Aarre löytyy lopulta vanhan puun alta.

Because löytyy comes from löytyä, which means to be found, to turn up, or to be discovered.

• löytää = to find something
• löytyä = to be found

So:

• Löydän aarteen = I find the treasure
• Aarre löytyy = The treasure is found / turns up

In this sentence, the focus is on the treasure being discovered, not on who finds it.

No. Löytyy is not the Finnish passive.

It is the 3rd person singular present form of löytyä:

• minä löydyn
• sinä löydyt
• hän/se löytyy

It often corresponds to English is found, which looks passive in English, but in Finnish it is just a normal active form of a different verb.

Because aarre is the subject of the sentence, so it is in the nominative singular.

• aarre = treasure

Finnish usually does not use articles like a or the, so aarre can mean a treasure or the treasure, depending on context.

Lopulta is an adverb meaning eventually, finally, or in the end.

It tells you that the treasure is found after some time, effort, or delay.

Examples:

• Se onnistui lopulta. = It finally succeeded.
• Aarre löytyy lopulta... = The treasure is eventually found...

Because alta requires the noun before it to be in the genitive.

So:

• vanha puu = an/the old tree (basic form)
• vanhan puun = of an/the old tree / the form used before this postposition

The adjective must agree with the noun, so both change:

• vanha → vanhan
• puu → puun

Alta is a postposition meaning from under or from beneath.

In Finnish, postpositions come after the noun phrase:

• puun alta = from under the tree
• literally: tree-GEN from-under

That is why the sentence ends with vanhan puun alta.

This is a very common question, because English often uses under for both.

The difference is:

• alla = under, in a static location
• alta = from under, movement or emergence from below

Compare:

• Aarre on puun alla. = The treasure is under the tree.
• Aarre löytyy puun alta. = The treasure is found from under the tree.

In English, we often still translate puun alta simply as under the tree, but Finnish keeps the distinction clearer.

Finnish word order is more flexible than English word order.

This sentence has a neutral, natural structure:

• Aarre = topic/subject
• löytyy = verb
• lopulta = adverb
• vanhan puun alta = place

A different order is also possible, for example:

• Lopulta aarre löytyy vanhan puun alta.

That would put more emphasis on eventually.

So the original word order is natural, but not the only possible one.

Because in Finnish, adjectives normally come before the noun they describe, just as in English.

• vanha puu = old tree
• vanhan puun = of the old tree / under the old tree in this structure

The important thing to notice is that the adjective agrees with the noun in case and number.

More or less, yes:

• Aarre = treasure
• löytyy = is found / turns up
• lopulta = eventually
• vanhan puun alta = from under the old tree

A very literal version would be:

Treasure is found eventually from under the old tree.

Natural English would usually be:

The treasure is eventually found under the old tree.

or

The treasure is finally found beneath the old tree.