Counting k-ary unlabeled unordered trees with n nodes

Question

I wish to know the asymptotic estimate of counting the trees satisfying the following conditions:

1. a total of n vertices;
2. each vertex has at most k outdegree;
3. all vertices are neither labeled nor ordered, i.e., only the structure of the tree matters.

If there is no closed form for it, I wish I could know the primary part of the reason.

I know there are some similar questions about counting trees; however, this one does not duplicate them:

• This frequently asked question counts k-ary labeled tree; however, my question is about unlabeled tree.

• This question gave a detailed calculation. However, it did not assert whether the counting has a closed form, which prompted me to take this post.

I appreciate any advice and suggestions.