In Matula's bijection, every natural number is a rooted tree. Multiplication merges children. GCD finds shared structure. Addition? Addition creates chaos.
In the Matula world, multiplication is a structural operation. To multiply two numbers, you combine the children of their trees into one root. The structure of each operand is preserved — you can still find tree(a) and tree(b) inside tree(a × b) by looking at its branches.
GCD and LCM are similarly structural: GCD takes the intersection of prime factors (keep only shared branches, minimum multiplicity), while LCM takes the union (keep all branches, maximum multiplicity). Both produce trees whose shape is predictable from the inputs.
Addition breaks this completely. a + b creates a number whose prime factorization bears no systematic relationship to the factorizations of a and b. The tree for 6 + 10 = 16 = 2⁴ looks nothing like either tree(6) or tree(10). You can't find the addends in the sum.
This is the deep asymmetry of number theory made visible: the multiplicative structure of the integers is tree-shaped, clean, decomposable. The additive structure is something else entirely. Mathematicians have spent centuries trying to understand how these two structures interact — it's the heart of problems like Goldbach's conjecture, the twin prime conjecture, and the distribution of primes. The trees make this tension visible.