Halm looks fine to me, rossim.
"The birch" is a punishment given with (usually) a bunch of bare twigs or stalks so
"A small birch" might be a single stalk - a halm
I am not convinced - there is nothing in HALM to relate to the part of the clue "its leaves fall"
I'm leaning towards HILL where the letter L refers to leaf - but I'm still not confident at all !!
If the clue had the phrase "its leaves FELL" then I would be happy - a FELL is a HILL