- The fundamental component 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 extends to infinity. 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.

- If the value of the bud is
- 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.
- Every number in the tree is guaranteed to converge onto 1 when it undergoes the Collitz process, but does every natural number actually appear on the tree. I invite you to consider the conundrum as to just what characteristics would a number need to have for it not to be included on this tree.
- There is a very intimate connection between the structure of the Collatz Tree and the Collatz Signature
discussed at Introducing Signatures and Syllables.
This is illustrated by the number 4970949 whose Collatz Signature and Collatz series are tabulated as follows:-