Kevin Ryde > Math-PlanePath > Math::PlanePath::RationalsTree

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NAME ^

Math::PlanePath::RationalsTree -- rationals by tree

SYNOPSIS ^

 use Math::PlanePath::RationalsTree;
 my $path = Math::PlanePath::RationalsTree->new (tree_type => 'SB');
 my ($x, $y) = $path->n_to_xy (123);

DESCRIPTION ^

This path enumerates reduced rational fractions X/Y > 0, ie. X and Y having no common factor.

The rationals are traversed by rows of a binary tree which represents a coprime pair X,Y by steps of a subtraction-only greatest common divisor algorithm which proves them coprime. Or equivalently by bit runs with lengths which are the quotients in the division-based Euclidean GCD algorithm and which are also the terms in the continued fraction representation of X/Y.

The SB, CW, AYT, HCS, Bird and Drib trees all have the same set of X/Y rationals in a row, but in a different order due to different encodings of the N value. See the author's mathematical write-up for a proof that these are the only trees with a fixed set of matrices.

http://user42.tuxfamily.org/rationals/index.html

The bit runs mean that N values are quite large for relatively modest sized rationals. For example in the SB tree 167/3 is N=288230376151711741, a 58-bit number. The tendency is for the tree to make excursions out to large rationals while only slowly filling in small ones. The worst is the integer X/1 for which N has X many bits, and similarly 1/Y is Y bits.

See examples/rationals-tree.pl in the Math-PlanePath sources for a printout of all the trees.

Stern-Brocot Tree

The default tree_type=>"SB" is the tree of Moritz Stern and Achille Brocot.

    depth    N                                                  
    -----  -------                                              
      0      1                         1/1                      
                                 ------   ------
      1    2 to 3             1/2               2/1             
                             /    \            /   \
      2    4 to 7         1/3      2/3      3/2      3/1        
                          | |      | |      | |      | |
      3    8 to 15     1/4  2/5  3/5 3/4  4/3 5/3  5/2 4/1      

Within a row the fractions increase in value. Each row of the tree is a repeat of the previous row as first X/(X+Y) and then (X+Y)/Y. For example

    depth=1    1/2, 2/1

    depth=2    1/3, 2/3    X/(X+Y) of previous row
               3/2, 3/1    (X+Y)/Y of previous row

Plotting the N values by X,Y is as follows. The unused X,Y positions are where X and Y have a common factor. For example X=6,Y=2 has common factor 2 so is never reached.

    tree_type => "SB"

    10  |    512        35                  44       767
     9  |    256   33        39   40        46  383       768
     8  |    128        18        21       191       384
     7  |     64   17   19   20   22   95       192   49   51
     6  |     32                  47        96
     5  |     16    9   10   23        48   25   26   55
     4  |      8        11        24        27        56
     3  |      4    5        12   13        28   29        60
     2  |      2         6        14        30        62
     1  |      1    3    7   15   31   63  127  255  511 1023
    Y=0 |
         ----------------------------------------------------
         X=0   1    2    3    4    5    6    7    8    9   10

The X=1 vertical is the fractions 1/Y which is at the left of each tree row, at N value

    Nstart = 2^depth

The Y=1 horizontal is the X/1 integers at the end each row which is

    Nend = 2^(depth+1)-1

Numbering nodes of the tree by rows starting from 1 means N without the high 1 bit is the offset into the row. For example binary N="1011" is "011"=3 into the row. Those bits after the high 1 are also the directions to follow down the tree to a node, with 0=left and 1=right. So N="1011" binary goes from the root 0=left then twice 1=right to reach X/Y=3/4 at N=11 decimal.

Stern-Brocot Mediant

Writing the parents between the children as an "in-order" tree traversal to a given depth has all values in increasing order (the same as each row individually is in increasing order).

                 1/1
         1/2      |      2/1
     1/3  |  2/3  |  3/2  |  3/1
      |   |   |   |   |   |   |

     1/3 1/2 2/3 1/1 3/2 2/1 3/1
                    ^
                    |
                    next level (1+3)/(1+2) = 4/3 mediant

New values at the next level of this flattening are a "mediant" (x1+x2)/(y1+y2) formed from the left and right parent. So the next level 4/3 shown is left parent 1/1 and right parent 3/2 giving mediant (1+3)/(1+2)=4/3. At the left end a preceding 0/1 is imagined. At the right end a following 1/0 is imagined, so as to have 1/(depth+1) and (depth+1)/1 at the ends for a total 2^depth many new values.

The turn sequence left or right along the row depth >= 2 is by a repeating LRRL pattern, except the first and last are always R. (See the author's mathematical write-up for details.)

    RRRL,LRRL,LRRL,LRRL,LRRL,LRRL,LRRL,LRRR   # row N=32 to N=63

Calkin-Wilf Tree

tree_type=>"CW" selects the tree of Calkin and Wilf,

Neil Calkin and Herbert Wilf, "Recounting the Rationals", American Mathematical Monthly, volume 107, number 4, April 2000, pages 360-363.

http://www.math.upenn.edu/~wilf/reprints.html http://www.math.upenn.edu/~wilf/website/recounting.pdf http://www.jstor.org/stable/2589182

As noted above, the values within each row are the same as the Stern-Brocot, but in a different order.

    N=1                             1/1
                              ------   ------
    N=2 to N=3             1/2               2/1
                          /    \            /    \
    N=4 to N=7         1/3      3/2      2/3      3/1
                       | |      | |      | |      | |
    N=8 to N=15     1/4  4/3  3/5 5/2  2/5 5/3  3/4 4/1

Going by rows the denominator of one value becomes the numerator of the next. So at 4/3 the denominator 3 becomes the numerator of 3/5 to the right. These values are Stern's diatomic sequence.

Each row is symmetric in reciprocals, ie. reading from right to left is the reciprocals of reading left to right. The numerators read left to right are the denominators read right to left.

A node descends as

          X/Y
        /     \
    X/(X+Y)  (X+Y)/Y

Taking these formulas in reverse up the tree shows how it relates to a subtraction-only greatest common divisor. At a given node the smaller of P or Q is subtracted from the bigger,

       P/(Q-P)         (P-Q)/P
      /          or        \
    P/Q                    P/Q

Plotting the N values by X,Y is as follows. The X=1 vertical and Y=1 horizontal are the same as the SB above, but the values in between are re-ordered.

    tree_type => "CW"

    10  |      512        56                  38      1022
     9  |      256   48        60   34        46  510       513
     8  |      128        20        26       254       257
     7  |       64   24   28   18   22  126       129   49   57
     6  |       32                  62        65
     5  |       16   12   10   30        33   25   21   61
     4  |        8        14        17        29        35
     3  |        4    6         9   13        19   27        39
     2  |        2         5        11        23        47
     1  |        1    3    7   15   31   63  127  255  511 1023
    Y=0 |
         -------------------------------------------------------------
           X=0   1    2    3    4    5    6    7    8    9   10

At each node the left leg is X/(X+Y) < 1 and the right leg is (X+Y)/Y > 1, which means N is even above the X=Y diagonal and odd below. In general each right leg increments the integer part of the fraction,

    X/Y                       right leg each time
    (X+Y)/Y   = 1 + X/Y
    (X+2Y)/Y  = 2 + X/Y
    (X+3Y)/Y  = 3 + X/Y
    etc

This means the integer part is the trailing 1-bits of N,

    floor(X/Y) = count trailing 1-bits of N
    eg. 7/2 is at N=23 binary "10111"
        which has 3 trailing 1-bits for floor(7/2)=3

N values for the SB and CW trees are converted by reversing bits except the highest. So at a given X,Y position

    SB  N = 1abcde         SB <-> CW by reversing bits
    CW  N = 1edcba         except the high 1-bit

For example at X=3,Y=4 the SB tree has N=11 = "1011" binary and the CW has N=14 binary "1110", a reversal of the bits below the high 1.

N to X/Y in the CW tree can be calculated keeping track of just an X,Y pair and descending to X/(X+Y) or (X+Y)/Y using the bits of N from high to low. The relationship between the SB and CW N's means the same can be used to calculate the SB tree by taking the bits of N from low to high instead.

See also Math::PlanePath::ChanTree for a generalization of CW to ternary or higher trees, ie. descending to 3 or more children at each node.

Yu-Ting and Andreev Tree

tree_type=>"AYT" selects the tree described independently by Yu-Ting and Andreev.

Shen Yu-Ting, "A Natural Enumeration of Non-Negative Rational Numbers -- An Informal Discussion", American Mathematical Monthly, 87, 1980, pages 25-29. http://www.jstor.org/stable/2320374

D. N. Andreev, "On a Wonderful Numbering of Positive Rational Numbers", Matematicheskoe Prosveshchenie, Ser. 3, 1, 1997, pages 126-134 http://mi.mathnet.ru/mp12

Their constructions are a one-to-one mapping between integer N and rational X/Y as a way of enumerating the rationals. This is not designed to be a tree as such, but the result is the same 2^level rows as the above trees. The X/Y values within each row are the same, but in a different order.

    N=1                             1/1
                              ------   ------
    N=2 to N=3             2/1               1/2
                          /    \            /    \
    N=4 to N=7         3/1      1/3      3/2      2/3
                       | |      | |      | |      | |
    N=8 to N=15     4/1  1/4  4/3 3/4  5/2 2/5  5/3 3/5

Each fraction descends as follows. The left is an increment and the right is reciprocal of the increment.

            X/Y
          /     \
    X/Y + 1     1/(X/Y + 1)

which means

          X/Y
        /     \
    (X+Y)/Y  Y/(X+Y)

The left leg (X+Y)/Y is the same the CW has on its right leg. But Y/(X+Y) is not the same as the CW (the other there being X/(X+Y)).

The left leg increments the integer part, so the integer part is given by (in a fashion similar to CW 1-bits above)

    floor(X/Y) = count trailing 0-bits of N
                 plus one extra if N=2^k

N=2^k is one extra because its trailing 0-bits started from N=1 where floor(1/1)=1 whereas any other odd N starts from some floor(X/Y)=0.

The Y/(X+Y) right leg forms the Fibonacci numbers F(k)/F(k+1) at the end of each row, ie. at Nend=2^(level+1)-1. And as noted by Andreev, successive right leg fractions N=4k+1 and N=4k+3 add up to 1,

    X/Y at N=4k+1  +  X/Y at N=4k+3  =  1
    Eg. 2/5 at N=13 and 3/5 at N=15 add up to 1

Plotting the N values by X,Y gives

    tree_type => "AYT"

    10  |     513        41                  43       515
     9  |     257   49        37   39        51  259       514
     8  |     129        29        31       131       258
     7  |      65   25   21   23   27   67       130   50   42
     6  |      33                  35        66
     5  |      17   13   15   19        34   26   30   38
     4  |       9        11        18        22        36
     3  |       5    7        10   14        20   28        40
     2  |       3         6        12        24        48
     1  |       1    2    4    8   16   32   64  128  256  512
    Y=0 |
         ----------------------------------------------------
          X=0   1    2    3    4    5    6    7    8    9   10

N=1,2,4,8,etc on the Y=1 horizontal is the X/1 integers at Nstart=2^level=2^X. N=1,3,5,9,etc in the X=1 vertical is the 1/Y fractions. Those fractions always immediately follow the corresponding integer, so N=Nstart+1=2^(Y-1)+1 in that column.

In each node the left leg (X+Y)/Y > 1 and the right leg Y/(X+Y) < 1, which means odd N is above the X=Y diagonal and even N is below.

The tree structure corresponds to Johannes Kepler's tree of fractions (see Math::PlanePath::FractionsTree). That tree starts from 1/2 and makes fractions A/B with A<B by descending to A/(A+B) and B/(A+B). Those descents are the same as the AYT tree and the two are related simply by

    A = Y        AYT denominator is Kepler numerator
    B = X+Y      AYT sum num+den is the Kepler denominator

    X = B-A      inverse
    Y = A

HCS Continued Fraction

tree_type=>"HCS" selects continued fraction terms coded as bit runs 1000...00 from high to low, as per Paul D. Hanna and independently Czyz and Self.

http://oeis.org/A071766

Jerzy Czyz and William Self, "The Rationals Are Countable: Euclid's Proof", The College Mathematics Journal, volume 34, number 5, November 2003, page 367. http://www.jstor.org/stable/3595818

http://www.cut-the-knot.org/do_you_know/countRatsCF.shtml http://www.dm.unito.it/~cerruti/doc-html/tremattine/tre_mattine.pdf

This arises also in a radix=1 variation of Jeffrey Shallit's digit-based continued fraction encoding. See "Radix 1" in Math::PlanePath::CfracDigits.

If the continued fraction of X/Y is

                 1
    X/Y = a + ------------             a >= 0
                     1
              b + -----------         b,c,etc >= 1
                        1
                  c + -------
                    ... +  1
                          ---          z >= 2
                           z

then the N value is bit runs of lengths a,b,c etc.

    N = 1000 1000 1000 ... 1000
        \--/ \--/ \--/     \--/
         a+1   b    c       z-1

Each group is 1 or more bits. The +1 in "a+1" makes the first group 1 or more bits, since a=0 occurs for any X/Y<=1. The -1 in "z-1" makes the last group 1 or more since z>=2.

    N=1                             1/1
                              ------   ------
    N=2 to N=3             2/1               1/2
                          /    \            /    \
    N=4 to N=7         3/1      3/2      1/3      2/3
                       | |      | |      | |      | |
    N=8 to N=15      4/1 5/2  4/3 5/3  1/4 2/5  3/4 3/5

The result is a bit reversal of the N values in the AYT tree.

    AYT  N = binary "1abcde"      AYT <-> HCS bit reversal
    HCS  N = binary "1edcba"

For example at X=4,Y=7 the AYT tree is N=11 binary "10111" whereas HCS there has N=30 binary "11110", a reversal of the bits below the high 1.

Plotting by X,Y gives

    tree_type => "HCS"

    10  |     768        50                  58       896
     9  |     384   49        52   60        57  448       640
     8  |     192        27        31       224       320
     7  |      96   25   26   30   29  112       160   41   42
     6  |      48                  56        80
     5  |      24   13   15   28        40   21   23   44
     4  |      12        14        20        22        36
     3  |       6    7        10   11        18   19        34
     2  |       3         5         9        17        33
     1  |       1    2    4    8   16   32   64  128  256  512
    Y=0 |
        +-----------------------------------------------------
          X=0   1    2    3    4    5    6    7    8    9   10

N=1,2,4,etc in the row Y=1 are powers-of-2, being integers X/1 having just a single group of bits N=1000..000.

N=1,3,6,12,etc in the column X=1 are 3*2^(Y-1) corresponding to continued fraction 0 + 1/Y so terms 0,Y making runs 1,Y-1 and so bits N=11000...00.

The turn sequence left or right following successive X,Y points is the Thue-Morse sequence. A proof of this can be found in the author's mathematical write-up (above).

    count 1-bits in N+1      turn at N
    -------------------      ---------
           odd                 right
           even                left

Bird Tree

tree_type=>"Bird" selects the Bird tree,

Ralf Hinze, "Functional Pearls: The Bird tree", Journal of Functional Programming, volume 19, issue 5, September 2009, pages 491-508. DOI 10.1017/S0956796809990116 http://www.cs.ox.ac.uk/ralf.hinze/publications/Bird.pdf

It's expressed recursively, illustrating Haskell programming features. The left subtree is the tree plus one and take the reciprocal. The right subtree is conversely the reciprocal first then add one,

       1             1
    --------  and  ---- + 1
    tree + 1       tree

which means Y/(X+Y) and (X+Y)/X taking N bits low to high.

    N=1                             1/1
                              ------   ------
    N=2 to N=3             1/2               2/1
                          /    \            /    \
    N=4 to N=7         2/3      1/3      3/1      3/2
                       | |      | |      | |      | |
    N=8 to N=15     3/5  3/4  1/4 2/5  5/2 4/1  4/3 5/3

Plotting by X,Y gives

    tree_type => "Bird"

    10  |     682        41                  38       597
     9  |     341   43        45   34        36  298       938
     8  |     170        23        16       149       469
     7  |      85   20   22   17   19   74       234   59   57
     6  |      42                  37       117
     5  |      21   11    8   18        58   28   31   61
     4  |      10         9        29        30        50
     3  |       5    4        14   15        25   24        54
     2  |       2         7        12        27        52
     1  |       1    3    6   13   26   53  106  213  426  853
    Y=0 |
         ----------------------------------------------------
          X=0   1    2    3    4    5    6    7    8    9   10

Notice that unlike the other trees N=1,2,5,10,etc in the X=1 vertical for fractions 1/Y is not the row start or end, but instead are on a zigzag through the middle of the tree binary N=1010...etc alternate 1 and 0 bits. The integers X/1 in the Y=1 vertical are similar, but N=11010...etc starting the alternation from a 1 in the second highest bit, since those integers are in the right hand half of the tree.

The Bird tree N values are related to the SB tree by inverting every second bit starting from the second after the high 1-bit,

    Bird N=1abcdefg..    binary
             101010..    xor, so b,d,f etc flip 0<->1
    SB   N=1aBcDeFg..         to make B,D,F

For example 3/4 in the SB tree is at N=11 = binary 1011. Xor with 0010 for binary 1001 N=9 which is 3/4 in the Bird tree. The same xor goes back the other way Bird tree to SB tree.

This xoring is a mirroring in the tree, swapping left and right at each level. Only every second bit is inverted because mirroring twice puts it back to the ordinary way on even rows.

Drib Tree

tree_type=>"Drib" selects the Drib tree by Ralf Hinze.

http://oeis.org/A162911

It reverses the bits of N in the Bird tree (in a similar way that the SB and CW are bit reversals of each other).

    N=1                             1/1
                              ------   ------
    N=2 to N=3             1/2               2/1
                          /    \            /    \
    N=4 to N=7         2/3      3/1      1/3      3/2
                       | |      | |      | |      | |
    N=8 to N=15     3/5  5/2  1/4 4/3  3/4 4/1  2/5 5/3

The descendants of each node are

          X/Y
        /     \
    Y/(X+Y)  (X+Y)/X

The endmost fractions of each row are Fibonacci numbers, F(k)/F(k+1) on the left and F(k+1)/F(k) on the right.

    tree_type => "Drib"

    10  |     682        50                  44       852
     9  |     426   58        54   40        36  340       683
     8  |     170        30        16       212       427
     7  |     106   18   22   24   28   84       171   59   51
     6  |      42                  52       107
     5  |      26   14    8   20        43   19   31   55
     4  |      10        12        27        23        41
     3  |       6    4        11   15        25   17        45
     2  |       2         7         9        29        37
     1  |       1    3    5   13   21   53   85  213  341  853
    Y=0 |
         -------------------------------------------------------
         X=0    1    2    3    4    5    6    7    8    9   10

In each node descent the left Y/(X+Y) < 1 and the right (X+Y)/X > 1, which means even N is above the X=Y diagonal and odd N is below.

Because Drib/Bird are bit reversals like CW/SB are bit reversals, the xor procedure described above which relates Bird<->SB applies to Drib<->CW, but working from the second lowest bit upwards, ie. xor binary "0..01010". For example 4/1 is at N=15 binary 1111 in the CW tree. Xor with 0010 for 1101 N=13 which is 4/1 in the Drib tree.

L Tree

tree_type=>"L" selects the L-tree by Peter Luschny.

http://www.oeis.org/wiki/User:Peter_Luschny/SternsDiatomic

It's a row-reversal of the CW tree with a shift to include zero as 0/1.

    N=0                             0/1
                              ------   ------
    N=1 to N=2             1/2               1/1
                          /    \            /    \
    N=3 to N=8         2/3      3/2      1/3      2/1
                       | |      | |      | |      | |
    N=9 to N=16     3/4  5/3  2/5 5/2  3/5 4/3  1/4 3/1

Notice in the N=9 to N=16 row rationals 3/4 to 1/4 are the same as in the CW tree but read right-to-left.

    tree_type => "L"

    10  |    1021        37                  55       511
     9  |     509   45        33   59        47  255      1020
     8  |     253        25        19       127       508
     7  |     125   21   17   27   23   63       252   44   36
     6  |      61                  31       124
     5  |      29    9   11   15        60   20   24   32
     4  |      13         7        28        16        58
     3  |       5    3        12    8        26   18        54
     2  |       1         4        10        22        46
     1  |  0    2    6   14   30   62  126  254  510 1022 2046
    Y=0 |
         -------------------------------------------------------
         X=0    1    2    3    4    5    6    7    8    9   10

N=0,2,6,14,30,etc along the row at Y=1 are powers 2^(X+1)-2. N=1,5,13,29,etc in the column at X=1 are similar powers 2^Y-3.

Common Characteristics

The SB, CW, Bird, Drib, AYT and HCS trees have the same set of rationals in each row, just in different orders. The properties of Stern's diatomic sequence mean that within a row the totals are

    row N=2^depth to N=2^(depth+1)-1 inclusive

      sum X/Y     = (3 * 2^depth - 1) / 2
      sum X       = 3^depth
      sum 1/(X*Y) = 1

For example the SB tree depth=2, N=4 to N=7,

    sum X/Y     = 1/3 + 2/3 + 3/2 + 3/1 = 11/2 = (3*2^2-1)/2
    sum X       = 1+2+3+3 = 9 = 3^2
    sum 1/(X*Y) = 1/(1*3) + 1/(2*3) + 1/(3*2) + 1/(3*1) = 1

Many permutations are conceivable within a row, but the ones here have some relationship to X/Y descendants, tree sub-forms or continued fractions. As an encoding of continued fraction terms by bit runs the combinations are

     bit encoding           high to low    low to high
    ----------------        -----------    -----------
    0000 1111 runs              SB             CW
    0101 1010 alternating       Bird           Drib
    1000 1000 runs              HCS            AYT

A run of alternating 101010 ends where the next bit is the oppose of the expected alternating 0,1. This is a doubled bit 00 or 11. An electrical engineer would think of it as a phase shift.

Minkowski Question Mark

The Minkowski question mark function is a sum of the terms in the continued fraction representation of a real number. If q0,q1,q2,etc are those terms then the question mark function "?(r)" is

                     1           1           1
    ?(r) = 2 * (1 - ---- * (1 - ---- * (1 - ---- * (1 - ...
                    2^q0        2^q1        2^q2

                     1         1            1
         = 2 * (1 - ---- + --------- - ------------ + ... )
                    2^q0   2^(q0+q1)   2^(q0+q1+q2)

For rational r the continued fraction q0,q1,q2,etc is finite and so the ?(r) sum is finite and rational. The pattern of + and - in the terms gives runs of bits the same as the N values in the Stern-Brocot tree. The RationalsTree code can calculate the ?(r) function by

    rational r=X/Y
    N = xy_to_n(X,Y) tree_type=>"SB"
    depth = floor(log2(N))       # row containing N (depth=0 at top)
    Ndepth = 2^depth             # start of row containing N

           2*(N-Ndepth) + 1
    ?(r) = ----------------
                Ndepth

The effect of N-Ndepth is to remove the high 1-bit, leaving an offset into the row. 2*(..)+1 appends an extra 1-bit at the end. The division by Ndepth scales down from integer N to a fraction.

    N    = 1abcdef      integer, in binary
    ?(r) = a.bcdef1     binary fraction

For example ?(2/3) is X=2,Y=3 which is N=5 in the SB tree. It is at depth=2, Ndepth=2^2=4, and so ?(2/3)=(2*(5-4)+1)/4=3/4. Or written in binary N=101 gives Ndepth=100 and N-Ndepth=01 so 2*(N-Ndepth)+1=011 and divide by Ndepth=100 for ?=0.11.

In practice this is not a very efficient way to handle the question function, since the bit runs in the N values may become quite large for relatively modest fractions. (Math::ContinuedFraction may be better, and also allows repeating terms from quadratic irrationals to be represented exactly.)

Pythagorean Triples

Pythagorean triples A^2+B^2=C^2 can be generated by A=P^2-Q^2, B=2*P*Q. If P>Q>1 with P,Q no common factor and one odd the other even then this gives all primitive triples, being primitive in the sense of A,B,C no common factor ("PQ Coordinates" in Math::PlanePath::PythagoreanTree).

In the Calkin-Wilf tree the parity of X,Y pairs are as follows. Pairs X,Y with one odd the other even are N=0 or 2 mod 3.

    CW tree           X/Y
                   --------
    N=0 mod 3      even/odd
    N=1 mod 3      odd/odd
    N=2 mod 3      odd/even

This occurs because the numerators are the Stern diatomic sequence and the denominators likewise but offset by 1. The Stern diatomic sequence is a repeating pattern even,odd,odd (eg. per "Odd and Even" in Math::NumSeq::SternDiatomic).

The X>Y pairs in the CW tree are the right leg of each node, which is N odd. so

    CW tree N=3 or 5 mod 6   gives X>Y one odd the other even

    index t=1,2,3,etc to enumerate such pairs
    N = 3*t   if t odd
        3*t-1 if t even

2 of each 6 points are used. In a given row it's width/3 but rounded up or down according to where the 3,5mod6 falls on the N=2^depth start, which is either floor or ceil according to depth odd or even,

    NumPQ(depth) = floor(2^depth / 3) for depth=even
                   ceil (2^depth / 3) for depth=odd
      = 0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, ...

These are the Jacobsthal numbers, which in binary are 101010...101 and 1010...1011.

For the other tree types the various bit transformations which map N positions between the trees can be applied to the above N=3or5 mod 6. The simplest is the L tree where the N offset and row reversal gives N=0or4 mod 6.

The SB tree is a bit reversal of the CW, as described above, and for the Pythagorean N this gives

    SB tree N=0 or 2 mod 2 and N="11...." in binary
     gives X>Y one odd the other even

N="11..." binary is the bit reversal of the CW N=odd "1...1" condition. This bit pattern is those N in the second half of each row, which is where the X/Y > 1 rationals occur. The N=0or2 mod 3 condition is unchanged by the bit reversal. N=0or2 mod 3 precisely when bitreverse(N)=0or2 mod 3.

For SB whether it's odd/even or even/odd at N=0or2 mod 3 alternates between rows. The two are both wanted, they just happen to switch in each row.

    SB tree X/Y    depth=even     depth=odd
                   ----------     ---------
    N=0 mod 3      odd/even       even/odd
    N=1 mod 3      odd/odd        odd/odd    <- exclude for Pythagorean
    N=2 mod 3      even/odd       odd/even

FUNCTIONS ^

See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.

$path = Math::PlanePath::RationalsTree->new ()
$path = Math::PlanePath::RationalsTree->new (tree_type => $str)

Create and return a new path object. tree_type (a string) can be

    "SB"      Stern-Brocot
    "CW"      Calkin-Wilf
    "AYT"     Yu-Ting, Andreev
    "HCS"
    "Bird"
    "Drib"
    "L"
$n = $path->n_start()

Return the first N in the path. This is 1 for SB, CW, AYT, HCS, Bird and Drib, but 0 for L.

($n_lo, $n_hi) = $path->rect_to_n_range ($x1,$y1, $x2,$y2)

Return a range of N values which occur in a rectangle with corners at $x1,$y1 and $x2,$y2. The range is inclusive.

For reference, $n_hi can be quite large because within each row there's only one new X/1 integer and 1/Y fraction. So if X=1 or Y=1 is included then roughly $n_hi = 2**max(x,y). If min(x,y) is bigger than 1 then it reduces a little to roughly 2**(max/min + min).

Tree Methods

Each point has 2 children, so the path is a complete binary tree.

@n_children = $path->tree_n_children($n)

Return the two children of $n, or an empty list if $n < 1 (ie. before the start of the path).

This is simply 2*$n, 2*$n+1. Written in binary the children are $n with an extra bit appended, a 0-bit or a 1-bit.

$num = $path->tree_n_num_children($n)

Return 2, since every node has two children. If $n<1, ie. before the start of the path, then return undef.

$n_parent = $path->tree_n_parent($n)

Return the parent node of $n. Or return undef if $n <= 1 (the top of the tree).

This is simply Nparent = floor(N/2), ie. strip the least significant bit from $n. (Undo what tree_n_children() appends.)

$depth = $path->tree_n_to_depth($n)

Return the depth of node $n, or undef if there's no point $n. The top of the tree at N=1 is depth=0, then its children depth=1, etc.

This is simply floor(log2(N)) since the tree has 2 nodes per point. For example N=4 through N=7 are all depth=2.

The L tree starts at N=0 and the calculation becomes floor(log2(N+1)) there.

$n = $path->tree_depth_to_n($depth)
$n = $path->tree_depth_to_n_end($depth)

Return the first or last N at tree level $depth in the path, or undef if nothing at that depth or not a tree. The top of the tree is depth=0.

The structure of the tree means the first N is at 2**$depth, or for the L tree 2**$depth - 1. The last N is 2**($depth+1)-1, or for the L tree 2**($depth+1).

Tree Descriptive Methods

$num = $path->tree_num_children_minimum()
$num = $path->tree_num_children_maximum()

Return 2 since every node has 2 children so that's both the minimum and maximum.

$bool = $path->tree_any_leaf()

Return false, since there are no leaf nodes in the tree.

OEIS ^

The trees are in Sloane's Online Encyclopedia of Integer Sequences in various forms,

http://oeis.org/A007305 (etc)

    tree_type=SB
      A007305   X, extra initial 0,1
      A047679   Y
      A057431   X,Y pairs (initial extra 0/1,1/0)
      A007306   X+Y sum, Farey 0 to 1 part (extra 1,1)
      A153036   int(X/Y), integer part
      A088696   length of continued fraction SB left half (X/Y<1)

    tree_type=CW
      A002487   X and Y, Stern diatomic sequence (extra 0)
      A070990   Y-X diff, Stern diatomic first diffs (less 0)
      A070871   X*Y product
      A007814   int(X/Y), integer part, count trailing 1-bits
                  which is count trailing 0-bits of N+1
      A086893   N position of Fibonacci F[n+1]/F[n], N = binary 1010..101
      A061547   N position of Fibonacci F[n]/F[n+1], N = binary 11010..10
      A047270   3or5 mod 6, being N positions of X>Y not both odd
                  which can generate primitive Pythagorean triples

    tree_type=AYT
      A020650   X
      A020651   Y (Kepler numerator)
      A086592   X+Y sum (Kepler denominator)
      A135523   int(X/Y), integer part,
                   count trailing 0-bits plus 1 extra if N=2^k

    tree_type=HCS
      A229742   X, extra initial 0/1
      A071766   Y
      A071585   X+Y sum

    tree_type=Bird
      A162909   X
      A162910   Y
      A081254   N of row Y=1,    N = binary 1101010...10
      A000975   N of column X=1, N = binary  101010...10

    tree_type=Drib
      A162911   X
      A162912   Y
      A086893   N of row Y=1,    N = binary 110101..0101 (ending 1)
      A061547   N of column X=1, N = binary  110101..010 (ending 0)

    tree_type=L
      A174981   X
      A002487   Y, same as CW X,Y, Stern diatomic
      A047233   0or4 mod 6, being N positions of X>Y not both odd
                  which can generate primitive Pythagorean triples

    tree_type=SB,CW,AYT,HCS,Bird,Drib,L
      A008776   total X+Y in row, being 2*3^depth

    A000523  tree_n_to_depth(), being floor(log2(N))

    A059893  permutation SB<->CW, AYT<->HCS, Bird<->Drib
               reverse bits below highest
    A153153  permutation CW->AYT, reverse and un-Gray
    A153154  permutation AYT->CW, reverse and Gray code
    A154437  permutation AYT->Drib, Lamplighter low to high
    A154438  permutation Drib->AYT, un-Lamplighter low to high
    A003188  permutation SB->HCS, Gray code shift+xor
    A006068  permutation HCS->SB, Gray code inverse
    A154435  permutation HCS->Bird, Lamplighter bit flips
    A154436  permutation Bird->HCS, Lamplighter variant

    A054429  permutation SB,CW,Bird,Drib N at transpose Y/X,
               (mirror binary tree, runs 0b11..11 down to 0b10..00)
    A004442  permutation AYT N at transpose Y/X, from N=2 onwards
               (xor 1, ie. flip least significant bit)
    A063946  permutation HCS N at transpose Y/X, extra initial 0
               (xor 2, ie. flip second least significant bit)

    A054424  permutation DiagonalRationals -> SB
    A054426  permutation SB -> DiagonalRationals
    A054425  DiagonalRationals -> SB with 0s at non-coprimes
    A054427  permutation coprimes -> SB right hand X/Y>1

    A044051  N+1 of those N where SB and CW have same X,Y
               same Bird<->Drib and HCS<->AYT
               begin N+1 of N binary palindrome below high 1-bit

The sequences marked "extra ..." have one or two extra initial values over what the RationalsTree here gives, but are the same after that. And the Stern first differences "less ..." means it has one less term than what the code here gives.

SEE ALSO ^

Math::PlanePath, Math::PlanePath::FractionsTree, Math::PlanePath::CfracDigits, Math::PlanePath::ChanTree

Math::PlanePath::CoprimeColumns, Math::PlanePath::DiagonalRationals, Math::PlanePath::FactorRationals, Math::PlanePath::GcdRationals, Math::PlanePath::PythagoreanTree

Math::NumSeq::SternDiatomic, Math::ContinuedFraction

HOME PAGE ^

http://user42.tuxfamily.org/math-planepath/index.html

LICENSE ^

Copyright 2011, 2012, 2013 Kevin Ryde

This file is part of Math-PlanePath.

Math-PlanePath is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 3, or (at your option) any later version.

Math-PlanePath is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

You should have received a copy of the GNU General Public License along with Math-PlanePath. If not, see <http://www.gnu.org/licenses/>.

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