How Does the Number Tree Grow?
As the numbers descend from the heavens and attach themselves to the growing tree, they look for a specific
number on a branch point. The largest divisor (factor) is what they choose. So, for instance, take the number 20. It has
the following divisors: (1, 2, 4, 5, and 10). (The number 20 itself is not considered because it can't branch from itself :)
So, it chooses 10 as its branch point - as you can see in the tree. Another example: the number 21 has (1, 3 and 7)
as its divisors, and so it chooses 7. The number 22 chooses 11, and the number 23 (a lonely prime) has only
one choice: 1.
Hey Cool - Did You Notice?
Did you notice that the tree is bushier on the left side? Also, did you notice that on the right side you can easiy see a line
of prime numbers? In fact, the tree seems to be like a brain - it has a "left brain" and a "right brain". While the
right side has a concentration of primes, the left side is rich with highly-composite numbers - numbers that have
a lot of divisors (and that's why it's bushy). How do you think this bushiness will change as the tree grows more numbers?
The answer to this question has to do with the nature of prime and composite numbers.
Prime Number Ordering of Branches
Notice that the branch at the lower-left has no sub-branches. It consists of a single series of numbers
which are powers of 2: 1x2=2; 2x2=4; 4x2=8, and so-on. This branch will remain a single thread forever, as
more numbers are added to the tree, and that's due to the nature of the rules of growth. Notice also that the
next branch above that starts with the number 3. This is called the "Order Three" branch. All numbers that
descend from it are multiples of 3. The next one is the "Order Five" branch. All numbers that branch from it
are multiples of 5. And so-on through all of the prime numbers.
When It Becomes a Mature Oak
If the number tree were to grow larger, you
would notice that the prime number branches to the right will start to grow more sub-branches. The
"bushiness" will start to make its way rightward. The tree doesn't actually show prime factorization, which is the basis for many other tree graphs that you
may have seen. This is a unique, and different kind of tree structure. It is a "family tree" of sorts, and if
you could watch it grow to the size of a magnificent oak, you would see that many new, and maybe
surprising, patterns will emerge.