There isn't a specific term for a botanist who specializes in trees. A tree botanist is usually just called a botanist who specializes in dendrology.

* Botany is the study of plants in general.

* Dendrology is the branch of botany that focuses specifically on trees.

So, while you could technically call someone a "tree botanist," it's more accurate and specific to say they are a dendrologist or a botanist specializing in dendrology.