Praise be to Allaah.
They should be cut because they cause harm to the people who come to visit the graves. Similarly any thorny plants that may be present should be removed to spare the visitors the harm and annoyance that they cause. It is not prescribed for anyone to plant any kind of trees or plants in the graveyard, for Allaah has not prescribed this.
Although the Prophet (peace and blessings of Allaah be upon him) planted two palm stalks on the graves of two people whom he knew were being punished, he did not plant anything on the graves of Madeenah or of al-Baqee’. The Sahaabah did not do any such thing either, so we know that this was only for the occupants of those two graves who were being punished. We ask Allaah to keep us safe and sound.