You can use this as a weapon against the Tree Spirit, but magic is suggested. You will need the bronze hatchet to cut the tree, but you can use the one in your toolbelt if you have one. When you have the branch go through the Magical Door and you will appear in Draynor village. Use a knife to make a Dramen staff of the branch.
When you have the Dramen Staff, equip it and go through the door of the shed in Lumbridge swamp. Your first time through it will complete the Lost City Quest. You will appear in the Lost City of Zanaris. You can cut as much branches as you want, and make as many Dramen Staffs as you want. If you lose your staff, you can just go to the Entrana Dungeon and cut a new branch; you won't have to kill the spirit again. You can enter the Lost City as much as you want.
To make cinnamon, just use the branch with a pestle and mortar.