Math::...hanTree(3pm)
✨ NAME ✨
okay so bestie 💖 this is the manual page for Math::PlanePath::ChanTree(3pm), from User Contributed Perl Documentation, and it is literally the tree of rationals.
Math::PlanePath::ChanTree— tree of rationals
🛠 SYNOPSIS 🛠
no cap, here's how you summon it:
use Math::PlanePath::ChanTree;
my $path = Math::PlanePath::ChanTree->new (k => 3, reduced => 0);
my ($x, $y) = $path->n_to_xy (123);🌈 DESCRIPTION 🌈
This path enumerates rationals X/Y in a tree as per
Song Heng Chan, "Analogs of the Stern Sequence", Integers 2011,
<http://www.integers-ejcnt.org/l26/l26.pdf>The default k=3 visits X,Y with one odd, one even, and perhaps a common factor 3^m.
14 | 728 20 12
13 | 53 11 77 27
12 | 242 14 18
11 |
10 | 80
9 | 17 23 9 15
8 | 26 78
7 |
6 | 8 24 28
5 | 5 3 19
4 | 2 6 10 22
3 |
2 | 0 4 16 52
1 | 1 7 25 79 241 727
Y=0 |
+--------------------------------------------------------
X=0 1 2 3 4 5 6 7 8 9 10 11 12 13There are 2 tree roots (so technically it's a "forest") and each node has 3 children. The points are numbered by rows starting from N=0. This numbering corresponds to powers in a polynomial product generating function.
N=0 to 1 1/2 2/1
/ | \ / | \
N=2 to 7 1/4 4/5 5/2 2/5 5/4 4/1
/ | \ ... ... ... ... / | \
N=8 to 25 1/6 6/9 9/4 ... ... 5/9 9/6 6/1
N=26 ...The children of each node are
X/Y
------------/ | \-----------
| | |
X/(2X+Y) (2X+Y)/(X+2Y) (X+2Y)/YWhich as X,Y coordinates means vertical, 45-degree diagonal, and horizontal.
X,Y+2X X+(X+Y),Y+(X+Y)
| /
| /
| /
| /
X,Y------- X+2Y,YThe slowest growth is on the far left of the tree 1/2, 1/4, 1/6, 1/8, etc advancing by just 2 at each level. Similarly on the far right 2/1, 4/1, 6/1, etc. This means that to cover such an X or Y requires a power-of-3, N=3^(max(X,Y)/2).
💖 GCD 💖
Chan shows that these top nodes and children visit all rationals X/Y with X,Y one odd, one even. But the X,Y are not in least terms, they may have a power-of-3 common factor GCD(X,Y)=3^m for some m.
The GCD is unchanged in the first and third children. The middle child GCD might gain an extra factor 3. This means the power is at most the number of middle legs taken, which is the count of ternary 1-digits of its position across the row.
GCD(X,Y) = 3^m
m <= count ternary 1-digits of N+1, excluding high digitAs per "N Start" below, N+1 in ternary has high digit 1 or 2 which indicates the tree root. Ignoring that high digit gives an offset into the row of that tree and the digits are 0,1,2 for left,middle,right.
For example the first GCD is at N=9 with X=6,Y=9 common factor GCD=3. N+1=10="101" ternary, which without the high digit is "01" which has a single "1" so GCD <= 3^1. The mirror image of this point is X=9,Y=6 at N=24 and there N+1=24+1=25="221" ternary which without the high digit is "21" with a single 1-digit likewise.
For various points the power m is equal to the count of 1-digits.
🎀 k Parameter 🎀
Parameter "k => $integer" controls the number of children and top nodes. There are k-1 top nodes and each node has k children. The top nodes are
k odd, k-1 many tops, with h=ceil(k/2)
1/2 2/3 3/4 ... (h-1)/h h/(h-1) ... 4/3 3/2 2/1
k even, k-1 many tops, with h=k/2
1/2 2/3 3/4 ... (h-1)/h h/h h/(h-1) ... 4/3 3/2 2/1Notice the list for k odd or k even is the same except that for k even there's an extra middle term h/h. The first few tops are as follows. The list in each row is spread to show how successive bigger h adds terms in the middle.
k X/Y top nodes
--- ---------------------------------
k=2 1/1
k=3 1/2 2/1
k=4 1/2 2/2 2/1
k=5 1/2 2/3 3/2 2/1
k=6 1/2 2/3 3/3 3/2 2/1
k=7 1/2 2/3 3/4 4/3 3/2 2/1
k=8 1/2 2/3 3/4 4/4 4/3 3/2 2/1As X,Y coordinates these tops are a run up along X=Y-1 and back down along X=Y+1, with a middle X=Y point if k even. For example,
7 | 5 k=13 top nodes N=0 to N=11
6 | 4 6 total 12 top nodes
5 | 3 7
4 | 2 8
3 | 1 9
2 | 0 10
1 | 11
Y=0 |
+------------------------------
X=0 1 2 3 4 5 6 7
k=14 top nodes N=0 to N=12
7 | 5 6 total 13 top nodes
6 | 4 7
5 | 3 8 N=6 is the 7/7 middle term
4 | 2 9
3 | 1 10
2 | 0 11
1 | 12
Y=0 |
+------------------------------
X=0 1 2 3 4 5 6 7Each node has k children. The formulas for the children can be seen from sample cases k=5 and k=6. A node X/Y descends to
k=5 k=6
1X+0Y / 2X+1Y 1X+0Y / 2X+1Y
2X+1Y / 3X+2Y 2X+1Y / 3X+2Y
3X+2Y / 2X+3Y 3X+2Y / 3X+3Y
2X+3Y / 1X+2Y 3X+3Y / 2X+3Y
1X+2Y / 0X+1Y 2X+3Y / 1X+2Y
1X+2Y / 0X+1YThe coefficients of X and Y run up to h=ceil(k/2) starting from either 0, 1 or 2 and ending 2, 1 or 0. When k is even there's two h coeffs in the middle. When k is odd there's just one. The resulting tree for example with k=4 is
k=4
1/2 2/2 2/1
/ \ / \ / \
1/4 4/6 6/5 5/2 2/6 6/8 8/6 6/2 2/5 5/6 6/4 4/1Chan shows that this combination of top nodes and children visits
if k odd: rationals X/Y with X,Y one odd, one even
possible GCD(X,Y)=k^m for some integer m
if k even: all rationals X/Y
possible GCD(X,Y) a divisor of (k/2)^mWhen k odd, GCD(X,Y) is a power of k, so for example as described above k=3 gives GCD=3^m. When k even GCD(X,Y) is a divisor of (k/2)^m but not necessarily a full such power. For example with k=12 the first such non-power GCD is at N=17 where X=16,Y=18 has GCD(16,18)=2 which is only a divisor of k/2=6, not a power of 6.
🦋 N Start 🦋
The "n_start => $n" option can select a different initial N. The tree structure is unchanged, just the numbering shifted. As noted above the default Nstart=0 corresponds to powers in a generating function.
"n_start=>1" makes the numbering correspond to digits of N written in base-k. For example k=10 corresponds to N written in decimal,
N=1 to 9 1/2 ... ... 2/1
N=10 to 99 1/4 4/7 ... ... 7/4 4/1
N=100 to 999 1/6 6/11 ... ... 11/6 6/1In general "n_start=>1" makes the tree
N written in base-k digits
depth = numdigits(N)-1
NdepthStart = k^depth
= 100..000 base-k, high 1 in high digit position of N
N-NdepthStart = position across whole row of all top treesAnd the high digit of N selects which top-level tree the given N is under, so
N written in base-k digits
top tree = high digit of N
(1 to k, selecting the k-1 many top nodes)
Nrem = digits of N after the highest
= position across row within the high-digit tree
depth = numdigits(Nrem) # top node depth=0
= numdigits(N)-1🌸 Diatomic Sequence 🌸
Chan shows that each denominator Y becomes the numerator X in the next point. The last Y of a row becomes the first X of the next row. This is a generalization of Stern's diatomic sequence and of the Calkin-Wilf tree of rationals. (See Math::NumSeq::SternDiatomic and "Calkin-Wilf Tree" in Math::PlanePath::RationalsTree.)
The case k=2 is precisely the Calkin-Wilf tree. There's just one top node 1/1, being the even k "middle" form h/h with h=k/2=1 as described above. Then there's two children of each node (the "middle" pair of the even k case),
k=2, Calkin-Wilf tree
X/Y
/ \
(1X+0Y)/(1X+1Y) (1X+1Y)/(0X+1Y)
= X/(X+Y) = (X+Y)/Y✨ FUNCTIONS ✨
Okay bestie, heads up: the general behaviour that all the path classes share lives over in "FUNCTIONS" in Math::PlanePath, so go peek there if you want the universal stuff. These ones here are ChanTree's own. 💖
$path = Math::PlanePath::ChanTree->new ()$path = Math::PlanePath::ChanTree->new (k => $k, n_start => $n)
Makes you a shiny new path object and hands it over. The defaults are k=3 and n_start=0. Slay, that's it. 🎀
$n = $path->n_start()
Gives you back the first N in the path. It's 0 by default, or whatever you passed as the n_start parameter. Simple as.
$n = $path->xy_to_n ($x,$y)
Returns the point number for the coordinates $x,$y. And if there's literally nothing at $x,$y? It returns undef. 💀
🌸 Tree Methods 🌸
Every point has k children, so the path is a complete k-ary tree. It's giving very full, very balanced, very demure. 👑
@n_children = $path->tree_n_children($n)
Returns the children of $n, or an empty list if $n < n_start(), aka before the start of the path. No kids yet, she's not ready. 🙈
$num = $path->tree_n_num_children($n)
Returns k, because every node has k children, no exceptions, periodt. Or returns undef if $n < n_start(), again meaning before the start of the path.
$n_parent = $path->tree_n_parent($n)
Returns the parent node of $n, or undef if $n has no parent — either because it's a top node or because it's before n_start().
$n_root = $path->tree_n_root ($n)
Returns the N that is the root node of $n. Finding your roots, iconic. 🌱
$depth = $path->tree_n_to_depth($n)
Returns the depth of node $n, or undef if there's no point $n. The tree tops are depth=0, then their children are depth=1, etc. Down and down we go. 🦋
$n = $path->tree_depth_to_n($depth)$n = $path->tree_depth_to_n_end($depth)
Returns the first or last N at tree level $depth in the path. The top of the tree is depth=0. First and last, the whole row, we love a complete set. 💅
💜 Tree Descriptive Methods 💜
$num = $path->tree_num_roots ()
Returns the number of root nodes in $path, which is k-1. For example the default k=3 returns 2, because there are two root nodes. Two roots, two icons. 👑
@n_list = $path->tree_root_n_list ()
Returns a list of the N values which are the root nodes of $path. This is n_start() through n_start()+k-2 inclusive — so the first k-1 many points. For example in the default k=2 and Nstart=0 the return is two values (0,1). Cute little pair fr fr.
$num = $path->tree_num_children_minimum()$num = $path->tree_num_children_maximum()
Returns k, since every node has k many children — which makes that both the minimum and the maximum. Consistent queen behaviour. ✨
$bool = $path->tree_any_leaf()
Returns false, since there are no leaf nodes in the tree. Not a single leaf. She never ends, she just keeps going. 😭
🔥 FORMULAS 🔥
🧮 N Children
For the default k=3 the children are
3N+2, 3N+3, 3N+4 n_start=0If n_start=>1 then instead it's
3N, 3N+1, 3N+2 n_start=1For this n_start=1 the children are found by appending an extra ternary digit — or base-k digit for arbitrary k, iykyk.
k*N, k*N+1, ... , k*N+(k-1) n_start=1In general, for k and Nstart, the children are
kN - (k-1)*(Nstart-1) + 0
...
kN - (k-1)*(Nstart-1) + k-1🧮 N Parent
The parent node reverses the children calculation above. The simplest case is n_start=1 where it's a division to remove the lowest base-k digit:
parent = floor(N/k) when n_start=1For other n_start, adjust before and after to an n_start=1 basis:
parent = floor((N-(Nstart-1)) / k) + Nstart-1For example in the default k=0 Nstart=1 the parent of N=3 is floor((3-(1-1))/3)=1. Math ate, honestly. 🍓
The post-adjustment can be worked into the formula with (k-1)*(Nstart-1), similar to the children above:
parent = floor((N + (k-1)*(Nstart-1)) / k)But the first style is more convenient to compare, so you can see that N is past the top nodes and therefore has a parent:
N-(Nstart-1) >= k to check N is past top-nodes🧮 N Root
As described under "N Start" above, if Nstart=1 then the tree root is simply the most significant base-k digit of N. For other Nstart an adjustment is made to N=1 style and back again.
adjust = Nstart-1
Nroot(N) = high_base_k_digit(N-adjust) + adjust🧮 N to Depth
The structure of the tree means
depth = floor(logk(N+1)) for n_start=0For example if k=3 then all of N=8 through N=25 inclusive have depth=floor(log3(N+1))=2. With an n_start it becomes
depth = floor(logk(N-(Nstart-1)))And n_start=1 is the simplest case of all — it's just the length of N written in base-k digits:
depth = floor(logk(N)) for n_start=1🌈 OEIS 🌈
This tree is in Sloane's Online Encyclopedia of Integer Sequences, as
<http://oeis.org/A191379> (etc)
k=3, n_start=0 (the defaults)
A191379 X coordinate, and Y=X(N+n)And as noted above, k=2 is the Calkin-Wilf tree. So go see "OEIS" in Math::PlanePath::RationalsTree for the CW related sequences. 🦋
👀 SEE ALSO 👀
Math::PlanePathMath::PlanePath::RationalsTreeMath::PlanePath::PythagoreanTree
🏠 HOME PAGE 🏠
<http://user42.tuxfamily.org/math-planepath/index.html>
📜 LICENSE 📜
Copyright 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 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/>.
Bestie that's a wrap on the whole page 💖 this one'sperl v5.32.0, dated2021-01-23, from the Math::...hanTree(3pm) manual — go forth and tree, you absolute legend. 👑✨
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