Po burzy jedna gałąź spadła na trawnik.

Because po here requires the locative case.

The base form is burza = storm.
After po in the sense of after, it changes to burzy.

So:

• burza = nominative, dictionary form
• po burzy = after the storm

This is a very common pattern:

• po obiedzie = after lunch
• po pracy = after work
• po deszczu = after the rain

So po burzy is not an irregular phrase you just memorize on its own; it follows the normal case rule.

Here po means after.

In Polish, po can have several meanings depending on context, for example:

• after → po burzy = after the storm
• along/around/on in some other expressions
• sometimes part of idiomatic phrases

In this sentence, the time meaning is the important one:
Po burzy = After the storm

So the sentence sets the scene first, and then tells you what happened.

Because gałąź is a feminine noun, and jeden (one) has to agree with it.

The forms are:

• jeden for masculine
• jedna for feminine
• jedno for neuter

Since gałąź is feminine, you say:

• jedna gałąź = one branch

Compare:

• jeden dom = one house
• jedna książka = one book
• jedno okno = one window

So jedna is just normal gender agreement.

You mainly know it from how the noun behaves grammatically.

Although many feminine nouns end in -a, some feminine nouns end in a soft consonant, and gałąź is one of them.

Clues that it is feminine:

• the adjective/number word is jedna, not jeden
• the past verb is spadła, not spadł

So the sentence itself gives you two signals that gałąź is feminine.

Because Polish past tense agrees with the gender and number of the subject.

The subject is jedna gałąź, which is:

• singular
• feminine

So the verb must also be:

• singular
• feminine

That gives:

• spadła = fell (feminine singular)

Compare:

• gałąź spadła = the branch fell
• liść spadł = the leaf fell
• jabłko spadło = the apple fell

In English, fell never changes. In Polish, it does.

Because this sentence describes movement onto the lawn, not location on the lawn.

With na, Polish often works like this:

• na + accusative = movement onto
• na + locative = location on

So:

• spadła na trawnik = it fell onto the lawn
• leży na trawniku = it is lying on the lawn

That is a very important contrast in Polish:

• Idę na stół would mean movement onto the table
• Jestem na stole means being on the table

In your sentence, the branch moved and landed there, so na trawnik is correct.

It actually is in the accusative, but for this kind of masculine inanimate noun, the accusative singular looks the same as the nominative singular.

So:

• nominative: trawnik
• accusative: trawnik

They look identical, but grammatically the case is different.

This is common with masculine inanimate nouns in Polish.

Yes. This is an aspect difference.

• spadła = perfective → the branch fell as a completed event
• spadała = imperfective → the branch was falling / kept falling

In this sentence, Polish uses spadła because it describes a single completed event: the branch came down and landed.

So:

• Po burzy jedna gałąź spadła na trawnik.
  = a branch fell onto the lawn

But:

• Gałąź spadała przez kilka sekund.
  = the branch was falling for a few seconds

English often uses time structure to show this difference; Polish often uses verb aspect.

Yes, Polish word order is fairly flexible, although different orders sound slightly different in emphasis.

The neutral order here is:

• Po burzy jedna gałąź spadła na trawnik.

You could also hear:

• Jedna gałąź spadła na trawnik po burzy.
• Na trawnik spadła jedna gałąź po burzy.

These all express basically the same event, but the focus changes:

• starting with Po burzy highlights the time/background
• starting with Jedna gałąź highlights the subject
• starting with Na trawnik highlights where it landed

So the original sentence is a natural, neutral way to present the information.

It can do both, depending on context.

Literally, jedna means one. But in many sentences it can sound close to a / one in English, especially when introducing something into the story.

So:

• jedna gałąź can mean one branch
• but in context it may also feel like a branch

Polish has no articles like a or the, so words like jeden/jedna/jedno are sometimes used when the speaker wants to make the noun more specific or countable.

Still, jedna gałąź usually gives at least some sense of one single branch.

Because Polish does not have articles like English.

So where English says:

• a branch
• the branch

Polish often just says:

• gałąź

The listener understands from context whether it is indefinite or definite.

If needed, Polish can add other words to make things clearer, such as:

• ta gałąź = that/the branch
• jedna gałąź = one branch / a certain branch

But normally no article is required.

A rough pronunciation guide:

• gałąź ≈ ga-wonzh
• spadła ≈ spad-wa

A few key points:

• ł sounds like English w
• ą is a nasal vowel; before some consonants it often sounds close to on/om
• ź is a soft sound, somewhat like a very soft zh

And in spadła:

• the dł cluster is pronounced with ł = w, so it sounds closer to spadwa than to spad-la

Stress in Polish is usually on the second-to-last syllable:

• BU-rzy
• SPAD-ła
• TRAW-nik

No. Po burza is simply incorrect in standard Polish.

Even in casual speech, Polish still keeps the case after po here:

• correct: po burzy
• incorrect: po burza

So this is not an optional ending; it is part of the grammar.

A useful breakdown is:

• Po burzy = time/background phrase, after the storm
• jedna gałąź = subject, one branch
• spadła = verb, fell
• na trawnik = direction/result of movement, onto the lawn

So the structure is basically:

After the storm / one branch / fell / onto the lawn.

That makes it easier to see why the cases and verb forms are what they are.