Enumeration of Matchings in Complete $q$-ary Trees

We study the asymptotic behavior of the parameters $m(T_{q,n})$ and $im(T_{q,n})$, that equal the number of matchings and independent matchings in a complete $q$-ary tree $T_{q,n}$ of height $n$. We show that, for any $q\ge 2$, there exists a $b_q>1$ such that, as $n\to+\infty$, the following asymptotic equality holds:
$$ m(T_{q,n})\sim\biggl(\frac{1+\sqrt{1+4\cdot q}}2\,\biggr)^{-1/(q-1)}\cdot(b_q)^{q^n}. $$
We also show that, for any $q\in \{1,2,3\}$, there exist numbers $a'_q$ and $b'_q>1$ such that $im(T_{q,n})\sim a'_q\cdot (b'_q)^{q^{n}}$ as $n\to+\infty$, and also, for any sufficiently large $q$, there exist numbers $a^{1}_q\ne a^{2}_q$ and $b'_q>1$ such that, as $n\to+\infty$, the following asymptotic equalities hold:
\begin{gather*} im(T_{q,3n})\sim a^{1}_q\cdot (b'_q)^{q^{3n}}, \\ im(T_{q,3n+1})\sim a^{2}_q\cdot (b'_q)^{q^{3n+1}},\qquad im(T_{q,3n+2})\sim a^{1}_q\cdot (b'_q)^{q^{3n+2}}. \end{gather*}