Learn Hardcore Hungarian: A hegy körül erdő van, és sok gyerek fut a fák között. - Around the mountain there is a forest, and many children run among the trees.

A hegy körül erdő van.
= There is forest around the mountain. (Just stating existence in general.)
A hegy körül egy erdő van.
= There is a (single) forest around the mountain. (More like “one forest”, countable, a bit more specific.)
A hegy körül az erdő van.
= The forest is around the mountain. (We already know which forest we’re talking about.)

So the sentence chooses the most neutral, general version: erdő without article.

Why do we say A hegy körül and not körül a hegy?

Hungarian uses postpositions, not prepositions, in many cases. That means the word like körül (“around”) usually comes after the noun:

a hegy körül = around the mountain
(literally: the mountain around)

You might occasionally see körül before a pronoun (e.g. körülöttem = around me), but with a regular noun phrase the normal order is:

[article] + [noun] + körül
a hegy körül, a ház körül, a tó körül, etc.

So körül a hegy is ungrammatical; körül must follow a hegy here.

What exactly does van do here, and why can’t it be left out?

Van is the 3rd person singular form of lenni (“to be”).

Hungarian has a special rule: in the present tense, you usually omit van/vannak when they simply link a subject to:

a noun: Ő tanár. = He is a teacher.
an adjective: Ő magas. = He is tall.
a place: Ő otthon. = He is at home.

But when you are expressing existence – the English “there is / there are” – you must use van/vannak:

Van egy könyv az asztalon. = There is a book on the table.
A hegy körül erdő van. = There is forest around the mountain.

So here erdő van is an existential construction, not just “the forest is X”, and van cannot be dropped.

Why is the word order A hegy körül erdő van and not Van erdő a hegy körül? Are both correct?

Both are grammatically correct, but the focus and topic are slightly different.

A hegy körül erdő van.
Topic: a hegy körül (“around the mountain”).
You’re saying something about that place: As for around the mountain, there is forest (there).
Van erdő a hegy körül.
Neutral word order, often an answer to “Mi van a hegy körül?” (“What is around the mountain?”):
There is a forest around the mountain.

In everyday speech, both can appear in similar contexts. The first one slightly emphasizes the location (“around the mountain”), the second slightly emphasizes the existence of a forest there.

Why is it sok gyerek and not sok gyerekek?

With numerals and most quantity words, Hungarian uses the singular form of the noun, even though the meaning is plural:

egy gyerek = one child
két gyerek = two children
három gyerek = three children
sok gyerek = many children
kevés gyerek = few children

So sok gyerek is the standard form. sok gyerekek is incorrect in standard Hungarian.

The plural marker -k (as in gyerekek) is used when you speak of a group on its own, without a numeral/quantifier:

A gyerekek futnak. = The children are running.
Látok gyerekeket. = I see children.

Should it be sok gyerek fut or sok gyerek futnak?

In standard, neutral Hungarian, with words like sok, kevés, három, etc., the verb is singular:

Sok gyerek fut.
Három kutya ugat.
Kevés ember dolgozik.

So sok gyerek fut is the recommended, textbook form.

In everyday spoken Hungarian, you will sometimes hear a plural verb:

Sok gyerek futnak.

This is very common in informal speech, but is considered non‑standard. If you’re learning, it’s safest to use the singular verb:

Sok gyerek fut a fák között.

Why does fa become fák in a fák között?

The base word is fa (“tree”). The regular plural ending is -k, but often the word’s vowel becomes long when you add -k:

fa → fák (tree → trees)
ló → lovak (horse → horses) – here the consonant changes
kutya → kutyák (dog → dogs)

For fa, the plural is:

fa + -k → fák

So a fák között literally means “among the trees” / “between the trees”.

Morphologically:

a = the
fá- = stem from fa (with long vowel in plural)
-k = plural
között = between / among

What does között mean exactly, and how is a fák között different from using -ban/-ben, like a fákban?

Között basically means “between / among”.

a fák között = among the trees, between the trees
(You are moving or located in the spaces between them.)
a fákban = inside the trees
(Literally “in the trees”, which is usually strange unless you mean inside their trunks, holes, etc.)

So for “Many children are running between / among the trees”, you need:

sok gyerek fut a fák között

not a fákban, unless you really want to say something like “inside the trees” (which would be unusual).

Why is there an article before hegy and fák, but not before erdő and gyerek?

Articles in the sentence:

a hegy = the mountain
a fák = the trees
erdő (no article) = forest (some forest)
sok gyerek (no egy or a) = many children

Reasons:

a hegy – a specific mountain is being talked about.
a fák – specific, known trees around that mountain.
erdő – presented indefinitely, as some forest that exists around the mountain; not yet specific, just stating existence.
sok gyerek – here sok (“many”) already gives a quantity; egy or a would be incorrect.

Possible alternatives with articles, with different nuances:

A hegy körül egy erdő van.
There is one forest around the mountain. (countable, more specific idea)
A hegy körül erdők vannak.
There are forests around the mountain. (plural, more than one forest)

The original sentence uses the most neutral pattern of definiteness versus indefiniteness.

Can we move a fák között earlier, like Sok gyerek a fák között fut? Does the meaning change?

Yes, you can say:

Sok gyerek fut a fák között.
Sok gyerek a fák között fut.

Both mean essentially the same: Many children are running among the trees.

Nuance:

Sok gyerek fut a fák között. – completely neutral; location comes last.
Sok gyerek a fák között fut. – a bit more emphasis on where they run (as opposed to somewhere else).

Hungarian word order is flexible and often used to express focus rather than grammatical relations (those are mostly shown by endings and postpositions). Here, both orders are natural.

What is the difference between körül and köré?

Both relate to the idea of “around”, but:

körül = “around” as a static location (no motion implied)
- A hegy körül erdő van. = There is forest around the mountain.
- A ház körül fű nő. = Grass grows around the house.
köré = “around” as a direction (movement towards forming a circle or surrounding something)
- A gyerekek a fa köré futnak. = The children run to around the tree / run to form a circle around the tree.
- A katonák a tűz köré gyűlnek. = The soldiers gather around the fire.

So in your sentence, the forest simply exists around the mountain (no movement), so körül is correct, not köré.

Is körül a case ending, like -ban / -ben, or is it a separate word?

Körül is a separate word: a postposition, not an ending.

Case endings attach to the noun:
- hegyen (on the mountain)
- hegyben (in the mountain)
- hegytől (from the mountain)
Postpositions come after the full noun phrase, as separate words:
- a hegy körül (around the mountain)
- a ház előtt (in front of the house)
- a bolt mellett (beside the shop)

So körül works like előtt, mögött, alatt, fölött, mellett, etc.: all are postpositions that follow the noun.