This proverb is a reference to Claudius killing his brother. The poison (the axe) was only the cause of old Hamlet's (the tree) death. He still remembered that his brother killed him and he came back as a ghost to tell his son to avenge him.
May the bridges I burn light the way. This is a powerful quote. I don’t advocate burning bridges just to burn them, but sometimes it’s best to cut all ties. Tactically burn bridges and use them as learning experiences for future endeavors.