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Each natural number can be associated with some tree graph. Namely, a naturalnumber $n$ can be factorized as $$ n = p_1^{\alpha_1}\ldots p_k^{\alpha_k},$$where $p_i$ are distinct prime numbers. Since $\alpha_i$ are naturals, they canbe factorized in such a manner as well. This process may be continued, buildingthe "factorization tree" until all the top numbers are $1$. Let $H(n)$ be theheight of the tree corresponding to the number $n$, and let the symbol$\uparrow\uparrow$ denote tetration. In this paper, we derive the asymptoticformulas for the sums $$\mathcal{M}(x) = \sum_{p\leqslant x} H(p-1),\ \\mathcal{H}(x) = \sum_{n\leqslant x}2\uparrow\uparrow H(n),$$ and$$\mathcal{L}(x) = \sum_{n\leqslant x}\dfrac{2\uparrow\uparrowH(n)}{2\uparrow\uparrow H(n+1)},$$ where the summation in the first sum istaken over primes.

On The Tree Structure of Natural Numbers, II