Min søster har flest bøger i familien, men min bror har færrest planter på sin altan.

They are the superlative forms used for quantity with countable nouns.

• mange = many
• flere = more
• flest = most

And:

• få = few
• færre = fewer
• færrest = fewest

So:

• flest bøger = the most books
• færrest planter = the fewest plants

In this sentence, they compare family members:

• your sister has the highest number of books
• your brother has the lowest number of plants

In Danish, flest and færrest can work by themselves as quantity words, so you do not need a separate the the way English does in the most or the fewest.

So Danish says:

• har flest bøger = has the most books
• har færrest planter = has the fewest plants

This is completely normal Danish.

This is a very common question.

• flest usually means the most in a comparison
• de fleste usually means most or most of the

So:

• Min søster har flest bøger i familien
  = My sister has the most books in the family.

But:

• De fleste bøger
  = most books / most of the books

That second expression is not mainly about comparing people; it is about a large part of a group of books.

Because the sentence is talking about more than one book and more than one plant.

The singular forms are:

• bog = book
• plante = plant

The plural forms are:

• bøger = books
• planter = plants

planter is a fairly regular plural with -r.

bøger is less regular, because the vowel changes:

• bog → bøger

That kind of vowel change happens in some Danish nouns, so it is worth memorizing.

i familien literally means in the family, but in natural English it often corresponds to in the family or within the family.

Here it shows the group inside which the comparison is being made. In other words, your sister has the most books among the family members.

So i familien tells us the comparison set.

Because Danish normally uses på with altan.

• på altanen = on the balcony

This is the standard idiomatic preposition. Even though English often says on the balcony too, learners sometimes expect i because a balcony can feel like a small enclosed space. But Danish uses på here.

Because sin is the reflexive possessive. It refers back to the subject of the same clause.

In the clause:

• min bror har færrest planter på sin altan

the subject is min bror, so sin altan means his own balcony.

If you said hans altan, it would usually mean someone else’s balcony, not the brother’s own.

So:

• sin altan = his own balcony
• hans altan = another male person’s balcony

No. Sin normally refers to the subject of its own clause, not just any earlier noun.

The second clause is:

• men min bror har færrest planter på sin altan

The subject of that clause is min bror, so sin refers to min bror.

The earlier clause about the sister does not control sin here.

The form depends on the noun that follows:

• sin for common gender singular nouns
• sit for neuter singular nouns
• sine for plural nouns

Here the noun is:

• altan = a common gender singular noun

So the correct form is:

• sin altan

Examples:

• sin bog = his/her own book
• sit hus = his/her own house
• sine planter = his/her own plants

Because men is a coordinating conjunction, like but in English. It links two main clauses.

So the second clause keeps normal main-clause word order:

• min bror har færrest planter ...

Subject + verb + rest

There is no special inversion here just because of men.

Because both søster and bror are common gender singular nouns.

In Danish, my changes according to gender and number:

• min for common gender singular
• mit for neuter singular
• mine for plural

So:

• min søster
• min bror

But for comparison:

• mit hus = my house
• mine bøger = my books

Yes. Here har means has in the sense of possessing or having.

• Min søster har flest bøger = My sister has the most books.
• Min bror har færrest planter = My brother has the fewest plants.

It is the ordinary present tense of at have = to have.

Yes, that is close to the Danish structure, but it is not natural English. English normally needs:

• the most books
• the fewest plants

So a more natural English version is:

• My sister has the most books in the family, but my brother has the fewest plants on his balcony.

The Danish sentence does not need a separate word corresponding to English the in those superlative quantity expressions.