Avia dicit radices aquam petere, sed ramos solem quaerere.

Because after a verb of saying like dicit (says), Latin often uses an indirect statement construction.

That construction works like this:

• the subject of the reported statement goes into the accusative
• the verb of the reported statement goes into the infinitive

So:

• radices aquam petere = that the roots seek water
• ramos solem quaerere = that the branches seek the sun

This is a very common Latin pattern, often called the accusative-and-infinitive construction.

Latin usually does not need a separate word for that in this kind of sentence.

English says:

• Grandmother says that the roots seek water

Latin says:

• Avia dicit radices aquam petere

The idea of that is built into the indirect statement construction itself, so no extra word is necessary.

The main verb is dicit = says.

The basic structure is:

• Avia = Grandmother
• dicit = says
• then two things she says:
  • radices aquam petere
  • sed ramos solem quaerere

So the sentence is built around dicit, and the rest depends on it.

It is the subject in meaning, but in an indirect statement Latin puts that subject into the accusative.

So in:

• ramos solem quaerere

ramos means the branches, and it is the subject of quaerere, but grammatically it is accusative because of the indirect statement after dicit.

In other words:

• English: the branches seek the sun
• Latin in indirect statement: ramos solem quaerere

This feels strange to English speakers at first, but it is one of the most important Latin sentence patterns to learn.

For radix, radicis (root), the form radices can be either:

• nominative plural
• or accusative plural

Here, because it is inside an indirect statement after dicit, it is understood as accusative plural, even though it looks the same as the nominative plural.

So:

• radices aquam petere = that the roots seek water

A learner may not be able to tell the case from the form alone here, but the construction tells you how to read it.

Because they are the direct objects of petere and quaerere.

So in the first part:

• radices = subject of the infinitive
• aquam = object of petere

And in the second part:

• ramos = subject of the infinitive
• solem = object of quaerere

That means each infinitive phrase has:

• an accusative subject
• and an accusative object

This is completely normal in Latin indirect statement.

You tell them apart by meaning and structure, not just by case ending.

You use a combination of:

• vocabulary meaning
• sentence structure
• common sense
• parallelism

For example:

• radices aquam petere
  roots are the thing doing the seeking, and water is the thing sought

• ramos solem quaerere
  branches are the thing doing the seeking, and sun is the thing sought

Also, the sentence is neatly parallel:

• radices ... aquam
• ramos ... solem

That symmetry helps you see the pattern.

Yes, both can be translated as seek, but they are not exactly the same.

Very roughly:

• petere can mean aim at, head for, seek, try to obtain
• quaerere can mean look for, search for, seek

So petere often suggests moving toward or striving for something, while quaerere often suggests searching for something.

In this sentence, the two verbs create a nice stylistic variation:

• roots seek for / reach toward water
• branches search for / seek the sun

In simple classroom translation, it is fine to render both as seek.

Because solem is the accusative singular form of sol.

Dictionary form:

• sol, solis = sun

In the sentence, the sun is the object of quaerere, so it must be accusative:

• sol = nominative
• solem = accusative

The same thing happens with:

• aqua → aquam

Because Latin has no articles like the or a/an.

So solem can mean, depending on context:

• the sun
• or simply sun

English usually needs an article, so we naturally say the sun.

The same is true for many Latin nouns. Latin leaves that part to context.

Latin word order is much freer than English word order.

English depends heavily on position:

• The roots seek water

Latin depends much more on endings, so it can arrange words more flexibly.

Here the order does a few nice things:

• Avia dicit sets up the statement
• radices aquam petere gives the first idea
• sed introduces the contrast
• ramos solem quaerere gives the matching second idea

The sentence is very balanced:

• roots / water
• branches / sun

That parallel structure is more important than following strict English-style order.

Sed means but.

It marks a contrast between the two halves:

• roots seek water
• but branches seek the sun

So it helps the sentence set up a natural opposition:

• below / above
• water / sun
• roots / branches

They are present active infinitives.

In an indirect statement, the present infinitive usually shows action happening at the same time as the main verb.

So:

• Avia dicit radices aquam petere
  means Grandmother says that the roots are seeking / seek water

The English translation may use either simple present or progressive present depending on style, but Latin is using the present infinitive.

Here are the main forms:

• avia → avia, aviae = grandmother
• dicit → dico, dicere = say
• radices → radix, radicis = root
• aquam → aqua, aquae = water
• petere → peto, petere = seek, head for, aim at
• sed = but
• ramos → ramus, rami = branch
• solem → sol, solis = sun
• quaerere → quaero, quaerere = seek, look for

Knowing the dictionary forms helps you recognize why the words change shape in the sentence.

Yes, but it would no longer be the same construction.

For example, a direct statement would look more like:

• Radices aquam petunt, sed rami solem quaerunt.
  The roots seek water, but the branches seek the sun.

But once Latin puts the statement after dicit, it commonly switches to indirect statement:

• Avia dicit radices aquam petere, sed ramos solem quaerere.

So the infinitives are not random; they are there because the sentence is reporting what someone says.