The most common and most likely answer to this crossword clue is the 7 letter word DEVIATE
I've seen this crossword clue in The Guardian
D
E
V
I
A
T
E
More clues leading to the result DEVIATE
- … crank up Greek character I’ve seen with intro of drums
- Climbing, setter’s cutting palm tree branch off
- I’ve turned into someone to take out stray
- Idea vet developed to change
- Turn away and cleverly evade it
- Go off fruit that’s mostly bad inside
- Go off course entering Belgrade via Tehran
- To digress, I’ve backtracked in court
- Stray fruit I have cut up inside
- ... crank up Greek character I've seen with intro of drums
Buddy explains!

The straight part
We know that "branch off" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case branch off = deviate.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does DEVIATE mean?
DEVIATE noun- A person with deviant behaviour; a deviant, degenerate or pervert.
- A value equal to the difference between a measured variable factor and a fixed or algorithmic reference value.
- To go off course from; to change course; to change plans.
- To fall outside of, or part from, some norm; to stray.
- To cause to diverge.