  Each natural number can be associated with some tree graph. Namely, a natural
number $n$ 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 can
be factorized in such a manner as well. This process may be continued, building
the "factorization tree" until all the top numbers are $1$. Let $H(n)$ be the
height of the tree corresponding to the number $n$, and let the symbol
$\uparrow\uparrow$ denote tetration. In this paper, we derive the asymptotic
formulas 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
$$\mathcal{L}(x) = \sum_{n\leqslant x}\dfrac{2\uparrow\uparrow
H(n)}{2\uparrow\uparrow H(n+1)},$$ where the summation in the first sum is
taken over primes.
