Learn Swedish: Är den nya parfymen blommig eller träig?

In this lesson, we learned to use the indefinite noun after den, like den parfym. Why does this sentence use the definite parfymen instead?

The rule you are learning in this lesson applies to the determinative construction, which points to a relative clause (like den parfym som du köpte - the perfume that you bought). Because this sentence does not have a relative clause and just says 'the new perfume', it uses the standard double definite rule: den + adjective (nya) + definite noun (parfymen).

Since the subject is definite (den nya parfymen), why don't the adjectives blommig and träig get the definite -a ending?

Adjectives only take the definite -a ending when they come directly before the definite noun (like nya does). When adjectives come after the verb to describe the subject (acting as predicate adjectives), they use the indefinite form. Since parfym is an en-word, they take the standard en-word forms blommig and träig.

How do the words blommig and träig relate to the Swedish nouns for flower and wood?

Adding the suffix -ig to a noun is a very common way to form an adjective in Swedish. It works exactly like the English suffix '-y'. The noun en blomma (a flower) drops its final vowel to become blommig (floral or flowery), and trä (wood) becomes träig (woody).