Ich bin fast über die Wurzel neben der Hecke gestolpert, als ich den Ast vom Weg weggetragen habe.

German uses two different auxiliaries in the perfect tense: sein and haben.

Here:

• ich bin gestolpert uses sein
• ich habe weggetragen uses haben

A common rule of thumb is:

• many verbs of movement or change of state use sein
• most other verbs, especially verbs that take a direct object, use haben

So:

• stolpern is an intransitive movement event here, so sein is natural
• wegtragen takes a direct object, den Ast, so it uses haben

That is why the sentence has both bin and habe.

Because the verb expression is über etwas stolpern, and that expression takes the accusative.

So:

• die Wurzel is accusative singular
• der Wurzel would be dative singular

Even though über is often taught as a two-way preposition, in this sentence it is best understood as part of the verb phrase über etwas stolpern = to trip over something. With this meaning, German uses the accusative.

So:

• über die Wurzel stolpern = correct
• über der Wurzel stolpern = not correct here

Because this part describes location, not movement.

neben is a two-way preposition:

• dative for location
• accusative for direction toward somewhere

Here the root is simply next to the hedge, so German uses the dative:

• neben der Hecke

If there were movement toward the hedge, you could get accusative:

• Ich lege den Ast neben die Hecke. = I put the branch next to the hedge.

But in your sentence, nothing is moving toward the hedge. It is just telling you where the root is.

als is used for a single event in the past and means when in that context.

So here:

• als ich den Ast vom Weg weggetragen habe = when I carried the branch away from the path

Also, als introduces a subordinate clause. In German subordinate clauses, the conjugated verb goes to the end. That is why habe comes last.

Compare:

• main clause: Ich habe den Ast weggetragen.
• subordinate clause: ..., als ich den Ast weggetragen habe.

That final-verb position is one of the most important patterns in German.

fast means almost, and here it modifies the whole action über die Wurzel gestolpert.

Placing fast before that part is very natural in German:

• Ich bin fast über die Wurzel gestolpert.

It tells you that the tripping almost happened.

German adverbs like fast often come before the element they affect. You may also hear similar alternatives such as beinahe.

Because wegtragen is a separable-prefix verb:

• infinitive: wegtragen
• present: ich trage den Ast weg
• perfect participle: weggetragen

With separable verbs in the participle, ge- goes between the prefix and the verb stem:

• weg + ge + tragen → weggetragen

So it is one verb, not two separate words.

vom is the contraction of von dem.

So:

• vom Weg = von dem Weg

In English, this usually means something like:

• from the path
• off the path

This contraction is very common in German:

• am = an dem
• im = in dem
• zum = zu dem
• vom = von dem

It is a little repetitive in meaning, but it is still natural.

The two parts do different jobs:

• vom Weg tells you the starting place: from the path
• weggetragen tells you the action/result: carried away

So together, the sentence emphasizes that the branch was removed from the path.

English might often use just one element, but German can comfortably use both:

• den Ast vom Weg weggetragen

It sounds like: carried the branch away from the path.

Because Ast is the direct object of wegtragen, and direct objects are usually in the accusative.

The noun is masculine:

• nominative: der Ast
• accusative: den Ast

So:

• ich trage den Ast weg
• ich habe den Ast weggetragen

That is why the article changes from der to den.

Yes. The sentence has a main clause and a subordinate clause.

Main clause:

• Ich bin fast über die Wurzel neben der Hecke gestolpert

Subordinate clause:

• als ich den Ast vom Weg weggetragen habe

In the main clause:

• the finite verb bin is in second position
• the participle gestolpert goes near the end

In the subordinate clause:

• als introduces the clause
• the finite verb habe moves to the end

So the overall pattern is very typical German:

• finite verb second in main clauses
• finite verb last in subordinate clauses

Here it is basically when, but it can suggest a time background similar to while in English depending on context.

The important point is that als is used for a one-time past situation. It sets the time frame in which the near-tripping happened.

So this sentence means that the almost-tripping happened at the time when the speaker was carrying the branch away.

If you wanted a more strongly continuous while feeling, German might sometimes use während, but als is perfectly normal here for a single past event.

Yes, there are several natural alternatives with only small differences in style or emphasis.

For example:

• Als ich den Ast vom Weg weggetragen habe, bin ich fast über die Wurzel neben der Hecke gestolpert.
• Ich bin, als ich den Ast vom Weg weggetragen habe, fast über die Wurzel neben der Hecke gestolpert.

These all keep the same basic meaning.

German word order is flexible, but the core rules still apply:

• main clause verb in second position
• subordinate clause verb at the end

So you can move parts around for emphasis, but you cannot ignore those verb-position rules.