Learn English – How to tell which word a prepositional phrase is modifying

Although this question almost feels like three questions in one, I will try to answer all three parts — and the general question concerning the whole sentence as well.

book

In the noun phrase book collector, the noun book is a noun adjunct, it is used attributively. A common way of expressing this is to say that book is used as an adjective, but this does not mean that the noun is an adjective. It just plays a similar role.

Looking up attributive noun, we find, for example, Merriam-Webster telling us about the label attributive used in dictionaries:

The italicized label often attributive placed after the functional label noun indicates that the noun is often used as an adjective equivalent in attributive position before another noun:
¹bot·tle . . . noun, often attributive
busi·ness . . . noun, often attributive
Examples of the attributive use of these nouns are bottle opener and business ethics.

Nouns used in this way are often said to be adjectives or act as an adjective. However, they are not. Adjectives do things that these nouns do not:

A great collector => a greater collector — We can form comparatives from adjectives.
A book collector => *a booker collector — Nouns do not like it when we do that.

A great collector => the collector is great — We can use a copula followed by an adjective.
A book collector => *the collector is book — Nouns do not like it when we do that!

In the answers to this question some more arguments are listed.

happy

This is an adjective. It modifies the noun (phrase) book collector. As we can see in this explanation from perfect English grammar about using make and let, this is a very simple construction where we use subject + make + object + adjective to mean “cause the object to be the adjective”:

We can also use subject + make + object + adjective. This means 'cause the object to be the adjective' (the adjective can be good or bad):
• Her story made me really happy.
• The traffic jam made us late.

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the other day

This is not an adverb, but a noun phrase that is used as an time adverbial.

The British Council teaches us about adverbials of time:

We use adverbials of time to say:

• when something happened
• for how long
• how often (frequency)

We often use a noun phrase as a time adverbial

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the whole sentence

Bob made a book collector happy the other day.

Bob — a noun functioning as the subject in the sentence.
made — a verb, functioning as the main verb describing the action that the subject executes.
a book collector — a noun phrase, functioning as the object of the sentence.
happy — an adjective, modifying the object of the sentence.
the other day — a noun phrase functioning as a time adverbial modifying the verb.

The noun phrases can be further taken apart as follows:

a — indefinite article
book — noun adjunct, or attributively used noun, modifying collector
collector — noun, head of this noun phrase

the — definite article
other — adjective modifying day
day — noun, head of this noun phrase

The prepositional phrase here is describing the type of bird by giving its location. If the sentence were to be phrased "The bird sang happily in the tree", it would be answering the question where as you said. It would then be supporting the verb sang. However, currently, the prepositional phrase is describing the bird, and the question "Where?" is an adverbial answer, and therefore cannot describe the noun bird.