Learn Swedish: Fågeln sjunger i det höga trädet. - The bird sings in the high tree.

This is the Swedish double definiteness rule! When you want to say 'the' plus an adjective and a noun, you need a standalone article (det or den) AND the definite ending on the noun itself (-et or -en). So det höga trädet literally translates to 'the high tree-the'.

Why does the adjective hög have an -a at the end here?

In a definite phrase like this one, the adjective must also take its definite form. For almost all Swedish adjectives, you make the definite form simply by adding an -a to the end of the base word. So hög becomes höga.

How do I know whether to use det or den before the adjective?

You look at the gender of the noun. Because tree is an ett-word (ett träd, trädet), we use the ett-word article det. If the sentence was about the bird, which is an en-word, we would use den, like in den lilla fågeln (the little bird).

Since fågeln means 'the bird', why doesn't it have den in front of it?