Lost City

Attempt to chop the swamp tree south of the campsite - There will be an option to "chop" tree instead of "chop down" tree, and a quest icon on the tooltip.

Talk to Shamus.

Bring items to craft a weapon on Entrana. If you plan on using magic, you must have a magic weapon; you cannot cast spells without one. You can also remove tool belt items once you enter the dungeon, e.g. a mithril hatchet, for use in combat.

Talk to the monks on the northern pier of Port Sarim to be transported to Entrana.

Do not craft your weapons now or you'll be thrown out of Entrana.

Run north-west though the wooded area to the dungeon entrance ladder.

Enter the dungeon.

Craft your weapons now.

Run to the southern-most part of the dungeon.

Chop the dramen tree.

Kill the dramen tree spirit.

Make sure to let the dialogue appear before chopping again, or you may spawn a second spirit.

Chop the dramen tree again.

Craft the dramen branch using a knife.

If you have not completed the quest Recipe for Disaster: Freeing Sir Amik Varze yet, you may want to get a second dramen branch and craft it with the pestle and mortar. This gives an item required for the quest, cinnamon, and saves an otherwise necessary second trip to the island.

Enter the door to the east or teleport out.

Wield your dramen staff.

Enter the shed in the centre of the Lumbridge swamp (south-east of the quest start).

Quest complete!