D.W. MATULA,
A one-to-one correspondence between the space of rooted trees and the positive integers is given by the following recursive procedure. Let the positive integer n correspond to the root node of a tree, the prime factors of n (each occurring as often as its multiplicity) correspond to the edges incident to the root node, and for each such edge incident to the root corresponds to the prime p let the branch of the tree incident to the edge be the rooted tree corresponding to the number pi(p) (the prime number function pi). The terminal condition of this recursion arises quite naturally as unity (which has no prime factors) is thus associated with each pendent vertex. On the other hand, the number associated with a given rooted tree may be recursively determined as follows: If the numbers associated with branches of a rooted tree T are n1, n2, … , nm, then the unique number associated with T is the product p(n1) ´ p(n2) ´ … ´ p(nm), where p(ni) is the ni th prime. Certain interesting relationships between the theories of primes and trees will be developed utilizing this "natural" correspondence.