Background
Recently, I asked a question about the average growth rate of the solutions to the following sequence of Egyptian fraction equation:
$$ \sum_{k=1}^{n} \frac{1}{x_{k}} = q. \tag{1} $$
Here, we have $x_i \in \mathbb{N}_{>0} $ for all $ 1 \leq i \leq n $, and $ \ 0 <x_{1}<x_{2}<\cdots<x_{n} $.
So far, the question hasn't received any answers. What is known, however, are bounds for the amount of solutions to $(1)$. If we define $$\mathsf{E}_{n}(q)=\left\{ \{x_1 , x_2 , \dots , x_n \} \in \mathbb{N}^n_{>1}: \sum_{k=1}^{n} \frac{1}{x_{k}} = q, \ 0 <x_{1}<x_{2}<\cdots<x_{n} \ \right\}, $$ then according to p.8 of this article we have $$ e^{c \frac{n^{3}}{\log(n)}} < | \mathsf{E}_{n}(1) | < c_{0}^{(1+\epsilon)2^{n-1} }, \tag{2} $$ where $c_{0} \approx 1.264$.
Question
Let's leave aside the specific case of Egyptian fractions (as described above) for a moment, and consider a more general case. Let $a(n)$ be a set consisting of sequences of solutions to a sequence of diophantine equations $d(n)$, with $|a(n)| = o(f(n))$ and $|a(n)| = O (g(n))$.
Can anything be said, in this general case, about the growth rate of the average size of the solutions to $d(n)$, and how this growth rate relates to $f(\cdot)$ and $g(\cdot)$? Are there any results on this question, or any conjectures?