Hello!
It seems to me that people prefer tree generation to enumeration, so I will stick to generating homeomorphically irreducible trees with graphical presentations. As I have not generated the graphical presentations for h14 and h15, I will be doing the case h14 now and the case h15 in the next post. This is so important that I have to delay the case h16 for later posts.
Here are the trees for the case h14:
I will see you in the next post!