It seems, from the top two answers here by Brett and Oerkelens, that the functions and parts of speech of this sentence are two completely different types of thing . So, we can have nouns, for example, functioning as subjects, objects or temporal adjuncts (read "adverbials"). It should be possible therefore to do an analysis of the sentence solely in terms of its functions, or solely in terms of the parts of speech. Here, then, is my own attempt. I have capitalized the names of functions, in order to distinguish them from parts of speech. [Sorry about the lack of binary branching in the trees, I couldn't get the software to play ball!]
Functions

The Sentence:
- Bob [Subject] made a book collector happy the other day [Predicate]
The verb phrase: made a book collector happy the other day
- made [predicator], a book collector [object], happy [predicative
complement], the other day [(temporal) adjunct]
The noun phrase: a book collector:
- a [determinative], book [complement], collector [head]
The noun phrase: the other day:
- the [determinative], other [modifier], day [head]
Evidence: We can show that Bob is the Subject, because if we make an emphatic version of the sentence, the auxiliary did will appear directly after the Subject. Secondly if we then make a yes/no question the Subject and auxiliary will change places:
- Bob did make a book collector happy the other day
- Did Bob make a book collector happy ...
By default the Predicate is everything else in the sentence. Within the Predicate, we know that made must be the predicator, because in English this function is reserved only for verbs. We can show that a book collector is an object because if we passivise the sentence it will become the Subject of the new sentence:
- A book collector was made happy the other day (awkward but grammatical)
We can show that happy is a Complement and that the other day is not a Complement but an Adjunct. If we replace the verb phrase with the pro-phrase did it, then did it should replace the verb plus all its Complements. If we repeat any Complements, the result will be ungrammatical, but if we repeat Adjuncts, it will be ok:
- *Bob did it happy. (ungrammatical)
- Bob did it the other day. (grammatical)
Because happy describes another complement of the verb ( - the book collector), we take it to be a Predicative Complement.
Within the two noun phrases, a and the are words that usually in Determinative function. But we can show for example that a is a Determinative, because singular count noun phrases must have a Determinative. Without a the phrase is ungrammatical. Similarly if we make the head noun plural, no determinative is needed:
- *Bob made book collector happy. (ungrammatical)
- Bob made book collectors happy. (grammatical)
We can show that other is a modifier (adjunct in some grammars) and not a complement, because the noun phrase is perfectly well-formed without this word:
Parts of speech:
Bob [noun]
made [verb], a [determiner], book [noun], collector [noun], happy
[adjective]
the [determiner], other [determiner] day [noun]
Evidence: Bob is a name, and therefore a proper noun. Like other nouns it can be genitively inflected:
Made is inflected for tense and must therefore be a verb. The determiner a, like other central determiners doesn't inflect, is monosyllabic, and most importantly cannot occur with other central determiners:
Book is a noun, not an adjective. Adjectives are freely modifiable by adverbs. Nouns are not. Nouns, on the other hand are freely modifiable by adjectives, whilst in general, adjectives are not modifiable by other adjectives:
- an extremely keen collector. (adjective modified by adverb - grammatical)
- *an extremely book collector. (noun modified by adverb - ungrammatical)
- a rare book collector. (noun modified by adjective - grammatical)
- *a rarely book collector. (again, noun modified by adverb - ungrammatical).
In addition, adjectives usually have comparative forms, whilst nouns do not:
- a keener collector would have chosen ... (comparative adjective - grammatical)
- *a booker collector would have chosen ... ('comparative' noun - ungrammatical)
Notice as well that book can function freely as the head word of a subject or object phrase.
Collector is uncontroversially a noun. But we can show that it inflects for number etc: collectors.
We can use the same tests for the that we used for a. The word other is a bit controversial. We have shown that it is functioning as a modifier. It looks like an adjective here because it occurs after the Determinative and before the Head of the noun phrase. However, it cannot be modified by adverbs: the extremely other day. Also it cannot function as a predicative complement:
- The day was other (ungrammatical)
Like other determiners, it seems to have slightly deictic properties.
Day is uncontroversially a noun. It inflects for number, appears with determiners such as every or other. It is modifiable by adjectives, postmodifiable by preposition phrases and so forth:
- Bob made the book collector happy every other sunny day.
Here we see day modified by the adjective sunny, and occurring with determiners. We could also have:
- Bob made the book collector happy several days per week.
Here we see day inflected for grammatical number and post-modified by the preposition phrase per week.
Notice that the function that the noun book has ( - namely modifier or adjunct in a noun phrase) does not affect the part of speech. Neither does the other day become an adverb or adverb phrase, just because it is an adjunct (read "adverbial"). The head word in the phrase is the noun day, and it is therefore, of course, a NOUN PHRASE.