Two results about Matula numbers

In Matula (1968), D.W. Matula described a bijection between N and the set of rooted trees; the number is called the Matula number of the rooted tree. The Gutman–Ivić–Matula (GIM) function g ( n ) computes the number of edges of the tree with Matula number n . Since there is a prefix-free code for the set of prime numbers such that the codelength of each prime p is 2 g ( p ) , we show how some results about the GIM function can be obtained trivially from coding theorems.