Consider the set of rooted labeled trees in which each node may have up to $d$ children. It is well known that the number of such trees with $\leq v$ vertices is (approximately) $(ed)^v$ and that the number of such trees of height $\leq h$ is (approximately) $2^{d^h}$.
I would like to estimate, for a given $v,h$ the number of such trees which have both properties. I am interested in the case in which $v$ is roughly comparable to $h$, say $v \approx \text{poly}(h)$. (Note that it is possible for $v$ to become as large as $d^h$.) I am also interested primarily but not exclusively in the case in which $d$ is small, say even constant.
Are there any good estimates for the number of trees satisfying these criteria?