Børnene løber rundt om træet i haven.

Because Danish usually puts the definite article at the end of the noun.

• barn = child
• børn = children
• børnene = the children

Two things are happening here:

1. barn has an irregular plural: børn
2. the definite plural ending is added: -ene

So børnene is not a separate word for the plus children; it is one word meaning the children.

In ordinary Danish, if there is no adjective, the most natural way to say the children is børnene.

• børnene = the children

You often use de when:

• there is an adjective: de små børn = the small children
• you mean those children: de børn derovre = those children over there
• the context especially calls for that structure

So in this sentence, børnene is the normal choice.

Danish does not usually make a separate progressive form like English are running.

So:

• løber can mean run, runs, or are running
• the exact meaning comes from context

Here, Børnene løber ... naturally means The children are running ...

Also:

• infinitive: at løbe = to run
• present tense: løber = run / are running

rundt om means around, especially with movement around something.

• rundt = around / in a circular way
• om = around / about / concerning, depending on context

Together, rundt om træet means around the tree.

In this sentence, rundt om is the natural expression because the children are moving around the tree. Danish often uses this two-word combination where English just has one word: around.

A close alternative is omkring, but rundt om often feels more concrete and physical here.

Because the two nouns have different grammatical genders.

• et træ = a tree
• en have = a garden

In the singular definite form:

• neuter nouns usually take -et
• common-gender nouns usually take -en

So:

• træ → træet = the tree
• have → haven = the garden

This is a very common pattern in Danish.

Because Danish usually uses a suffix definite article instead of a separate word.

Compare English and Danish:

• the tree → træet
• the garden → haven

So the idea of the is already built into the noun itself.

You do sometimes see a separate article in Danish, but usually with adjectives or certain other structures, for example:

• det store træ = the big tree
• den gamle have = the old garden

But in your sentence, plain træet and haven are exactly what you would expect.

The sentence follows a very normal Danish main-clause pattern:

• Børnene = subject
• løber = verb
• rundt om træet = prepositional phrase
• i haven = prepositional phrase

So the structure is basically:

Subject + Verb + the rest

That is the standard order in a simple statement.

A useful Danish rule is that in main clauses, the finite verb is normally in second position. For example, if you move i haven to the front, you get:

• I haven løber børnene rundt om træet.

Notice that løber still stays in second position.

In practice, it gives the location of the scene: in the garden.

In a sentence like this, Danish can be a little flexible, just like English. So i haven can be understood as telling you where the children are running, and it also fits naturally with the tree in the garden.

Usually context makes this clear, and native speakers do not find it confusing.

If you wanted to make it more explicit that the tree is in the garden, you could say something like:

• Børnene løber rundt om træet, som står i haven.
  = The children are running around the tree that is in the garden.

The letter ø has no exact English equivalent, which is why English speakers often find it tricky.

A good starting point:

• make a sound somewhat like the vowel in British English bird
• but round your lips while saying it

That gets you closer to Danish ø.

Very roughly:

• løber sounds something like LUR-ber, but with rounded lips on the first vowel
• børnene starts with a similar type of vowel, again with rounded lips

A few notes:

• Do not pronounce ø like English oh
• Do not pronounce it like oo
• Lip rounding is important

If you want the easiest practical habit: think say an “eh/uh”-type vowel while rounding your lips. That will already sound much more Danish.