English Grammar Sentence Structure

I (subject) am (predicate) happy (predicate) to see you here (predicative complement). The verb be is followed by predicates. am happy forms a table structure.

I guess your confusion lies in the fact that predicates also have complements? Yes, complements are supplementary explanations. to see you here for additional explanation happy. In fact, in addition to this, there are also object complements and subject complements.

To do as an adverbial has only two functions: 1. To express purpose and future; 2. To express unexpected results. (For example: I hurried to the train station, only to find the train had gone.)

This sentence is obviously not an adverbial of purpose, because "I am happy to see you here" does not make sense. So it is not an adverbial of purpose. And the adverbial about unexpected results doesn't make sense, so to do is not used as an adverbial.