=>[.,[.,[.,[.,[[.,.],[[.,[[.,.],.]],.]]]]]]
[1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,1,0,0,1,0]=>[.,[.,[.,[[[.,.],[.,[[.,[.,.]],.]]],.]]]]
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0]=>[.,[.,[.,[[.,.],[[.,[[.,[.,.]],.]],.]]]]]
[1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]=>[[[[[.,.],.],[.,.]],[[.,.],[[.,.],.]]],.]
[1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,1,0]=>[.,[[.,[.,[.,[.,[.,[.,.]]]]]],[.,[.,.]]]]
[1,0,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0]=>[.,[.,[[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]]]
[1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]=>[[.,.],[.,[.,[.,[.,[[[.,[.,.]],.],.]]]]]]
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0,0]=>[[.,[.,[.,[.,[.,[.,.]]]]]],[[.,.],[.,.]]]
[1,1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]=>[[.,.],[[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]]
[1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0,1,1,0,0]=>[[[.,[.,.]],.],[[.,[[.,[[.,.],.]],.]],.]]
[1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]=>[[.,[[.,.],[[.,[[.,[[.,.],.]],.]],.]]],.]
[1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]=>[[.,[[[.,[.,.]],.],[[.,[[.,.],.]],.]]],.]
[1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]=>[[.,.],[[[.,[.,.]],.],[[.,[[.,.],.]],.]]]
[1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0,1,1,0,0]=>[[[.,[[.,[.,.]],.]],.],[[.,[[.,.],.]],.]]
[1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0,1,1,0,0]=>[[[[.,[.,.]],.],[[.,[[.,.],.]],.]],[.,.]]
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]=>[[.,[[.,[[.,.],[[.,[[.,.],.]],.]]],.]],.]
[1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]=>[[.,.],[[.,[[.,.],[[.,[[.,.],.]],.]]],.]]
[1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,1,0,0]=>[[[.,[.,.]],.],[[.,.],[[.,[[.,.],.]],.]]]
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]=>[[.,[[.,.],[[.,.],[[.,[[.,.],.]],.]]]],.]
[1,1,0,0,1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]=>[[.,[[.,[[[.,[.,.]],.],[[.,.],.]]],.]],.]
[1,1,1,0,0,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]=>[[.,.],[[.,[[[.,[.,.]],.],[[.,.],.]]],.]]
[1,1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,0,0]=>[[[.,[.,.]],.],[[[.,[.,.]],.],[[.,.],.]]]
[1,1,0,0,1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]=>[[.,[[.,.],[[[.,[.,.]],.],[[.,.],.]]]],.]
[1,1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]=>[[.,.],[[.,.],[[[.,[.,.]],.],[[.,.],.]]]]
[1,1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,0,0]=>[[[.,[[.,[[.,[.,.]],.]],.]],.],[[.,.],.]]
[1,1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,0,0]=>[[[[.,[[.,[.,.]],.]],.],[[.,.],.]],[.,.]]
[1,1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0,1,0,0]=>[[[.,[[[.,[.,.]],.],[[.,.],.]]],.],[.,.]]
[1,1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0,1,0,0]=>[[[[[.,[.,.]],.],[[.,.],.]],[.,.]],[.,.]]
[1,1,0,0,1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]=>[[.,[[[.,[[.,[.,.]],.]],.],[[.,.],.]]],.]
[1,1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]=>[[.,.],[[[.,[[.,[.,.]],.]],.],[[.,.],.]]]
[1,1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0]=>[[[[.,[.,.]],.],[[.,.],.]],[[.,[.,.]],.]]
[1,1,0,0,1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]=>[[.,[[[[.,[.,.]],.],[[.,.],.]],[.,.]]],.]
[1,1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]=>[[.,.],[[[[.,[.,.]],.],[[.,.],.]],[.,.]]]
[1,1,1,0,0,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]=>[[.,.],[[.,[[.,[[.,.],[[.,.],.]]],.]],.]]
[1,1,1,1,0,0,1,0,0,1,0,0,1,1,1,0,0,1,0,0]=>[[[.,[.,.]],.],[[.,[[.,.],[[.,.],.]]],.]]
[1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]=>[[.,[[.,.],[[.,[[.,.],[[.,.],.]]],.]]],.]
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[1,1,0,0,1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]=>[[.,[[[.,[.,.]],.],[[.,.],[[.,.],.]]]],.]
[1,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]=>[[.,.],[[[.,[.,.]],.],[[.,.],[[.,.],.]]]]
[1,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,0,0]=>[[[.,[[.,[.,.]],.]],.],[[.,.],[[.,.],.]]]
[1,1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0,1,0,0]=>[[[[.,[.,.]],.],[[.,.],[[.,.],.]]],[.,.]]
[1,1,1,0,0,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]=>[[.,.],[[.,[[.,.],[[.,.],[[.,.],.]]]],.]]
[1,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,0,0]=>[[[.,[.,.]],.],[[.,.],[[.,.],[[.,.],.]]]]
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0,1,0]=>[[[[[[[.,[[.,[.,.]],.]],.],.],.],.],.],.]
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[1,1,0,1,0,1,0,0,1,1,1,0,0,1,0,1,0,0,1,0]=>[[[[.,[[.,.],[[[.,[.,.]],.],.]]],.],.],.]
[1,1,0,1,0,0,1,1,1,0,0,1,0,1,0,1,0,0,1,0]=>[[[.,[[.,.],[[[[.,[.,.]],.],.],.]]],.],.]
[1,1,0,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0,1,0]=>[[.,[[.,.],[[[[[.,[.,.]],.],.],.],.]]],.]
[1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0,1,0]=>[.,[[.,.],[[[[[[.,[.,.]],.],.],.],.],.]]]
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[1,1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0,0]=>[[[.,[[.,[[.,[[.,.],.]],.]],.]],.],[.,.]]
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[1,1,1,1,0,1,1,0,1,0,1,0,1,0,0,1,0,0,0,0]=>[[[[[.,.],.],[.,.]],.],[.,[.,[.,[.,.]]]]]
[1,1,1,1,0,1,0,1,1,0,1,0,1,0,0,0,1,0,0,0]=>[[[[[.,[.,.]],[.,.]],.],.],[.,[.,[.,.]]]]
[1,1,1,1,0,1,0,1,0,1,1,0,1,0,0,0,0,1,0,0]=>[[[[[[.,.],[[.,.],.]],.],.],.],[.,[.,.]]]
[1,1,1,1,0,1,0,1,1,1,0,1,0,0,0,0,1,0,0,0]=>[[[[[.,[.,.]],[.,.]],.],.],[[[.,.],.],.]]
[1,1,1,1,0,1,1,1,0,1,0,1,0,0,0,1,0,0,0,0]=>[[[[[.,.],.],[.,.]],.],[[[[.,.],.],.],.]]
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,1,0,0]=>[[[[.,.],.],[.,.]],[[.,[.,.]],[[.,.],.]]]
[1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0,1,0,0]=>[[[.,.],[.,.]],[[.,[.,.]],[[[.,.],.],.]]]
[1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0,1,0,0]=>[[[.,[.,.]],[[[[.,.],.],.],.]],[[.,.],.]]
[1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0,1,0,0]=>[[[.,[.,.]],[[[[[.,.],.],.],.],.]],[.,.]]
[1,1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]=>[[[.,[.,.]],.],[[[[[[.,.],.],.],.],.],.]]
[1,1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0,1,0,0]=>[[[[[[[.,[.,.]],.],.],.],.],.],[[.,.],.]]
[1,1,1,1,1,0,1,0,1,0,0,1,0,0,1,0,0,1,0,0]=>[[[[.,[.,.]],.],[[.,.],.]],[.,[.,[.,.]]]]
[1,1,1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]=>[[.,[[[.,[.,.]],.],[.,.]]],[[.,.],[.,.]]]
[1,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]=>[[.,[[[.,.],.],[.,.]]],[[.,.],[[.,.],.]]]
[1,1,1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0,0,0]=>[[[.,[[.,[.,.]],.]],[.,.]],[[.,.],[.,.]]]
[1,1,1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0,0,0]=>[[[[.,.],.],[[.,.],.]],[[[.,.],[.,.]],.]]
[1,1,1,1,1,0,1,1,1,1,0,0,0,0,1,0,0,0,0,0]=>[[[[.,.],.],[.,[.,.]]],[.,[[.,.],[.,.]]]]
[1,1,1,1,0,0,0,1,1,0,0,0,1,1,0,0,1,1,0,0]=>[[[.,.],.],[[.,.],[.,[[.,[[.,.],.]],.]]]]
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[1,1,1,0,1,1,1,1,0,0,0,0,1,1,0,0,0,1,0,0]=>[[.,[[.,[[.,.],.]],[[.,.],.]]],[[.,.],.]]
[1,1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0,0]=>[[.,[[.,[[.,[[.,.],.]],.]],.]],[[.,.],.]]
[1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,1,1,0,0,0]=>[[.,[[.,[[[.,.],[[.,.],[.,.]]],.]],.]],.]
[1,1,0,0,1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0]=>[[.,[[[.,.],[[.,.],[[.,.],[.,.]]]],.]],.]
[1,1,0,0,1,1,1,0,1,0,1,1,0,0,1,0,1,0,0,0]=>[[.,[[.,[.,[[.,[.,[.,.]]],.]]],[.,.]]],.]
[1,1,1,1,0,0,1,0,1,0,0,1,1,1,0,0,0,1,0,0]=>[[[.,[.,[.,.]]],.],[[[.,.],.],[[.,.],.]]]
[1,1,1,0,1,0,1,1,0,0,1,0,1,0,0,0,1,1,0,0]=>[[.,[.,[[.,[.,[.,.]]],.]]],[.,[[.,.],.]]]
[1,0,1,1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0,0]=>[.,[[[.,.],[.,.]],[[.,[.,[.,[.,.]]]],.]]]
[1,0,1,1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0,0]=>[.,[[.,[[[.,.],[.,.]],.]],[.,[[.,.],.]]]]
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,1,0,1,0,0]=>[[[[[[.,[.,[[[.,.],.],.]]],.],.],.],.],.]
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0]=>[[[[[[.,[.,[.,[[.,.],.]]]],.],.],.],.],.]
[1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0,1,0,1,0]=>[[[[[.,.],.],.],.],[.,[.,[.,[.,[.,.]]]]]]
[1,0,1,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0]=>[.,[[[[[[[.,.],[.,.]],.],.],.],.],[.,.]]]
[1,0,1,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0,0]=>[.,[[[.,[.,[.,[.,[.,[.,.]]]]]],.],[.,.]]]
[1,0,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,0]=>[.,[[.,.],[[.,[[.,[[.,[.,.]],.]],.]],.]]]
[1,1,1,0,1,1,0,1,1,0,0,0,0,1,1,0,1,0,0,0]=>[[.,[[[.,.],[[.,[.,.]],[.,.]]],.]],[.,.]]
[1,1,1,0,1,1,0,1,1,0,0,0,0,1,0,0,1,0,1,0]=>[[.,[[[.,.],[[.,[.,[.,.]]],.]],.]],[.,.]]
[1,1,0,1,1,1,0,1,1,0,0,0,1,1,0,1,0,0,0,0]=>[[[[[.,[[[.,.],.],.]],[.,.]],.],[.,.]],.]
[1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0,1,0,1,0]=>[[[[[.,[.,.]],[.,[.,[.,.]]]],.],[.,.]],.]
[1,0,1,1,1,1,0,1,1,0,0,0,1,0,0,1,0,0,1,0]=>[.,[[[[.,[.,.]],[[.,[.,.]],.]],.],[.,.]]]
[1,1,1,0,1,0,0,1,1,0,0,0,1,1,0,1,0,0,1,0]=>[[.,[.,.]],[[.,.],[.,[[[.,[.,.]],.],.]]]]
[1,1,1,0,1,0,0,1,1,0,0,0,1,0,1,1,0,1,0,0]=>[[.,[.,.]],[[.,.],[.,[.,[[[.,.],.],.]]]]]
[1,1,0,1,1,0,1,1,1,0,0,1,1,0,1,0,0,0,0,0]=>[[[.,[[[.,.],[.,.]],[.,[.,.]]]],[.,.]],.]
[1,1,0,1,0,0,1,1,1,0,0,1,1,0,1,0,0,0,1,0]=>[[[.,[[.,.],[[.,[.,.]],[.,[.,.]]]]],.],.]
[1,1,0,1,1,0,1,1,1,0,0,1,0,0,0,0,1,0,1,0]=>[[[.,[[[.,.],.],[.,[.,[.,.]]]]],[.,.]],.]
[1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0,1,0]=>[.,[[.,[[[.,.],.],[[.,[.,.]],.]]],[.,.]]]
[1,1,0,1,0,0,1,1,1,0,0,1,0,0,1,1,0,1,0,0]=>[[[.,[[.,.],[[.,[[[.,.],.],.]],.]]],.],.]
[1,0,1,1,0,1,1,1,1,0,0,1,0,0,1,0,0,0,1,0]=>[.,[[[[[.,[.,.]],.],[.,[.,.]]],[.,.]],.]]
[1,0,1,0,1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]=>[.,[.,[[[[.,[.,.]],.],[[.,.],.]],[.,.]]]]
[1,1,0,1,1,1,0,1,0,0,0,0,1,1,0,1,0,0,1,0]=>[[[[[.,.],.],.],[.,[[[.,[.,.]],.],.]]],.]
[1,1,0,1,1,1,0,1,0,0,0,0,1,0,1,1,0,1,0,0]=>[[[[[.,.],.],.],[.,[.,[[[.,.],.],.]]]],.]
[1,0,1,1,1,1,0,1,0,0,0,1,1,0,1,0,0,0,1,0]=>[.,[[[[.,.],.],.],[[.,[.,.]],[.,[.,.]]]]]
[1,0,1,1,1,1,0,1,0,0,0,1,0,0,1,1,0,1,0,0]=>[.,[[[[.,.],.],.],[[.,[[[.,.],.],.]],.]]]
[1,1,1,1,1,1,0,1,0,0,0,1,0,0,1,0,0,1,0,0]=>[[[[.,[.,.]],.],[[.,.],.]],[[[.,.],.],.]]
[1,1,0,1,1,0,1,1,0,1,1,1,0,1,0,0,0,0,0,0]=>[[[.,[[[.,.],[.,.]],.]],[[[.,.],.],.]],.]
[1,1,0,1,0,0,1,1,0,1,1,1,0,1,0,0,0,0,1,0]=>[[[.,[[[[[.,.],.],.],[.,[.,.]]],.]],.],.]
[1,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,0,0,1,0]=>[.,[[.,[.,.]],[[[[.,.],.],.],[.,[.,.]]]]]
[1,1,0,1,0,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]=>[[[.,[.,[[[[.,.],.],.],[[.,.],.]]]],.],.]
[1,0,1,1,0,1,0,0,1,1,1,1,0,1,0,0,0,1,0,0]=>[.,[[[.,[[[[.,.],.],.],[[.,.],.]]],.],.]]
[1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0,1,0,1,0]=>[[[.,[[[.,.],[.,[.,[.,.]]]],.]],[.,.]],.]
[1,0,1,1,1,0,1,1,0,1,1,0,0,0,0,1,0,0,1,0]=>[.,[[.,[[[.,.],[[.,[.,.]],.]],.]],[.,.]]]
[1,1,0,1,0,0,1,1,0,1,1,0,0,0,1,1,0,1,0,0]=>[[[.,[[[.,.],[.,[[[.,.],.],.]]],.]],.],.]
[1,0,1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0,1,0]=>[.,[[[[[.,[.,.]],[.,[.,.]]],.],[.,.]],.]]
[1,0,1,0,1,1,1,1,0,1,1,0,0,0,1,0,0,1,0,0]=>[.,[.,[[[[.,[.,.]],[[.,.],.]],.],[.,.]]]]
[1,0,1,1,1,0,1,0,0,1,1,0,0,0,1,1,0,1,0,0]=>[.,[[.,[.,.]],[[.,.],[.,[[[.,.],.],.]]]]]
[1,1,1,1,1,0,1,0,0,1,1,0,0,0,1,0,0,1,0,0]=>[[[[.,[.,.]],[[.,.],.]],[.,.]],[.,[.,.]]]
[1,1,0,1,0,0,1,0,1,1,1,0,0,1,1,0,1,0,0,0]=>[[[.,[.,[[.,.],[[.,[.,.]],[.,.]]]]],.],.]
[1,0,1,1,0,1,0,0,1,1,1,0,0,1,1,0,1,0,0,0]=>[.,[[[.,[[.,.],[[.,[.,.]],[.,.]]]],.],.]]
[1,0,1,1,0,1,1,0,1,1,1,0,0,1,0,0,0,0,1,0]=>[.,[[[.,[[[.,.],.],[.,[.,.]]]],[.,.]],.]]
[1,1,1,1,0,1,0,0,1,1,1,0,0,1,0,0,0,1,0,0]=>[[[[.,[[[.,.],.],[[.,.],.]]],.],.],[.,.]]
[1,0,1,0,1,1,0,1,1,1,1,0,0,1,0,0,1,0,0,0]=>[.,[.,[[[[[.,[.,.]],.],[.,.]],[.,.]],.]]]
[1,1,1,0,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0,0]=>[[.,[.,.]],[[[[.,[.,.]],.],[.,.]],[.,.]]]
[1,1,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0,0]=>[[.,[.,[.,[[.,.],[.,.]]]]],[.,[.,[.,.]]]]
[1,1,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0,0]=>[[.,[.,[[.,.],[.,.]]]],[.,[.,[.,[.,.]]]]]
[1,1,1,0,1,0,1,0,1,0,1,0,0,1,1,1,0,0,0,0]=>[[.,[.,[.,[.,[.,.]]]]],[[[.,.],.],[.,.]]]
[1,1,1,0,1,0,1,0,1,0,0,1,1,1,1,0,0,0,0,0]=>[[.,[.,[.,[.,.]]]],[[[.,.],[.,.]],[.,.]]]
[1,1,1,0,1,0,1,0,0,1,1,1,1,0,1,0,0,0,0,0]=>[[.,[.,[.,.]]],[[[.,.],[.,.]],[.,[.,.]]]]
[1,1,1,0,1,0,0,1,1,1,1,0,1,0,1,0,0,0,0,0]=>[[.,[.,.]],[[[.,.],[.,.]],[.,[.,[.,.]]]]]
[1,1,1,0,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]=>[[.,.],[[[.,.],[.,.]],[.,[.,[.,[.,.]]]]]]
[1,1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0,0]=>[[[[[.,[.,.]],[.,.]],.],[[.,[.,.]],.]],.]
[1,1,0,1,1,1,0,1,1,0,0,1,1,0,1,0,0,0,0,0]=>[[[[[[.,.],[.,.]],[.,[.,.]]],.],[.,.]],.]
[1,1,0,1,1,1,0,0,1,1,1,1,0,0,0,0,1,0,0,0]=>[[[[.,[[.,[.,.]],[.,.]]],.],[[.,.],.]],.]
[1,1,0,1,1,1,0,1,1,0,1,0,0,1,1,0,0,0,0,0]=>[[[[[[.,.],[.,.]],[.,.]],.],[.,[.,.]]],.]
[1,1,0,1,1,1,0,0,1,1,1,0,1,1,0,0,0,0,0,0]=>[[[[.,[[.,.],[.,.]]],.],[.,[[.,.],.]]],.]
[1,1,0,1,1,0,1,1,0,1,0,1,1,0,0,0,1,0,0,0]=>[[[.,[[[[.,[.,.]],[.,.]],.],.]],[.,.]],.]
[1,1,0,1,1,0,1,1,0,1,0,0,1,1,1,0,0,0,0,0]=>[[[.,[[[.,[[.,.],[.,.]]],.],.]],[.,.]],.]
[1,1,0,1,1,1,1,0,0,0,1,1,0,1,0,0,1,0,0,0]=>[[[[.,[[[.,[.,.]],.],.]],[.,.]],[.,.]],.]
[1,1,0,1,1,0,1,1,1,0,0,1,0,1,0,0,1,0,0,0]=>[[[.,[[[[.,[.,.]],.],.],[.,.]]],[.,.]],.]
[1,1,0,1,1,1,1,1,0,0,1,0,0,0,1,1,0,0,0,0]=>[[[[.,[[.,.],.]],[.,.]],[[.,[.,.]],.]],.]
[1,1,0,1,1,1,1,0,0,1,1,0,1,0,0,1,0,0,0,0]=>[[[[[[.,.],.],[.,.]],[.,[.,.]]],[.,.]],.]
[1,1,0,1,1,0,1,1,1,1,0,0,0,0,1,1,0,0,0,0]=>[[[.,[[.,[[.,.],.]],[.,.]]],[[.,.],.]],.]
[1,1,0,1,1,1,1,0,1,0,0,1,1,0,0,1,0,0,0,0]=>[[[[[[.,.],.],[.,.]],[.,.]],[.,[.,.]]],.]
[1,1,0,1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0,0]=>[[[.,[[[.,.],.],[.,.]]],[.,[[.,.],.]]],.]
[1,1,1,0,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0]=>[[.,.],[[[[[.,[.,.]],[.,.]],.],[.,.]],.]]
[1,1,1,0,0,1,0,1,1,1,0,0,1,1,1,0,0,0,0,0]=>[[.,.],[[[[.,[[.,.],[.,.]]],.],[.,.]],.]]
[1,1,1,0,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0,0]=>[[.,.],[[[.,[[[.,[.,.]],.],.]],[.,.]],.]]
[1,1,1,0,0,1,0,1,1,1,1,0,0,0,1,1,0,0,0,0]=>[[.,.],[[[[.,[[.,.],.]],[.,.]],[.,.]],.]]
[1,1,1,0,0,1,0,1,1,0,1,1,1,0,0,1,0,0,0,0]=>[[.,.],[[[.,[[[.,.],.],[.,.]]],[.,.]],.]]
[1,1,1,1,0,0,1,0,0,1,0,1,1,1,0,0,1,0,0,0]=>[[[.,[.,.]],.],[[[[.,[.,.]],.],[.,.]],.]]
[1,1,1,1,0,0,1,0,0,1,0,1,1,0,1,1,0,0,0,0]=>[[[.,[.,.]],.],[[[.,[[.,.],.]],[.,.]],.]]
[1,1,1,0,1,1,0,0,0,1,0,1,1,1,0,0,1,0,0,0]=>[[.,[[.,.],.]],[[[[.,[.,.]],.],[.,.]],.]]
[1,1,1,0,1,1,0,0,0,1,0,1,1,0,1,1,0,0,0,0]=>[[.,[[.,.],.]],[[[.,[[.,.],.]],[.,.]],.]]
[1,1,1,1,0,1,1,0,0,0,1,0,0,1,0,1,1,0,0,0]=>[[[[.,[.,.]],[[[.,.],[.,.]],.]],.],[.,.]]
[1,1,1,1,0,0,1,1,1,0,0,0,0,1,0,1,1,0,0,0]=>[[[.,[[.,.],[[[.,.],[.,.]],.]]],.],[.,.]]
[1,1,1,0,1,1,0,1,0,0,1,0,0,1,0,1,1,0,0,0]=>[[.,[[[.,[.,.]],.],.]],[[[.,.],[.,.]],.]]
[1,1,1,1,1,0,0,0,1,1,0,0,0,1,0,1,1,0,0,0]=>[[[.,[[.,.],.]],[[[.,.],[.,.]],.]],[.,.]]
[1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,1,1,0,0,0]=>[[.,[[[.,.],.],[[[.,.],[.,.]],.]]],[.,.]]
[1,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0,1,0,0]=>[[[[.,[.,.]],[[.,.],.]],.],[[.,[.,.]],.]]
[1,1,1,1,0,1,1,0,0,1,1,0,1,0,0,0,0,1,0,0]=>[[[[[.,.],[[.,.],.]],[.,[.,.]]],.],[.,.]]
[1,1,1,1,0,0,1,1,1,1,0,0,0,0,1,0,0,1,0,0]=>[[[.,[[.,[.,.]],[[.,.],.]]],.],[[.,.],.]]
[1,1,1,1,0,1,1,0,1,0,0,1,1,0,0,0,0,1,0,0]=>[[[[[.,.],[[.,.],.]],[.,.]],.],[.,[.,.]]]
[1,1,1,1,0,0,1,1,1,0,1,1,0,0,0,0,0,1,0,0]=>[[[.,[[.,.],[[.,.],.]]],.],[.,[[.,.],.]]]
[1,1,1,0,1,1,0,1,0,1,1,0,0,0,1,0,0,1,0,0]=>[[.,[[[[.,[.,.]],[[.,.],.]],.],.]],[.,.]]
[1,1,1,0,1,1,0,1,0,0,1,1,1,0,0,0,0,1,0,0]=>[[.,[[[.,[[.,.],[[.,.],.]]],.],.]],[.,.]]
[1,1,1,1,1,0,0,0,1,1,0,1,0,0,1,0,0,1,0,0]=>[[[.,[[[.,[.,.]],.],.]],[[.,.],.]],[.,.]]
[1,1,1,0,1,1,1,0,0,1,0,1,0,0,1,0,0,1,0,0]=>[[.,[[[[.,[.,.]],.],.],[[.,.],.]]],[.,.]]
[1,1,1,1,1,1,0,0,1,0,0,0,1,1,0,0,0,1,0,0]=>[[[.,[[.,.],.]],[[.,.],.]],[[.,[.,.]],.]]
[1,1,1,1,1,0,0,1,1,0,1,0,0,1,0,0,0,1,0,0]=>[[[[[.,.],.],[[.,.],.]],[.,[.,.]]],[.,.]]
[1,1,1,1,1,0,1,0,0,1,1,0,0,1,0,0,0,1,0,0]=>[[[[[.,.],.],[[.,.],.]],[.,.]],[.,[.,.]]]
[1,1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0,1,0,0]=>[[.,[[[.,.],.],[[.,.],.]]],[.,[[.,.],.]]]
[1,1,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,0,0]=>[[[[.,[[.,[[.,.],.]],.]],.],[.,.]],[.,.]]
[1,0,1,1,1,0,0,1,0,1,1,1,0,0,1,1,0,0,0,0]=>[.,[[.,.],[[[[.,[[.,.],.]],.],[.,.]],.]]]
[1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0,0]=>[.,[[.,.],[[[[.,.],.],[[.,.],[.,.]]],.]]]
[1,0,1,1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0,0]=>[.,[[.,.],[[[[.,.],[[.,.],[.,.]]],.],.]]]
[1,0,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0,0]=>[.,[[.,.],[[[.,.],[[[.,.],[.,.]],.]],.]]]
[1,0,1,1,1,0,0,1,0,1,1,0,1,1,0,0,0,1,0,0]=>[.,[[.,.],[[[.,[[.,.],.]],[[.,.],.]],.]]]
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[1,1,1,1,1,1,0,0,0,0,0,1,1,1,0,1,0,0,0,0]=>[[[.,.],[[[[.,.],.],.],[.,.]]],[[.,.],.]]
[1,1,1,1,1,0,0,0,0,1,1,1,1,0,1,0,0,0,0,0]=>[[[.,.],[[[.,.],[.,.]],[.,[.,.]]]],[.,.]]
[1,1,1,1,0,0,0,1,1,1,1,1,0,1,0,0,0,0,0,0]=>[[[.,.],.],[[[.,.],[.,.]],[[[.,.],.],.]]]
[1,1,0,0,1,1,0,0,1,1,0,1,1,0,1,1,0,0,0,0]=>[[.,[[.,[[[.,[[.,.],.]],[.,.]],.]],.]],.]
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[1,1,0,0,1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]=>[[.,[[[.,.],[[.,[[.,.],.]],[.,.]]],.]],.]
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[1,1,1,1,0,1,0,1,1,0,0,0,1,0,0,0,1,1,0,0]=>[[[[[.,[.,.]],[.,[[.,.],.]]],.],.],[.,.]]
[1,1,1,0,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[[.,.],[[[[.,.],[.,.]],[[.,.],[.,.]]],.]]
[1,1,1,0,1,0,0,1,0,1,1,1,1,0,1,0,0,0,0,0]=>[[.,[.,.]],[[[[.,.],[.,.]],[.,[.,.]]],.]]
[1,1,1,1,1,0,0,0,0,1,0,1,1,1,1,0,0,0,0,0]=>[[[.,.],[[[[.,.],[.,.]],[.,.]],.]],[.,.]]
[1,1,1,1,1,0,1,0,0,0,0,1,0,1,1,0,1,0,0,0]=>[[[.,.],[[[.,[.,.]],[.,.]],.]],[.,[.,.]]]
[1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,1,0,0,0]=>[[[.,.],[.,.]],[[.,.],[[[.,.],[.,.]],.]]]
[1,1,1,1,1,0,1,0,1,0,1,0,0,1,1,0,0,0,0,0]=>[[[[.,.],[.,.]],[.,.]],[.,[.,[.,[.,.]]]]]
[1,1,1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0,0,0]=>[[[[[.,.],.],.],[.,.]],[.,[.,[.,[.,.]]]]]
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,0,1,0]=>[[.,[.,[.,[.,[.,[.,.]]]]]],[[.,[.,.]],.]]
[1,0,1,1,1,1,0,1,1,1,1,0,0,0,0,1,0,0,0,0]=>[.,[[[[.,.],[.,.]],.],[[[.,.],.],[.,.]]]]
[1,0,1,1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0,0]=>[.,[[.,[[.,.],[.,.]]],[[.,[.,.]],[.,.]]]]
[1,0,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]=>[.,[[[[.,.],[.,.]],[[.,.],[[.,.],.]]],.]]
[1,1,0,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]=>[[[[[.,[[.,[[.,[.,.]],.]],.]],.],.],.],.]
[1,1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,0,0]=>[[[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]],[.,.]]
[1,0,1,1,0,1,1,1,0,0,1,1,0,0,1,0,0,0,1,0]=>[.,[[[[.,[[.,[.,.]],.]],.],[.,[.,.]]],.]]
[1,0,1,1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]=>[.,[[[[.,[[.,[[.,.],.]],.]],.],[.,.]],.]]
[1,0,1,1,0,1,1,1,0,0,1,1,0,0,0,1,0,0,1,0]=>[.,[[[[.,[[.,.],.]],.],[[.,[.,.]],.]],.]]
[1,0,1,1,0,1,1,1,0,0,0,1,0,0,1,1,0,0,1,0]=>[.,[[[[.,.],.],[[.,[[.,[.,.]],.]],.]],.]]
[1,0,1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]=>[.,[[[[.,[[.,[[.,[.,.]],.]],.]],.],.],.]]
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]=>[[.,.],[[[[[[.,.],[[.,.],.]],.],.],.],.]]
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]=>[[.,.],[[[[[[.,[.,.]],[.,.]],.],.],.],.]]
[1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0,0]=>[[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]],[.,.]]
[1,1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,1,0,0,0]=>[[[[[.,[.,[.,.]]],.],[.,.]],[.,.]],[.,.]]
[1,1,1,1,0,0,1,1,1,0,0,1,0,1,0,0,0,1,0,0]=>[[[.,[[[[.,.],.],.],[[.,.],.]]],.],[.,.]]
[1,1,1,1,1,0,0,1,0,0,1,1,0,1,0,0,1,0,0,0]=>[[[[.,[[[.,[.,.]],.],.]],.],[.,.]],[.,.]]
[1,1,1,1,0,0,1,1,0,0,1,1,0,1,0,1,0,0,0,0]=>[[[.,[[.,[[[[.,.],.],.],.]],.]],.],[.,.]]
[1,1,1,0,0,1,1,1,1,0,1,0,0,0,1,0,0,1,0,0]=>[[.,.],[[[.,[.,.]],[[.,.],.]],[.,[.,.]]]]
[1,1,1,0,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0,0]=>[[.,.],[[[[.,[.,[.,.]]],[.,.]],.],[.,.]]]
[1,1,1,1,0,0,1,0,1,1,1,0,0,1,0,0,0,1,0,0]=>[[[.,[.,[[[.,.],.],[[.,.],.]]]],.],[.,.]]
[1,1,1,0,0,1,1,0,1,1,1,0,1,0,0,0,0,1,0,0]=>[[.,.],[[.,[[.,.],[[.,.],.]]],[.,[.,.]]]]
[1,1,1,1,1,0,0,1,0,1,0,0,1,1,0,0,1,0,0,0]=>[[[[[.,[[.,[.,.]],.]],.],.],[.,.]],[.,.]]
[1,1,1,1,0,0,1,1,0,1,0,0,1,1,0,1,0,0,0,0]=>[[[.,[[[.,[[[.,.],.],.]],.],.]],.],[.,.]]
[1,1,1,0,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0,0]=>[[.,.],[[[[[.,[.,.]],[.,.]],.],.],[.,.]]]
[1,1,1,1,0,0,1,0,1,1,0,1,1,0,0,1,0,0,0,0]=>[[[.,[.,[[[[.,.],.],[.,.]],.]]],.],[.,.]]
[1,1,1,0,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]=>[[.,.],[[.,[[[.,.],[.,.]],.]],[.,[.,.]]]]
[1,1,0,1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0,0]=>[[[[[.,[.,.]],.],[[.,.],.]],[[.,.],.]],.]
[1,1,0,1,1,1,1,0,0,0,1,1,0,0,1,0,1,0,0,0]=>[[[[.,[[.,[.,[.,.]]],.]],[.,.]],[.,.]],.]
[1,1,0,1,1,1,0,0,0,1,1,1,0,1,0,0,0,1,0,0]=>[[[[.,.],.],[[[[.,.],.],.],[[.,.],.]]],.]
[1,1,0,1,1,1,1,0,0,0,1,0,1,1,0,0,1,0,0,0]=>[[[[.,[.,[[.,[.,.]],.]]],[.,.]],[.,.]],.]
[1,1,0,1,1,1,0,0,0,1,1,0,1,1,0,1,0,0,0,0]=>[[[[.,.],.],[[.,[[[.,.],.],.]],[.,.]]],.]
[1,1,1,0,1,0,0,1,1,1,1,0,0,0,1,0,0,1,0,0]=>[[.,[.,.]],[[[.,[.,.]],[[.,.],.]],[.,.]]]
[1,1,1,0,1,0,0,1,1,1,0,0,0,1,1,0,1,0,0,0]=>[[.,[.,.]],[[[.,.],.],[[.,[.,.]],[.,.]]]]
[1,1,0,1,1,0,1,1,1,1,0,0,0,0,1,0,0,1,0,0]=>[[[.,[[.,[.,.]],[[.,.],.]]],[[.,.],.]],.]
[1,1,0,1,1,0,1,1,1,0,0,0,0,1,1,0,1,0,0,0]=>[[[.,[[.,.],[[.,[.,.]],[.,.]]]],[.,.]],.]
[1,1,1,1,0,1,0,0,1,0,0,1,1,1,0,0,0,1,0,0]=>[[[[.,[.,.]],.],.],[[[.,.],.],[[.,.],.]]]
[1,1,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0,1,0,0]=>[[.,[[.,[.,[[.,.],[[.,.],.]]]],.]],[.,.]]
[1,1,0,1,1,1,0,1,0,0,0,1,1,1,0,0,0,1,0,0]=>[[[[[.,.],.],.],[[[.,.],.],[[.,.],.]]],.]
[1,1,1,0,1,0,1,1,0,0,1,1,1,0,0,0,0,1,0,0]=>[[.,[.,[[.,[[.,.],[[.,.],.]]],.]]],[.,.]]
[1,1,0,1,1,0,1,1,0,1,1,1,0,0,0,0,0,1,0,0]=>[[[.,[[[.,.],[[.,.],.]],.]],[[.,.],.]],.]
[1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,1,0,0,0]=>[[[[.,[.,.]],.],[[.,.],[.,.]]],[.,[.,.]]]
[1,1,1,1,0,1,1,0,0,1,0,0,1,0,1,1,0,0,0,0]=>[[[[[.,[.,[[.,.],.]]],.],[.,.]],.],[.,.]]
[1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,1,1,0,0,0]=>[[.,[[[[.,.],.],.],[[.,.],[.,.]]]],[.,.]]
[1,1,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,0,0]=>[[[[.,[[[.,[[.,.],.]],.],.]],.],.],[.,.]]
[1,1,1,0,1,1,0,0,1,1,0,1,0,1,1,0,0,0,0,0]=>[[.,[[.,[[[[.,.],[.,.]],.],.]],.]],[.,.]]
[1,1,0,1,1,1,1,0,1,0,0,0,1,0,0,1,1,0,0,0]=>[[[[.,[.,.]],[[.,.],[.,.]]],[.,[.,.]]],.]
[1,1,0,1,1,1,0,1,1,0,0,0,1,0,1,1,0,0,0,0]=>[[[[[.,[.,[[.,.],.]]],[.,.]],.],[.,.]],.]
[1,1,1,0,1,0,1,1,1,0,0,1,0,0,0,1,1,0,0,0]=>[[.,[.,[[[.,.],.],[[.,.],[.,.]]]]],[.,.]]
[1,1,0,1,1,0,1,1,1,0,1,0,0,0,0,1,1,0,0,0]=>[[[.,[[.,.],[[.,.],[.,.]]]],[.,[.,.]]],.]
[1,1,1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0,0,0]=>[[[[[.,[[.,[[.,.],.]],.]],.],.],.],[.,.]]
[1,1,1,0,1,1,0,1,0,0,1,1,0,1,1,0,0,0,0,0]=>[[.,[[[.,[[[.,.],[.,.]],.]],.],.]],[.,.]]
[1,1,0,1,1,1,0,1,0,1,1,0,0,0,1,1,0,0,0,0]=>[[[[[[.,[[.,.],.]],[.,.]],.],.],[.,.]],.]
[1,1,1,0,1,0,1,1,0,1,1,0,0,1,1,0,0,0,0,0]=>[[.,[.,[[[[.,.],[.,.]],[.,.]],.]]],[.,.]]
[1,1,0,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]=>[[.,[[[.,.],[[.,.],.]],[[.,.],[.,.]]]],.]
[1,0,1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]=>[.,[.,[[[.,.],[.,[.,.]]],[[.,.],[.,.]]]]]
[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]=>[[.,[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]],.]
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0,0]=>[[.,[.,.]],[[[[[.,[.,.]],[.,.]],.],.],.]]
[1,1,0,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]=>[[.,[[[.,.],[.,.]],[[[[.,.],.],.],.]]],.]
[1,1,0,0,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]=>[[.,[[[[[[[.,.],.],.],.],.],.],[.,.]]],.]
[1,1,1,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0,0,0]=>[[[[.,.],[.,.]],.],[.,[.,[.,[[.,.],.]]]]]
[1,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,0,1,0,0]=>[[.,[[[[.,.],.],[[.,.],.]],[.,[.,.]]]],.]
[1,1,0,0,1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]=>[[.,[[[[.,[.,.]],[[[.,.],.],.]],.],.]],.]
[1,1,0,0,1,1,0,1,1,1,0,1,1,0,1,0,0,0,0,0]=>[[.,[[[[[.,.],[.,.]],.],[.,[.,.]]],.]],.]
[1,1,1,1,0,0,1,1,1,0,0,0,1,0,0,1,0,1,0,0]=>[[[.,[[.,[.,.]],[[[.,.],.],.]]],.],[.,.]]
[1,1,1,1,0,0,1,1,0,0,0,1,0,1,1,0,1,0,0,0]=>[[[.,[[.,.],.]],.],[[[.,[.,.]],[.,.]],.]]
[1,1,1,0,0,1,0,1,1,0,1,1,1,0,0,0,1,0,0,0]=>[[.,.],[[[.,[[.,[.,.]],[.,.]]],[.,.]],.]]
[1,1,1,1,0,0,1,1,1,0,0,1,0,1,0,0,0,0,1,0]=>[[[.,[[[[.,.],.],.],[.,[.,.]]]],.],[.,.]]
[1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,1,0,0,1,0]=>[[[[.,.],.],[[.,[.,.]],.]],[[.,[.,.]],.]]
[1,1,1,0,0,1,0,1,1,0,1,0,0,1,0,1,0,0,1,0]=>[[.,.],[[[.,[.,.]],[[[.,[.,.]],.],.]],.]]
[1,1,1,0,0,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]=>[[.,.],[[[[.,.],[.,[.,.]]],.],[.,[.,.]]]]
[1,1,1,1,1,0,0,1,0,1,1,0,0,0,1,0,0,0,1,0]=>[[[[[.,[.,.]],[.,[.,.]]],.],[.,.]],[.,.]]
[1,0,1,0,1,1,0,0,1,1,1,0,1,0,0,1,0,1,0,0]=>[.,[.,[[.,[[.,[.,.]],[[[.,.],.],.]]],.]]]
[1,0,1,0,1,1,0,0,1,1,0,1,0,1,1,0,1,0,0,0]=>[.,[.,[[.,[[[[.,[.,.]],[.,.]],.],.]],.]]]
[1,0,1,0,1,1,1,1,0,0,1,1,1,0,0,0,1,0,0,0]=>[.,[.,[[[.,[[.,[.,.]],[.,.]]],.],[.,.]]]]
[1,0,1,0,1,1,1,1,0,0,1,0,0,1,0,1,0,0,1,0]=>[.,[.,[[[.,[.,.]],.],[[[.,[.,.]],.],.]]]]
[1,0,1,0,1,1,1,0,0,1,0,1,1,0,1,0,0,0,1,0]=>[.,[.,[[.,.],[[[.,[.,.]],[.,[.,.]]],.]]]]
[1,0,1,1,1,0,1,1,0,0,0,0,1,1,1,0,1,0,0,0]=>[.,[[.,[[.,.],.]],[.,[[.,[.,.]],[.,.]]]]]
[1,0,1,1,1,0,1,1,1,1,0,0,0,0,1,0,0,0,1,0]=>[.,[[.,[[.,[.,.]],[.,[.,.]]]],[[.,.],.]]]
[1,1,1,0,1,0,0,0,1,1,1,1,0,0,1,0,0,0,1,0]=>[[.,[.,.]],[.,[[[.,[.,.]],.],[.,[.,.]]]]]
[1,0,1,1,1,0,1,0,0,1,0,1,1,0,0,0,1,0,1,0]=>[.,[[.,[.,.]],[[[.,.],[.,[.,[.,.]]]],.]]]
[1,0,1,1,0,1,0,1,1,0,1,1,0,0,0,0,1,0,1,0]=>[.,[[[[.,[[.,.],.]],[.,[.,[.,.]]]],.],.]]
[1,1,1,0,1,1,1,0,0,0,1,1,0,0,0,0,1,0,1,0]=>[[.,[[.,[[.,.],.]],[.,[.,[.,.]]]]],[.,.]]
[1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,0,1,0,1,0]=>[[.,[.,.]],[[[.,[[.,[.,[.,.]]],.]],.],.]]
[1,1,0,1,0,1,1,0,1,0,0,0,1,1,0,0,1,0,1,0]=>[[[[.,[.,.]],[.,[[.,[.,[.,.]]],.]]],.],.]
[1,0,1,1,1,0,1,1,1,0,0,1,0,1,1,0,0,0,0,0]=>[.,[[.,[[[[.,.],[.,.]],.],[.,.]]],[.,.]]]
[1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,1,1,0,0,0]=>[.,[[[[.,.],.],[[.,.],[.,.]]],[.,[.,.]]]]
[1,0,1,1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0,0]=>[.,[[[[.,[.,.]],[[[.,.],[.,.]],.]],.],.]]
[1,0,1,1,0,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]=>[.,[[[[[.,.],[.,.]],.],[.,[[.,.],.]]],.]]
[1,0,1,1,1,1,0,1,0,1,1,0,0,0,1,1,0,0,0,0]=>[.,[[[[[.,[[.,.],.]],[.,.]],.],.],[.,.]]]
[1,1,1,0,1,1,1,0,0,0,1,0,0,1,0,1,1,0,0,0]=>[[.,[[.,[.,.]],[[[.,.],[.,.]],.]]],[.,.]]
[1,1,1,0,1,0,0,1,0,1,1,1,0,0,1,1,0,0,0,0]=>[[.,[.,.]],[[[[.,[[.,.],.]],.],[.,.]],.]]
[1,1,0,1,0,1,1,0,1,1,1,0,0,0,1,1,0,0,0,0]=>[[[[.,[[.,[[.,.],.]],[.,.]]],[.,.]],.],.]
[1,1,1,1,1,0,1,0,0,1,0,0,0,1,0,0,1,1,0,0]=>[[[[.,.],.],[[.,[[.,.],.]],.]],[.,[.,.]]]
[1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0,1,1,0,0]=>[[[[.,[.,.]],[[[.,[[.,.],.]],.],.]],.],.]
[1,1,0,1,1,1,0,1,1,0,1,0,0,0,0,0,1,1,0,0]=>[[[[[.,.],[.,[[.,.],.]]],.],[.,[.,.]]],.]
[1,1,1,0,0,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]=>[[.,.],[[.,[[[.,.],.],[.,[.,[.,.]]]]],.]]
[1,1,0,0,1,0,1,1,0,1,1,1,0,0,0,0,1,1,0,0]=>[[.,[.,[[[[.,.],.],[.,[[.,.],.]]],.]]],.]
[1,1,1,0,0,1,0,1,0,0,1,1,1,1,0,0,0,0,1,0]=>[[.,.],[[[.,[[[.,.],.],[.,[.,.]]]],.],.]]
[1,1,0,1,0,0,1,1,0,1,1,1,0,0,0,0,1,1,0,0]=>[[[.,[[[[.,.],.],[.,[[.,.],.]]],.]],.],.]
[1,1,1,0,1,0,0,0,1,0,1,1,1,1,0,0,0,0,1,0]=>[[.,[.,.]],[.,[.,[[[.,.],.],[.,[.,.]]]]]]
[1,1,0,1,1,0,1,0,0,0,1,1,1,1,0,0,0,0,1,0]=>[[[.,[.,.]],[.,[[[.,.],.],[.,[.,.]]]]],.]
[1,1,1,0,0,0,1,0,1,1,0,1,1,1,0,0,0,1,0,0]=>[[.,.],[.,[.,[[[[.,.],.],[[.,.],.]],.]]]]
[1,1,1,0,1,0,0,0,1,0,1,1,1,1,0,0,0,1,0,0]=>[[.,[.,.]],[.,[.,[[[.,.],.],[[.,.],.]]]]]
[1,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0,1,1,0,0]=>[[[.,.],[[[[.,.],.],[.,[[.,.],.]]],.]],.]
[1,1,1,0,0,1,0,1,0,0,1,1,1,1,0,0,0,1,0,0]=>[[.,.],[[[.,[[[.,.],.],[[.,.],.]]],.],.]]
[1,1,0,1,1,0,1,0,0,1,1,1,0,0,0,0,1,1,0,0]=>[[[.,[.,.]],[[[.,.],.],[.,[[.,.],.]]]],.]
[1,1,1,0,0,1,0,0,1,1,1,1,0,0,0,1,0,1,0,0]=>[[.,.],[[.,[[[.,.],.],[[[.,.],.],.]]],.]]
[1,1,1,0,1,0,0,1,1,1,0,0,0,0,1,1,0,1,0,0]=>[[.,[.,.]],[[[.,.],.],[.,[[[.,.],.],.]]]]
[1,1,1,1,1,0,0,1,0,0,0,0,1,0,1,1,0,0,1,0]=>[[[[.,.],.],[.,[.,[[.,[.,.]],.]]]],[.,.]]
[1,1,0,1,1,1,1,0,0,1,0,0,0,0,1,1,0,0,1,0]=>[[[[[.,.],.],[.,[[.,[.,.]],.]]],[.,.]],.]
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[1,1,1,1,0,1,1,0,0,0,1,1,0,0,1,0,0,0,1,0]=>[[[[.,[[.,[.,.]],.]],[.,[.,.]]],.],[.,.]]
[1,1,1,1,1,0,0,0,1,0,1,1,0,1,0,0,0,0,1,0]=>[[[.,[.,[[[.,.],.],.]]],[.,[.,.]]],[.,.]]
[1,1,1,1,0,1,1,0,0,1,0,0,1,1,0,0,0,0,1,0]=>[[[[[.,[[.,.],.]],.],[.,[.,.]]],.],[.,.]]
[1,1,1,1,1,0,0,1,0,0,1,1,0,1,0,0,0,0,1,0]=>[[[[.,[[[.,.],.],.]],.],[.,[.,.]]],[.,.]]
[1,1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0,0]=>[[.,.],[[[[.,.],.],[[[.,.],[.,.]],.]],.]]
[1,1,1,1,0,0,1,0,0,1,1,1,0,0,0,1,0,1,0,0]=>[[[.,[.,.]],.],[[[.,.],.],[[[.,.],.],.]]]
[1,1,1,0,0,1,0,1,1,0,1,1,0,0,0,1,1,0,0,0]=>[[.,.],[[[.,[[.,.],.]],[[.,.],[.,.]]],.]]
[1,1,1,0,1,0,0,1,1,0,1,1,0,0,0,1,1,0,0,0]=>[[.,[.,.]],[[.,[[.,.],.]],[[.,.],[.,.]]]]
[1,1,1,1,0,1,0,0,0,1,0,1,1,1,0,0,0,1,0,0]=>[[[[.,.],.],.],[[[[.,.],.],[[.,.],.]],.]]
[1,1,1,0,1,1,0,1,0,0,0,1,1,1,0,0,0,1,0,0]=>[[.,[[[.,.],.],.]],[[[.,.],.],[[.,.],.]]]
[1,1,1,1,0,0,0,1,0,1,1,0,1,1,0,0,1,0,0,0]=>[[[.,.],.],[[[.,[[.,[.,.]],.]],[.,.]],.]]
[1,1,1,1,0,1,0,0,0,1,0,1,1,1,0,0,1,0,0,0]=>[[[[.,.],.],.],[[[[.,[.,.]],.],[.,.]],.]]
[1,1,1,0,1,1,0,0,1,0,1,1,0,0,0,1,1,0,0,0]=>[[.,[[.,[.,[[.,.],.]]],.]],[[.,.],[.,.]]]
[1,1,1,1,0,0,1,0,1,0,0,1,1,1,0,0,1,0,0,0]=>[[[.,[.,[.,.]]],.],[[[.,[.,.]],.],[.,.]]]
[1,1,1,0,1,1,0,1,0,0,1,1,0,0,0,1,1,0,0,0]=>[[.,[[[.,[[.,.],.]],.],.]],[[.,.],[.,.]]]
[1,1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,1,0,0,0]=>[[[.,[.,.]],.],[[[.,[.,[.,.]]],.],[.,.]]]
[1,1,1,1,0,1,0,0,1,1,0,0,0,1,1,0,1,0,0,0]=>[[[[.,[[.,.],.]],.],.],[[.,[.,.]],[.,.]]]
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0,1,0,0]=>[[[.,.],[.,.]],[[[.,.],[[[.,.],.],.]],.]]
[1,1,1,1,1,0,1,1,1,0,0,0,0,0,1,0,1,0,0,0]=>[[[.,.],[.,.]],[.,[[.,[.,[.,.]]],[.,.]]]]
[1,1,1,1,0,1,1,1,1,0,0,0,0,1,0,1,0,0,0,0]=>[[[[.,.],[.,.]],.],[[[[.,.],.],.],[.,.]]]
[1,1,1,0,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0,0]=>[[.,[[.,.],[.,[.,[.,.]]]]],[[.,.],[.,.]]]
[1,1,1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0,0,0]=>[[[[.,.],.],[.,.]],[[[.,.],[.,.]],[.,.]]]
[1,1,1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0,0,0]=>[[[[.,.],.],[.,[[.,.],.]]],[[.,.],[.,.]]]
[1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0,0,0]=>[[[[.,[.,.]],.],[.,.]],[[.,[.,.]],[.,.]]]
[1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0,0,0]=>[[[.,[[.,.],.]],[.,.]],[[.,[.,.]],[.,.]]]
[1,1,1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0,0,0]=>[[[.,[.,.]],[[.,[.,.]],.]],[[.,.],[.,.]]]
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[1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]=>[[.,[[.,[[.,[[.,.],[[.,[[.,.],.]],.]]],.]],.]],.]
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[1,1,1,1,0,0,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]=>[[[.,[.,.]],.],[[.,[[.,.],[[.,[[.,.],.]],.]]],.]]
[1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]=>[[.,[[.,.],[[.,[[.,.],[[.,[[.,.],.]],.]]],.]]],.]
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[1,1,0,0,1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,1,0,0]=>[[.,[[[.,[.,.]],.],[[.,.],[[.,[[.,.],.]],.]]]],.]
[1,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,1,0,0]=>[[.,.],[[[.,[.,.]],.],[[.,.],[[.,[[.,.],.]],.]]]]
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[1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]=>[[.,[[.,[[.,[[[.,[.,.]],.],[[.,.],.]]],.]],.]],.]
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[1,1,1,1,0,0,1,0,0,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]=>[[[.,[.,.]],.],[[.,[[[.,[.,.]],.],[[.,.],.]]],.]]
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[1,1,0,0,1,1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,0,0]=>[[.,[[[.,[.,.]],.],[[[.,[.,.]],.],[[.,.],.]]]],.]
[1,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,0,0]=>[[.,.],[[[.,[.,.]],.],[[[.,[.,.]],.],[[.,.],.]]]]
[1,1,1,1,0,0,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,0,0]=>[[[.,[[.,[.,.]],.]],.],[[[.,[.,.]],.],[[.,.],.]]]
[1,1,1,1,1,0,0,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,0,0]=>[[[[.,[.,.]],.],[[[.,[.,.]],.],[[.,.],.]]],[.,.]]
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]=>[[.,[[.,[[.,.],[[[.,[.,.]],.],[[.,.],.]]]],.]],.]
[1,1,1,0,0,1,0,0,1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]=>[[.,.],[[.,[[.,.],[[[.,[.,.]],.],[[.,.],.]]]],.]]
[1,1,1,1,0,0,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]=>[[[.,[.,.]],.],[[.,.],[[[.,[.,.]],.],[[.,.],.]]]]
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]=>[[.,[[.,.],[[.,.],[[[.,[.,.]],.],[[.,.],.]]]]],.]
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[1,1,0,0,1,1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,0,0]=>[[.,[[[.,[[.,[[.,[.,.]],.]],.]],.],[[.,.],.]]],.]
[1,1,1,0,0,1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,0,0]=>[[.,.],[[[.,[[.,[[.,[.,.]],.]],.]],.],[[.,.],.]]]
[1,1,1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0,1,0,0,1,0,0]=>[[[[.,[[.,[.,.]],.]],.],[[.,.],.]],[[.,[.,.]],.]]
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[1,1,1,0,0,1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,0,0]=>[[.,.],[[[[.,[[.,[.,.]],.]],.],[[.,.],.]],[.,.]]]
[1,1,0,0,1,1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0,1,0,0]=>[[.,[[[.,[[[.,[.,.]],.],[[.,.],.]]],.],[.,.]]],.]
[1,1,1,0,0,1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0,1,0,0]=>[[.,.],[[[.,[[[.,[.,.]],.],[[.,.],.]]],.],[.,.]]]
[1,1,1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,0,0,1,0,0]=>[[[[[.,[.,.]],.],[[.,.],.]],[.,.]],[[.,[.,.]],.]]
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[1,1,1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0,1,0,0]=>[[[.,[[.,[[[.,[.,.]],.],[[.,.],.]]],.]],.],[.,.]]
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[1,1,1,1,0,0,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]=>[[[.,[.,.]],.],[[[.,[[.,[.,.]],.]],.],[[.,.],.]]]
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[1,1,1,0,1,0,0,1,0,0,1,1,0,0,1,1,1,0,1,0,0,1,0,0]=>[[.,[.,.]],[[.,[[.,[[.,[.,.]],[[.,.],.]]],.]],.]]
[1,1,1,0,1,0,0,1,0,0,1,1,1,0,1,0,0,0,1,1,0,1,0,0]=>[[.,[.,.]],[[.,[[.,[.,.]],[.,[[[.,.],.],.]]]],.]]
[1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0,1,0,0]=>[[[.,[[[.,.],[[.,.],.]],.]],[.,[[[.,.],.],.]]],.]
[1,1,0,1,1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0,1,0,0]=>[[[.,[[[.,.],[[.,.],.]],.]],[[[.,[.,.]],.],.]],.]
[1,1,0,1,1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0,1,0,0]=>[[[.,[[[.,.],[[.,.],.]],[.,[.,.]]]],[.,[.,.]]],.]
[1,1,0,1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0,1,0,0]=>[[[.,[[[.,.],.],[[.,.],.]]],[.,[[[.,.],.],.]]],.]
[1,1,0,1,1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0,1,0,0]=>[[[.,[[[.,.],.],[[.,.],.]]],[[[.,[.,.]],.],.]],.]
[1,1,0,1,1,1,0,1,0,0,1,1,0,1,1,0,1,0,0,0,0,1,0,0]=>[[[[[.,[[[.,.],[[.,.],.]],.]],.],.],[.,[.,.]]],.]
[1,1,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,0,0,1,0,0]=>[[[[[.,[[[.,.],.],[[.,.],.]]],.],.],[.,[.,.]]],.]
[1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,1,0,1,0,0,0,1,0,0]=>[[[[[.,[[[.,.],.],.]],[[.,.],.]],.],[.,[.,.]]],.]
[1,1,0,1,1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0,0,1,0,0]=>[[[[[.,[[[.,.],.],.]],.],[[.,.],.]],[.,[.,.]]],.]
[1,1,0,1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0,0,1,0,0]=>[[[[[.,[.,.]],[[.,.],.]],.],[.,[[[.,.],.],.]]],.]
[1,1,0,1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0,0,1,0,0]=>[[[[[.,[.,.]],[[.,.],.]],.],[[[.,[.,.]],.],.]],.]
[1,1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,0,0,1,0,0]=>[[[[[.,[.,.]],[[.,.],.]],[.,[.,.]]],[.,[.,.]]],.]
[1,1,0,1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0,0,1,0,0]=>[[[[[.,[.,.]],.],[[.,.],.]],[.,[[[.,.],.],.]]],.]
[1,1,1,0,1,0,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0,0]=>[[.,[.,.]],[[.,[[[.,.],[[.,.],.]],.]],[.,[.,.]]]]
[1,1,1,0,1,0,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0,0]=>[[.,[.,.]],[[.,[[[.,.],.],[[.,.],.]]],[.,[.,.]]]]
[1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,0,0]=>[[.,[.,.]],[[[[.,[[[.,.],.],.]],.],.],[[.,.],.]]]
[1,1,1,0,1,0,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0,0]=>[[.,[.,.]],[[[[.,[.,.]],[[.,.],.]],.],[.,[.,.]]]]
[1,1,1,0,1,0,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0]=>[[.,[.,.]],[[[[.,[.,.]],.],[[.,.],.]],[.,[.,.]]]]
[1,1,1,0,1,1,0,1,0,0,0,1,1,0,1,1,0,1,0,0,0,1,0,0]=>[[.,[[[.,.],.],.]],[[.,[[[.,.],.],.]],[[.,.],.]]]
[1,1,1,0,1,1,0,1,0,0,0,1,1,1,0,1,0,0,1,0,0,1,0,0]=>[[.,[[[.,.],.],.]],[[[[.,[.,.]],.],.],[[.,.],.]]]
[1,1,1,1,0,1,0,0,1,0,0,1,1,0,1,1,0,1,0,0,0,1,0,0]=>[[[[.,[.,.]],.],.],[[.,[[[.,.],.],.]],[[.,.],.]]]
[1,1,1,1,0,1,0,0,1,0,0,1,1,1,0,1,0,0,1,0,0,1,0,0]=>[[[[.,[.,.]],.],.],[[[[.,[.,.]],.],.],[[.,.],.]]]
[1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,1,0,1,0,0,1,0,0]=>[[.,[[[.,.],[[.,[.,.]],[[.,.],.]]],.]],[.,[.,.]]]
[1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,1,0,1,0,0,1,0,0]=>[[.,[[[.,.],.],[[.,[.,.]],[[.,.],.]]]],[.,[.,.]]]
[1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,1,0,1,0,0,1,0,0]=>[[[[.,[[[.,.],.],.]],.],.],[[.,[.,.]],[[.,.],.]]]
[1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,1,0,1,0,0,1,0,0]=>[[[[.,[.,.]],[[.,[.,.]],[[.,.],.]]],.],[.,[.,.]]]
[1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,1,0,1,0,0,1,0,0]=>[[[[.,[.,.]],.],[[.,[.,.]],[[.,.],.]]],[.,[.,.]]]
[1,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0,1,0,1,0,0]=>[[.,[[[.,.],[[[.,.],.],.]],.]],[.,[[[.,.],.],.]]]
[1,1,1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0,1,0,1,0,0]=>[[.,[[[.,.],[[[.,.],.],.]],.]],[[[.,[.,.]],.],.]]
[1,1,1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0,1,0,1,0,0]=>[[.,[[[.,.],[[[.,.],.],.]],[.,[.,.]]]],[.,[.,.]]]
[1,1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0,1,0,1,0,0]=>[[.,[[[.,.],.],[[[.,.],.],.]]],[.,[[[.,.],.],.]]]
[1,1,1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0,1,0,1,0,0]=>[[.,[[[.,.],.],[[[.,.],.],.]]],[[[.,[.,.]],.],.]]
[1,1,1,1,0,1,0,0,1,1,0,1,1,0,1,0,0,0,0,1,0,1,0,0]=>[[[[.,[[[.,.],[[[.,.],.],.]],.]],.],.],[.,[.,.]]]
[1,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0,0]=>[[[[.,[[[.,.],.],[[[.,.],.],.]]],.],.],[.,[.,.]]]
[1,1,1,1,0,1,1,0,1,0,0,0,1,1,0,1,0,0,0,1,0,1,0,0]=>[[[[.,[[[.,.],.],.]],[[[.,.],.],.]],.],[.,[.,.]]]
[1,1,1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0,0,1,0,1,0,0]=>[[[[.,[[[.,.],.],.]],.],[[[.,.],.],.]],[.,[.,.]]]
[1,1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0,0,1,0,1,0,0]=>[[[[.,[.,.]],[[[.,.],.],.]],.],[.,[[[.,.],.],.]]]
[1,1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0,0,1,0,1,0,0]=>[[[[.,[.,.]],[[[.,.],.],.]],.],[[[.,[.,.]],.],.]]
[1,1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,0,0,1,0,1,0,0]=>[[[[.,[.,.]],[[[.,.],.],.]],[.,[.,.]]],[.,[.,.]]]
[1,1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0,0,1,0,1,0,0]=>[[[[.,[.,.]],.],[[[.,.],.],.]],[.,[[[.,.],.],.]]]
[1,1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,1,0,0]=>[[[[.,[.,.]],.],[[[.,.],.],.]],[[[.,[.,.]],.],.]]
          

click to show experimental identities (only identities of compositions of up to three maps are shown)

Description

Francon's map from Dyck paths to binary trees.
This bijection sends the logarithmic height of the Dyck path, St000920The logarithmic height of a Dyck path., to the pruning number of the binary tree, St000396The register function (or Horton-Strahler number) of a binary tree.. The implementation is a literal translation of Knuth's [2].

References

[1] Françon, J. Sur le nombre de registres nécessaires à l'évaluation d'une expression arithmétique MathSciNet:0775838
[2] http://www-cs-faculty.stanford.edu/~knuth/programs/francon.w

Properties

bijective, graded

Sage code

def mapping(D):
    def LR_to_binary_tree(links, root=0):
        if links[root][0] != 0:
            left = LR_to_binary_tree(links, root=links[root][0])
        else:
            left = None

        if links[root][1] != 0:
            right = LR_to_binary_tree(links, root=links[root][1])
        else:
            right = None
        return BinaryTree([left, right])


    n = D.semilength()
    d = list(D) + [0]
    l = [0]*(n)
    r = [0]*(n)
    h = [0]*(2*n+3)
    q = [0]*(n+1)
    qm = [0]*(n+1)
    h[1] = m = 1; k = 2; p = 0
    while True:
        while h[k] != 0:
            k = k + 2
        kk = h[k//2]
        p += 1
        if d[p] == 1:
            l[kk-1] = m; m += 1; h[k] = m;
        else:
            l[kk-1] = 0
        p += 1
        if d[p] == 1:
            r[kk-1] = m; m += 1; h[k+1] = m;
        else:
            r[kk-1] = 0
        if h[k] != 0:
            if h[k+1] != 0:
                continue
            else:
                kk = k
        elif h[k+1] != 0:
            kk = k+1
        else:
            h[k//2] = 0; kk = (k//2) ^^ 1; k = kk & -2
            if k == 0:
                return LR_to_binary_tree(list(zip(l, r)))
        j = 0; jj = 1; q[0] = kk; qm[0] = 1
        while j < jj:
            kk = q[j]
            h[((kk//2)&(-qm[j]))+(kk&(qm[j]-1))] = h[kk]
            if h[2*kk] != 0:
                q[jj] = 2*kk
                q[jj+1] = 2*kk+1
                qm[jj] = qm[jj+1] = qm[j]*2
                jj = jj + 2
            else:
                h[kk] = 0
            j += 1

Weight

Created

Jan 31, 2020 at 19:19 by FindStatCrew

Updated

Jan 31, 2020 at 19:19 by Martin Rubey