Tree asymptotic densities in number theory

We study the asymptotic distribution of integers sharing the same rooted-tree structure that encodes their complete prime factorization tower. For each tree we derive an explicit density formula depending only on a pair $(m,k)$, the density signature of the tree, up to a suitable multiplicative scalar factor and introduce the corresponding tree zeta function, which generalizes the prime zeta function. Classical results such as the prime number theorem and later work by Landau appear as special cases.