In your sentence, "I saw a red fox under the willow tree in the backyard" there are three nouns, one pronoun, one verb, five adjectives and two prepositions. There are no adverbs.

Nouns: Tree, fox, and backyard are all common, singular nouns.

Pronouns: I is the first-person singular pronoun.

Verbs: Saw is an action verb expressed here in the past tense (note: the present tense is "see").

Adjectives: A, red, the, willow, and the all modify/qualify nouns in this sentence.

(note: "a" and "the" are both articles, which function as adjectives)

Adverbs: There are no adverbs in this sentence. Adverbs function to modify/qualify verbs, adjectives, or other adverbs. To insert one in this sentence, one could write:

"I clearly saw a red fox under the willow tree in the backyard."

Prepositions: Under and in denote a relationship between the fox and its environment.

“I saw a red fox under the willow tree in the backyard.”  Main clause

I – first person singular pronoun; subject of main clause.

saw – past tense (action) verb; predicate of main clause

a – indefinite article

red – adjective modifying noun “fox”

fox – noun; direct object of main clause

under – preposition

the – definite article

willow – noun modifier of “tree”

tree – noun, object of prepositional phrase “under the willow tree”; adverbial prepositional phrase.

in – preposition

the – definite article

backyard – noun, object of prepositional phrase “in the backyard”; adverbial prepositional phrase

No adverbs, no linking or auxiliary verbs or helping verbs, no clause other than the main clause.