UAD Tree¶

The UAD tree by Berggren (1934) and later independently by Barning (1963), Hall (1970), and other authors, uses three matrices U, A and D which can be multiplied onto an existing primitive triple to form three further new primitive triples.

    tree_type => "UAD"   (the default)
    Y=40 |          14
         |
         |
         |
         |                                              7
    Y=24 |        5
         |
    Y=20 |                      3
         |
    Y=12 |      2                             13
         |
         |                4
     Y=4 |    1
         |
         +--------------------------------------------------
            X=3         X=15  X=20           X=35      X=45

The UAD matrices are

        / 1 -2  2 \        / 1  2  2 \        / -1  2  2 \
    U = | 2 -1  2 |    A = | 2  1  2 |    D = | -2  1  2 |
        \ 2 -2  3 /        \ 2  2  3 /        \ -2  2  3 /

They're multiplied on the right of an (A,B,C) column vector, for example

         / 3 \     /  5 \
     U * | 4 |  =  | 12 |
         \ 5 /     \ 13 /

The starting point is N=1 at X=3,Y=4 which is the well-known triple

    3^2 + 4^2 = 5^2

From it three further points N=2, N=3 and N=4 are derived, then three more from each of those, etc,

   tree_type => "UAD"    coordinates A,B
               ______________ 3,4 _____________
              /                |               \
          5,12               21,20              15,8
         /  |  \            /  |  \           /   |  \
    7,24  55,48 45,28  39,80 119,120 77,36  33,56 65,72 35,12
    rows depth = 0    N=1
         depth = 1    N=2..4
         depth = 2    N=5..13
         depth = 3    N=14..

Counting N=1 as depth=0, each level has 3^depth many points and the first N of a level (tree_depth_to_n()) is at

    Nrow = 1 + (1 + 3 + 3^2 + ... + 3^(depth-1))
         = (3^depth + 1) / 2
         = 1, 2, 5, 14, 41, 122, 365, ...    (A007051)

The level numbering is like a mixed-radix representation of N where the high digit is binary (so always 1) and the digits below are ternary.

         +--------+---------+---------+--   --+---------+
    N =  | binary | ternary | ternary |  ...  | ternary |
         +--------+---------+---------+--   --+---------+
              1      0,1,2     0,1,2             0,1,2

The number of ternary digits is the "depth" and their value without the high binary 1 is the position in the row.

U Repeatedly¶

Taking the upper "U" matrix repeatedly gives

    3.4 -> 5,12 -> 7,24 -> 9,40 -> etc

with C=B+1 and A the odd numbers. These are the first of each level so at Nrow described above. The resulting triples are a sequence known to Pythagoras (Dickson's History of the Theory of Numbers, start of chapter IV).

    A = any odd integer, so A^2 any odd square
    B = (A^2-1)/2
    C = (A^2+1)/2
           / A^2-1 \       / A^2+1 \
    A^2 + | ------  |^2 = |  -----  |^2
           \   2   /       \   2   /

This is also described by Fibonacci (Liber Quadratorum) in terms of sums of odd numbers

    s = any odd square = A^2
    B^2 = 1 + 3 + 5 + ... + s-2      = ((s-1)/2)^2
    C^2 = 1 + 3 + 5 + ... + s-2 + s  = ((s+1)/2)^2
    so C^2 = A^2 + B^2
    eg. s=25=A^2  B^2=((25-1)/2)^2=144  so A=5,B=12

The geometric interpretation is that an existing square of side B is extended by a "gnomon" around two sides making a new larger square of side C=B+1. The length of the gnomon is odd and when it's an odd square then the new total area is the sum of two squares.

       ****gnomon*******     gnomon length an odd square = A^2
       +-------------+ *
       |             | *     so new bigger square area
       |   square    | *     C^2 = A^2 + B^2
       | with side B | *
       |             | *
       +-------------+ *

See Math::PlanePath::Corner for a path following such gnomons.

A Repeatedly¶

Taking the middle "A" matrix repeatedly gives

    3,4 -> 21,20 -> 119,120 -> 697,696 -> etc        A,B legs

which are the triples with legs A,B differing by 1 and so just above and below the X=Y leading diagonal. The N values are 1,3,9,27,etc = 3^depth.

D Repeatedly¶

Taking the lower "D" matrix repeatedly gives

   3,4 -> 15,8 -> 35,12 -> 63,16 -> etc              A,B legs

which is the primitives among a sequence of triples known to the ancients (Dickson's History of the Theory of Numbers, start of chapter IV),

     A = k^2 - 1       k even >= 2 for primitives
     B = 2*k
     C = k^2 + 1       so C=A+2

When k is even these are primitive. If k is odd then A and B are both even, ie. a common factor of 2, so not primitive. These points are the last of each level, so at N=(3^(depth+1)-1)/2 which is tree_depth_to_n_end().

UArD Tree¶

Option "tree_type => "UArD"" varies the UAD tree by applying a left-right reflection under each "A" matrix. The result is ternary reflected Gray code order. The 3 children under each node are unchanged, just their order.

   tree_type => "UArD"    coordinates A,B
               ______________ 3,4 _____________
              /                |               \
          5,12               21,20              15,8
         /  |  \            /  |  \           /   |  \
    7,24  55,48 45,28  77,36 119,120 39,80  33,56 65,72 35,12

Notice the middle points 77,36 and 39,80 are swapped relative to the UAD shown above. In general, the whole tree underneath an "A" is mirrored left <->right. If there's an even number of "A"s above then those mirrorings cancel out to be plain again.

This tree form is primarily of interest for "Digit Order Low to High" described below since it gives each row of points in order clockwise down from the Y axis.

In "PQ Coordinates" below, with the default digits high to low, UArD also makes successive steps across the row either horizontal or 45-degrees NE-SW.

In all cases, the Gray coding is applied to N first, then the resulting digits are interpreted either high to low (the default) or low to high ("LtoH" option).

FB Tree¶

Option "tree_type => "FB"" selects a tree independently by

Firstov finds this tree by semigroup transformations. Price finds it by expressing triples in certain "Fibonacci boxes" with a box of four values q',q,p,p' having p=q+q' and p'=p+q so each is the sum of the preceding two in a fashion similar to the Fibonacci sequence. A box where p and q have no common factor corresponds to a primitive triple. See "PQ Coordinates" and "FB Transformations" below.

    tree_type => "FB"
    Y=40 |         5
         |
         |
         |
         |                                             17
    Y=24 |       4
         |
         |                     8
         |
    Y=12 |     2                             6
         |
         |               3
    Y=4  |   1
         |
         +----------------------------------------------
           X=3         X=15   x=21         X=35

For a given box, three transformations can be applied to go to new boxes corresponding to new primitive triples. This visits all and only primitive triples, but in a different order to UAD above.

The first point N=1 is again at X=3,Y=4, from which three further points N=2,3,4 are derived, then three more from each of those, etc.

   tree_type => "FB"    coordinates A,B
               ______________ 3,4 _____________
              /                |               \
          5,12               15,8               7,24
         /  |  \            /  |  \           /   |  \
    9,40  35,12 11,60  21,20 55,48 39,80  13,84 63,16 15,112

UMT Tree¶

Option "tree_type => "UMT"" is a third tree type by Firstov (reference above). It is matrices U, M2, and a new third T = M1*D.

   tree_type => "UMT"    coordinates A,B            children
                                                     U,M2,T
              ______________ 3,4 _____________
             /                |               \
         5,12               15,8               21,20
        /  |  \            /  |  \           /   |  \
   7,24  35,12 65,72  33,56 55,48 45,28  39,80 91,60 105,88

The first "T" child 21,20 is the same as the "A" matrix, but it differs at further levels down. For example "T" twice is 105,88 (bottom most in the diagram) which is not the same as "A" twice 119,120.

Digit Order Low to High¶

Option "digit_order => 'LtoH'" applies matrices using the ternary digits of N taken from low to high. The points in each row are unchanged, as is the parent-child N numbering, but the X,Y values are rearranged within the row.

The UAD matrices send points to disjoint regions and the effect of LtoH is to keep the tree growing into those separate wedge regions. The arms grow roughly as follows

    tree_type => "UAD", digit_order => "LtoH"
    Y=80 |                  6                       UAD LtoH
         |                 /
         |                /
    Y=56 |               /   7     10  9
         |              /   /       / /
         |             /   /       | /  8
         |            /  _/       / /  /
         |           /  /        / /  /
    Y=24 |        5 /  /        | / _/        __--11
         |       / / _/         |/_/      __--
    Y=20 |      / / /         __3     __--       _____----12
         |      |/_/      __--   __---  ____-----
    Y=12 |      2     __--     _/___----  ____13
         |     /  __--     __-- _____-----
         |    /_--_____---4-----
     Y=4 |    1---
         |
         +--------------------------------------------------
            X=3         X=15  X=20           X=35        X=76

Notice the points of the second row N=5 to N=13 are almost clockwise down from the Y axis, except N=8,9,10 go upwards. Those N=8,9,10 go upwards because the A matrix has a reflection (its determinant is -1).

Option "tree_type => "UArD"" reverses the tree underneath each A, and that plus LtoH gives A,B points going clockwise in each row. P,Q coordinates likewise go clockwise.

AC Coordinates¶

Option "coordinates => 'AC'" gives the A and C legs of each triple as X=A,Y=C.

    coordinates => "AC"
     85 |        122                             10
        |
        |
     73 |                             6
        |
     65 |                  11             40
     61 |       41
        |
        |                        7
        |
        |
     41 |      14
        |                   13
     35 |
        |            3
     25 |     5
        |
     17 |         4
     13 |    2
        |
    Y=5 |   1
        |
        +-------------------------------------------
          X=3 7 9   21      35   45  55   63     77

Since A<C, the coordinates are X<Y all above the X=Y diagonal. The "D Repeatedly" triples described above have C=A+2 so they are the points Y=X+2 just above the diagonal.

For the FB tree the set of points visited is the same (of course), but a different N numbering.

    tree_type => "FB", coordinates => "AC"
     85 |        11                              35
        |
        |
     73 |                             9
        |
     65 |                  23             12
     61 |       7
        |
        |                        17
        |
        |
     41 |      5
        |                   6
     35 |
        |            8
     25 |     4
        |
     17 |         3
     13 |    2
        |
    Y=5 |   1
        |
        +-------------------------------------------
          X=3 7 9   21      35   45  55   63     77

BC Coordinates¶

Option "coordinates => 'BC'" gives the B and C legs of each triple as X=B,Y=C. This is the B=even and C=long legs of all primitive triples. This combination has points on 45-degree straight lines.

    coordinates => "BC"
    101 |           121
     97 |                                     12
        |
     89 |                                         8
     85 |                   10                      122
        |
        |
     73 |                         6
        |
     65 |         40                  11
     61 |                               41
        |
        |               7
        |
        |
     41 |                     14
        |       13
     35 |
        |           3
     25 |             5
        |
     17 |     4
     13 |       2
        |
    Y=5 |   1
        |
        +--------------------------------------------------
          X=4  12    24      40        60           84

Since B<C, the coordinates are X<Y above the X=Y leading diagonal. N=1,2,5,14,41,etc along the X=Y-1 diagonal are the "U Repeatedly" triples described above which have C=B+1 and are at the start of each tree row.

For the FB tree, the set of points visited is the same of course, but a different N numbering.

    tree_type => "FB", coordinates => "BC"
    101 |           15
     97 |                                     50
        |
     89 |                                         10
     85 |                   35                      11
        |
        |
     73 |                         9
        |
     65 |         12                  23
     61 |                               7
        |
        |               17
        |
        |
     41 |                     5
        |       6
     35 |
        |           8
     25 |             4
        |
     17 |     3
     13 |       2
        |
    Y=5 |   1
        |
        +----------------------------------------------
          X=4  12    24      40        60           84

As seen in the diagrams, B,C points fall on 45-degree straight lines going up from X=Y-1. This occurs because a primitive triple A,B,C with A odd and B even can be written

    A^2 = C^2 - B^2
        = (C+B)*(C-B)

Then gcd(A,B)=1 means also gcd(C+B,C-B)=1 and so since C+B and C-B have no common factor they must each be squares to give A^2. Call them s^2 and t^2,

    C+B = s^2    and conversely  C = (s^2 + t^2)/2
    C-B = t^2                    B = (s^2 - t^2)/2
      s = odd integer      s >= 3
      t = odd integer  s > t >= 1
      with gcd(s,t)=1 so that gcd(C+B,C-B)=1

When t=1, this is C=(s^2+1)/2 and B=(s^2-1)/2 which is the "U"-repeated points at Y=X+1 for each s. As t increases, the B,C coordinate combination makes a line upwards at 45-degrees from those t=1 positions,

     C + B = s^2      anti-diagonal 45-degrees,
                      position along diagonal determined by t

All primitive triples start from a C=B+1 with C=(s^2+1)/2 half an odd square, and go up from there. To ensure the triple is primitive, must have gcd(s,t)=1. Values of t where gcd(s,t)!=1 are gaps in the anti-diagonal lines.

PQ Coordinates¶

Primitive Pythagorean triples can be parameterized as follows for A odd and B even. This is per Euclid, Diophantus, and anonymous Arabic manuscript for constraining it to primitive triples (Dickson's History of the Theory of Numbers, start of chapter IV).

    A = P^2 - Q^2
    B = 2*P*Q
    C = P^2 + Q^2
    with P > Q >= 1, one odd, one even, and no common factor
    P = sqrt((C+A)/2)
    Q = sqrt((C-A)/2)

The first P=2,Q=1 is the triple A=3,B=4,C=5.

Option "coordinates => 'PQ'" gives these as X=P,Y=Q, for either "tree_type". Because P>Q>=1 the values fall in the eighth of the plane below the X=Y diagonal,

    tree_type => "UAD", coordinates => "PQ"
     10 |                                                   9842
      9 |                                              3281
      8 |                                         1094        23
      7 |                                     365        32
      6 |                                122                  38
      5 |                            41         8
      4 |                       14        11        12        15
      3 |                   5                   6        16
      2 |              2         3         7        10        22
      1 |         1         4        13        40       121
    Y=0 |
        +--------------------------------------------------------
        X=0  1    2    3    4    5    6    7    8    9   10   11

The diagonal N=1,2,5,14,41,etc is P=Q+1 as per "U Repeatedly" above.

The one-to-one correspondence between P,Q and A,B means all tree types visit all P,Q pairs, so all X,Y with no common factor and one odd one even. There's other ways to iterate through such coprime pairs and any such method would generate Pythagorean triples too, in a different order from the trees here.

The letters P and Q here are a little bit arbitrary. This parameterization is often written m,n or u,v but don't want "n" to be confused that with N point numbering or "u" to be confused with the U matrix (leg "A" is already too close to matrix "A"!).

SM Coordinates¶

Option "coordinates => 'SM'" gives the small and medium legs from each triple as X=small,Y=medium. This is like "AB" except that if A>B then they're swapped to X=B,Y=A so X<Y always. The effect is to mirror the AB points below the X=Y diagonal up to the upper half quadrant,

    coordinates => "SM"
     91 |                                16
     84 |        122
        |                     8
        |                    10
     72 |                                  12
        |
        |
     60 |       41 40
        |                  11
     55 |                          6
        |
        |                7
     40 |      14
        |
     35 |        13
        |
     24 |     5
     21 |            3
        |
     12 |    2 4
        |
    Y=4 |   1
        |
        +----------------------------------------
          X=3  8     20     33     48      60 65

SC Coordinates¶

Option "coordinates => 'SC'" gives the small leg and hypotenuse from each triple,

    coordinates => "SC"
     85 |        122         10
        |
        |
     73 |                          6
        |
        |          40      11
     61 |       41
        |
     53 |                7
        |
        |
     41 |      14
     37 |        13
        |
        |            3
     25 |     5
        |
        |      4
     13 |    2
        |
    Y=5 |   1
        |
        +-----------------------------
          X=3  8     20     33     48

The points are all X < sqrt(2)*Y since with X as the smaller leg must have X^2 < Y^2/2 so X < Y*1/sqrt(2).

MC Coordinates¶

Option "coordinates => 'MC'" gives the medium leg and hypotenuse from each triple,

    coordinates => "MC"
     65 |                             11 40
     61 |                               41
        |
     53 |                       7
        |
        |
     41 |                     14
     37 |                  13
        |
     29 |           3
     25 |             5
        |
     17 |        4
     13 |       2
        |
    Y=5 |   1
        |
        +-----------------------------------
          X=4   15   24    35 40      56 63

The points are in a wedge sqrt(2)*Y < X < Y. X is the bigger leg and X^2 > Y^2/2 so X > Y*1/sqrt(2).

UAD Coordinates AB, AC, PQ -- Turn Right¶

In the UAD tree with coordinates AB, AC or PQ the path always turns to the right. For example in AB coordinates at N=2 the path turns to the right to go towards N=3.

    coordinates => "AB"
    20 |                      3           N    X,Y
       |                                 --   ------
    12 |      2                           1    3,4
       |                                  2    5,12
       |                                  3   21,20
     4 |    1
       |                               turn towards the
       +-------------------------        right at N=2
            3 5              21

This can be proved from the transformations applied to seven cases, a triplet U,A,D, then four crossing a gap within a level, then two wrapping around at the end of a level. The initial N=1,2,3 can be treated as a wrap-around from the end of depth=0 (the last case D to U,A).

    U              triplet U,A,D
    A
    D
    U.D^k.A        crossing A,D to U
    U.D^k.D        across U->A gap
    A.U^k.U         k>=0
    A.D^k.A        crossing A,D to U
    A.D^k.D        across A->D gap
    D.U^k.U         k>=0
    U.D^k.D        crossing D to U,A
    U.U^k.U        across U->A gap
    A.U^k.A         k>=0
    A.D^k.D        crossing D to U,A
    A.U^k.U        across A->D gap
    D.U^k.A         k>=0
    D^k    .A      wraparound A,D to U
    D^k    .D       k>=0
    U^(k+1).U
    D^k            wraparound D to U,A
    U^k.U           k>=0
    U^k.A           (k=0 is initial N=1,N=2,N=3 for none,U,A)

The powers U^k and D^k are an arbitrary number of descents U or D. In P,Q coordinates, these powers are

    U^k    P,Q   ->  (k+1)*P-k*Q, k*P-(k-1)*Q
    D^k    P,Q   ->  P+2k*Q, Q

For AC coordinates, squaring to stretch to P^2,Q^2 doesn't change the turns. Then a rotate by -45 degrees to A=P^2-Q^2, C=P^2+Q^2 also doesn't change the turns.

UAD Coordinates BC -- Turn Left¶

In the UAD tree with coordinates BC the path always turns to the left. For example in BC coordinates at N=2 the path turns to the right to go towards N=3.

    coordinates => "BC"
    29 |           3                N    X,Y
       |                           --   ------
       |                            1    4,5
       |                            2   12,13
    13 |       2                    3   20,29
       |
     5 |   1                     turn towards the
       |                           left at N=2
       +---------------
           4  12   20

As per above, A,C turns to the right, which squared is A^2,C^2 to the right too, which equals C^2-B^2,C^2. Negating the X coordinate to B^2-C^2,C^2 mirrors to be a left turn always, and addition shearing to X+Y,Y doesn't change that, giving B^2,C^2 always left and so B,C always left.

Descriptive Methods¶

"$x = $path->x_minimum()"

"$y = $path->y_minimum()"

Return the minimum X or Y occurring in the path. The value goes according to the "coordinates" option,

    coordinate    minimum
    ----------    -------
        A,S          3
        B,M          4
         C           5
         P           2
         Q           1
    

Tree Methods¶

Each point has 3 children, so the path is a complete ternary tree.

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

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

This is simply "3*$n-1, 3*$n, 3*$n+1". This is appending an extra ternary digit 0, 1 or 2 to the mixed-radix form for N described above. Or staying all in ternary then appending to N+1 rather than N and adjusting back.

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

Return 3, since every node has three children, or return "undef" if "$n<1" (ie. before the start of the path).

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

Return the parent node of $n, or "undef" if "$n <= 1" (the top of the tree).

This is simply "floor(($n+1)/3)", reversing the tree_n_children() calculation above.

"$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.

The structure of the tree with 3 nodes per point means the depth is floor(log3(2N-1)), so for example N=5 through N=13 all have depth=2.

"$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.

Tree Descriptive Methods¶

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

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

Return 3 since every node has 3 children, making that both the minimum and maximum.

"$bool = $path->tree_any_leaf()"

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

UAD Matrices¶

Internally the code uses P,Q and calculates A,B at the end as necessary. The UAD transformations in P,Q coordinates are

    U     P -> 2P-Q            ( 2 -1 )
          Q -> P               ( 1  0 )
    A     P -> 2P+Q            ( 2  1 )
          Q -> P               ( 1  0 )
    D     P -> P+2Q            ( 1  2 )
          Q -> Q unchanged     ( 0  1 )

The advantage of P,Q for the calculation is that it's 2 values instead of 3. The transformations can be written with the 2x2 matrices shown, but explicit steps are enough for the code.

Repeatedly applying "U" gives

    U       2P-Q, P
    U^2     3P-2Q, 2P-Q
    U^3     4P-3Q, 3P-2Q
    ...
    U^k     (k+1)P-kQ, kP-(k-1)Q
          = P+k(P-Q),  Q+k*(P-Q)

If there's a run of k many high zeros in the Nrem = N-Nrow position in the level then they can be applied to the initial P=2,Q=1 as

    U^k    P=k+2, Q=k+1       start for k high zeros

FB Transformations¶

The FB tree is calculated in P,Q and converted to A,B at the end as necessary. Its three transformations are

    M1     P -> P+Q         ( 1  1 )
           Q -> 2Q          ( 0  2 )
    M2     P -> 2P          ( 2  0 )
           Q -> P-Q         ( 1 -1 )
    M3     P -> 2P          ( 2  0 )
           Q -> P+Q         ( 1  1 )

Price's paper shows rearrangements of a set of four values q',q,p,p'. Just the p and q are enough for the calculation. The set of four has some attractive geometric interpretations though.

X,Y to N -- UAD¶

xy_to_n() works in P,Q coordinates. An A,B or other input is converted to P,Q per the formulas in "PQ Coordinates" above. P,Q can be reversed up the UAD tree to its parent point

    if P > 3Q    reverse "D"   P -> P-2Q
                  digit=2      Q -> unchanged
    if P > 2Q    reverse "A"   P -> Q
                  digit=1      Q -> P-2Q
    otherwise    reverse "U"   P -> Q
                  digit=0      Q -> 2Q-P

This gives a ternary digit 2, 1, 0 respectively from low to high. Those plus a high "1" bit make N. The number of steps is the "depth" level.

If at any stage P,Q doesn't satisfy P>Q>=1, one odd, the other even, then it means the original point, however it was converted, was not a primitive triple. For a primitive triple the endpoint is always P=2,Q=1.

The conditions P>3Q or P>2Q work because each matrix sends its parent P,Q to one of three disjoint regions,

     Q                  P=Q                    P=2Q                P=3Q
     |                    *       U         ----     A        ++++++
     |                  *               ----            ++++++
     |                *             ----          ++++++
     |              *           ----        ++++++
     |            *         ----      ++++++
     |          *       ----    ++++++
     |        *     ----  ++++++                     D
     |      *   ----++++++
     |    * ----++++
     |  ----++
     |
     +------------------------------------------------- P

So U is the upper wedge, A the middle, and D the lower. The parent P,Q can be anywhere in P>Q>=1, the matrices always map to these regions.

X,Y to N -- FB¶

After converting to P,Q as necessary, a P,Q point can be reversed up the FB tree to its parent

    if P odd     reverse M1    P -> P-Q
    (Q even)                   Q -> Q/2
    if P > 2Q    reverse M2    P -> P/2
    (P even)                   Q -> P/2 - Q
    otherwise    reverse M3    P -> P/2
    (P even)                   Q -> Q - P/2

This is a little like the binary greatest common divisor algorithm, but designed for one value odd and the other even. Like the UAD ascent above, if at any stage P,Q doesn't satisfy P>Q>=1, one odd, the other even, then the initial point wasn't a primitive triple.

The M1 reversal works because M1 sends any parent P,Q to a child which has P odd. All odd P,Q come from M1. The M2 and M3 always make children with P even. Those children are divided between two disjoint regions above and below the line P=2Q.

     Q                  P=Q                     P=2Q
     |                    *     M3 P=even   ----
     |                  *               ----
     |                *             ----
     |              *           ----
     |            *         ----              M2 P=even
     |          *       ----
     |        *     ----
     |      *   ----
     |    * ----                 M1 P=odd anywhere
     |  ----
     |
     +------------------------------------------------- P

X,Y to N -- UMT¶

After converting to P,Q as necessary, a P,Q point can be reversed up the UMT tree to its parent

    if P > 2Q    reverse "U"     P -> Q
                  digit=0        Q -> 2Q-P
    if P even    reverse "M2"    P -> P/2
    (Q odd)                      Q -> P/2 - Q
    otherwise    reverse "T"     P -> P - 3 * Q/2
    (Q even)                     Q -> Q/2

These reversals work because U sends any parent P,Q to a child P>2Q whereas the M2 and T go below that line. M2 and T are distinguished by M2 giving P even whereas T gives P odd.

     Q                  P=Q                     P=2Q
     |                    *       U         ----
     |                  *               ----
     |                *             ----
     |              *           ----
     |            *         ----        M2 for P=even
     |          *       ----             T for P=odd
     |        *     ----
     |      *   ----
     |    * ----
     |  ----
     |
     +------------------------------------------------- P

Rectangle to N Range -- UAD¶

For the UAD tree, the smallest A,B within each level is found at the topmost "U" steps for the smallest A or the bottom-most "D" steps for the smallest B. For example in the table above of level=2 N=5..13 the smallest A is the top A=7,B=24, and the smallest B is in the bottom A=35,B=12. In general

    Amin = 2*level + 1
    Bmin = 4*level

In P,Q coordinates the same topmost line is the smallest P and bottom-most the smallest Q. The values are

    Pmin = level+1
    Qmin = 1

The fixed Q=1 arises from the way the "D" transformation sends Q->Q unchanged, so every level includes a Q=1. This means if you ask what range of N is needed to cover all Q < someQ then there isn't one, only a P < someP has an N to go up to.

Rectangle to N Range -- FB¶

For the FB tree, the smallest A,B within each level is found in the topmost two final positions. For example in the table above of level=2 N=5..13 the smallest A is in the top A=9,B=40, and the smallest B is in the next row A=35,B=12. In general,

    Amin = 2^level + 1
    Bmin = 2^level + 4

In P,Q coordinates a Q=1 is found in that second row which is the minimum B, and the smallest P is found by taking M1 steps half-way then a M2 step, then M1 steps for the balance. This is a slightly complicated

    Pmin = /  3*2^(k-1) + 1    if even level = 2*k
           \  2^(k+1) + 1      if odd level = 2*k+1
    Q = 1

The fixed Q=1 arises from the M1 steps giving

    P = 2 + 1+2+4+8+...+2^(level-2)
      = 2 + 2^(level-1) - 1
      = 2^(level-1) + 1
    Q = 2^(level-1)
    followed by M2 step
    Q -> P-Q
         = 1

As for the UAD above this means small Q's always remain no matter how big N gets, only a P range determines an N range.