I think this a compound-complex sentence - but I'm struggling to draw a tree diagram of the co-ordinating/sub-ordinating clause(s)

Solution 1:

Here is an analysis in which I find a problem, long known to me but for which I am still left to my own devices, and therfore, for that little part I mention a possible way out that I conceived. It is to be taken as food for thought until a more satisfactory resolution is found or proposed.

• And I saw a man with a newspaper and a bag of golf-clubs

  • go up to one of the doors of the train
  • |and| press a big button next to it

• |and| the doors were electronic

• |and| they slipped open

• |and| I liked that.

• 'And I saw a man […] with a newspaper and a bag of golf-clubs' = Independent, Main Clause

•'with a newspaper' = Subordinate Clause prepositional phrase, complement of noun "man"

•'and a bag of golf clubs' = This could stand independently, but its connected with a co-coordinating conjunction to the last sub-ordinate clause - so aren't they both co-ordinated but as subordinated clauses, to the first main clause? prepositional phrase, noun complement (man), coordinated to the preceding one, ellipsis of the preposition (with) which could be repeated but which usually isn't

•'a man go up to one of the doors' - Further confusion. If this sentence began with 'he' and was in the full past-tense like the majority of the novel is, then 'He went up to one of the doors' would be an independent. But because it just says 'go' I'm assuming it is a subordinated clause. bare infinitive clause, object of "saw", "man" (actually, exactly "man with a newspaper and a bag of golf-clubs") is the "partial object" of "saw" as I personally consider this matter and as well the subject of "go up" (as generally reckognized); therefore it is a word that has two functions according to the verb to which it is related. There is a sort of incongruity in this grammatical context: the independent clause has to borrow a word from the infinitive and vice versa or, in other words, there has to be a word that belongs to both clauses.

•'of the train' - subordinate clause. No, again merely a prepositional phrase with for function that of complement of the noun "doors"

•'press a big button next to it' = again, like with 'go', the use of the simple present tense 'press', means its subordinate? It could be independent if it were in the past tense, like: 'and he pressed a big button next to it', but even then it is unclear what 'it' is in anaphoric reference to - so still subordinate clause. No, much simpler: merely another bare infinitive clause coordinated to the preceding by "and" with the same "suject" (man) which figures elliptically and can't be repeated as otherwise one more man is introduced in the context

•'the doors were electronic' - main/independent clause (?) - again, this throws me - how would I draw this whole sentence in a tree diagram? It would be easier if this very long sentence were split into two and it feels like this could be the start of a whole new sentence, if arranged by parataxis: 'the doors were electronic. they slipped open. I liked that'.
This is a typical run-on sentence and something such as what you explain should be done. Nevertheless, run on sentences are not necessarily ungrammatical and that one is grammatically correct; therefore we have an independent clause coordinated to "I saw a man".

•'they slipped open' = they, referring anaphorically to the doors, so therefore sub-ordinated? Or could it stand alone - despite it being clear what the pronoun 'they' is referring to? No, another coordinated independent clause; pronouns (they) are replaceable; the subordination is not founded on semantics but solely on relational words; "They slipped open." is a correct sentence, therefore, as it is introduced by "and" on the "level" of the independents, it must be an independent.

•'I liked that' - A subordinate clause, with 'that' acting as a relative pronoun referring to the previous phrase 'the doors were electronic' and 'they slipped open'.
As above, again an independent (be careful, "that" is a demonstrative)

Solution 2:

I don't have access to that book, so I don't know what that author counts as a subordinate clause.  I do know that a clause in traditional English grammar is the joining of a subject and predicate, and that such a predicate must include a finite verb form.  Given that, the simple subject / finite verb pairings in your model sentence are these:

• I | saw
• doors | were
• they | slipped
• I | liked

There are two other verbs in the sentence.  These are non-finite forms which do not join with a subject.  Those verbs are go and press.  These are bare infinitives.  According to traditional English grammar, they form phrases rather than clauses.  In your model sentence, they form a compound subject complement.  The full phrase "^[to]go up to one of the doors of the train and ^[to]press a big button next to it" is the thing that the man was seen to do.

There are no subordinating conjunctions and no relative pronouns or adverbs present in the sentence.  In other words, there is nothing to make any of the clauses listed above into a subordinate clause.  Under this analysis, you are not looking at a complex sentence.  It is a compound sentence with four independent clauses.

More modern grammars consider even non-finite verbs to form clauses.  In that case, "go up to one of the doors of the train and press a big button next to it" is (I would assume) a compound predicative subordinate clause.  For all I know, this difference in labeling might make no difference in the tree that you're trying to draw.  Regardless of label, it is one of the two arguments of the verb saw. 

• I | saw / { him } \ { [ do / this ] and [ do / that ] }.

This { him } stands for the direct object "a man with a newspaper and a bag of golf-clubs".  This { [ do / this ] and [ do / that ] } stands for the object complement "go up to one of the doors of the train and press a big button next to it".

There are several prepositional phrases within this clause.  Some phrases are inside other phrases.  The phrase "with a newspaper and a bag of golf-clubs" modifies "a man".  The phrase "of golf clubs" modifies "a bag".  The phrase "to one of the doors of the train" modifies "go".  The phrase "of the doors of the train" modifies "one".  The phrase "of the train" modifies "doors".  The phrase "next to it" modifies "a big button".

The three remaining independent clauses are much shorter:

• the doors | were \ electronic
• they | slipped open
• I | liked / that

Finally, the "that" of "I liked that" is not a relative pronoun.  Here, it's just a demonstrative pronoun, serving as the direct object.  A relative "that" (even when a direct object) would appear at the beginning of the subordinate clause, as it does in the complex sentence "It was something that I liked."