The most essential part of any tree is the trunk. This is the the light brown section to the left of the table.
It is made up of the root of the tree (the eternal loop of 1, 2 and 4) plus the continuation of the list of powers
of 2 which is effectively unbounded. Being powers of 2, they will be either 1 mod 3 or 2 mod 3. Those
that are 1 mod 3 are distinguished by the attachment of a second number which is (power of 2 minus 1) divided
by 3. This number can be looked upon as a bud which will develop into a branch consisting of a series of numbers
which are the value of the bud multiplied by consecutive powers of 2. Those that are 2 mod 3 cannot give
rise to a branch and are therefore labeled Sterile.
Three variants in the structure of a branch can be identified:-
If the value of the bud is 2 mod 3 (such as 5, 341 and 21845) then the values on the branch will alternate
between 1 mod 3 and 2 mod 3 with the first candidate number being 1 mod 3. These branches
are colored light blue in the diagram. All of the 1 mod 3 candidates are buds which will spawn another new
generation of branches.
If the value of the bud is 1 mod 3 (such as 85 and 5461) then the values on the branch will alternate
between 1 mod 3 and 2 mod 3 with the first candidate number being 2 mod 3. These branches
are colored lime green in the diagram. Once again, all of the 1 mod 3 candidates are buds which will spawn
another new generation of branches.
If the value of the bud is 0 mod 3 (such as 21 and 1365) then the values on the branch will also be
0 mod 3 and cannot promote a new branch. These branches are colored pink in the diagram. It would be
appropriate to classify them as sterile branches.
The trunk is classified as a level 1 branch. Every other branch in the tree will have a level number which is
one more than the level number of the branch from which it arose.
There is a very intimate connection between the structure of the Hailstone Tree and the Hailstone Signature
introduced at Introduction to the Hailstone Problem. This is illustrated by
the number 4970949 whose Hailstone Signature and Hailstone series are tabulated as follows:-
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E
E
E
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4970949
14912848
7456424
3728212
1864106 |
Third level Branch.
The first Chunk of the Signature of 4970949 is shown here. It originates in a level 3
branch which is external to that part of the Hailstone Tree shown previously and connects with a level 2
branch at the bottom right corner where it meets the odd number 932053. Any Chunk will always be
contained entirely within a single branch, and will end at the point where that branch connects with the next
(lower level) branch.
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E
E
E
E
E
E
E |
932053
2796160
1398080
699040
349520
174760
87380
43690 |
Second level Branch.
Like all Signature Chunks, the second Chunk begins with an odd number which must be
multiplied by 3, and have 1 added to it giving 2796160. A series of divisions by 2 eventually leads us to
another odd number 21845 which is the beginning of the next Chunk, located in the only level 1 branch
(trunk) of the tree. |
O
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
|
21845
65536
32768
16384
8192
4096
2048
1024
512
256
128
64
32
6
8
4
2 |
First level Branch.
Here we have the third and final Signature Chunk of our original starting number. Its
initial (odd) number is 21845, which becomes 65536 after multiplication by 3 and the addition of 1. Now
this number happens to be an integral power of 2, and so again we see another series of divisions by 2 which
brings us inevitably to the terminating point of 1.
These discussions have stressed the fact that every Signature Chunk begins with an odd number. This is
important. It is even more important to remember that EVERY odd number heralds the beginning of another
branch, and that every branch is infinite in extent, having an infinite number of even numbers, one half of
which will equal 1 mod 3 and so will spawn yet more new branches.
This Signature is rather short with only 3 Chunks. If you refer to the information at Hailstone Tree Outliers you will see an example of a Signature having
48,126 Chunks. There is no upper limit to the number of Chunks a Signature can have.
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| O |
1 |
Zero level Branch.
This is the absolute base of the tree and could quite reasonably be referred to as the root. |
The Hailstone Tree is a vast and complex structure, and the path taken by a Signature on its journey through the tree is
virtually impossible to visualize. What follows is an attempt to depict the path taken by 79 as it makes this journey.
It is a very cut down version of the tree, and shows only those branches which were actually used.
^^^^^ ^^^^ ^^^
19456 8608 952
9728 4304 476
4864 2152 238 79 158 316 672 1344 >
2432 1076 119
^^^^ 1216 538 179 358 716 1432 2864 5728 11456 >
5632 608 269
2816 304 101 202 404 808 1616 3232 6464 12928 25856 51712 >
1408 152
^^^^ 704 76
3328 352 38
1664 176 19
832 88 29 58 116 232 464 928 1856 3712 >
^^^ 416 44
512 208 22
256 104 11
128 52 17 34 68 136 272 544 1088 2176 >
64 26
32 13
16 5 10 20 40 80 160 320 640 1280 2560 >
8
4
2
1
The numbers which constitute the path are colored red and blue. The color changes at the beginning of each new
Signature Chunk. Note that the first number of each Chunk is an odd number which is the bud of a branch.
The black numbers provide extensions to each branch. All of these numbers are powers of 2 multiplied by the value
of the bud of the branch.
The UP and RIGHT arrows remind you that all of these branches extend onward forever.
Remember that when an odd number is encountered, the next number (multiply by 3 and add 1) will be an even number
and will be located on a branch which is one level lower than the current branch.
You will note that the path uses only a small portion of each branch located right at the beginning of the
branch. Naturally this won't always be the case, but Signatures which use large chunks of a single branch will be a
rarity.
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