Arithmetic for Rooted Trees

We propose a new arithmetic for rooted unordered trees of n vertices and a method for their enumeration. We define three arithmetic operations on trees called addition, addition-plus, and multiplication and show how all trees can be generated by addition and addition-plus from a starting tree proving that both operations are needed. We show how a given tree can be obtained as the sum, sum-plus, or product of two trees, thus defining prime trees with respect to the three operations, and prove that primality can be decided in time polynomial in n. We suggest how these concepts can be useful and discuss the two lines of similar studies appeared in the literature