Interesting question! Here is an article describing the techniques used by arborists. The article describes a number of different techniques and different pieces of gear. I'll describe one specific method, using cheap gear, that is based on techniques that I've used in rock climbing.
- a short length (maybe 20 m) of 9-10 mm static climbing rope
- a small amount (maybe 10 m) of tubular webbing
- a small amount of 6 mm accessory cord
- a locking carabiner
- a rock-climbing/mountaineering helmet
Prepare a diaper harness made from tubular webbing, tied with a water knot. You can find instructions for this in various places, such as the mountaineering textbook Freedom of the Hills. Clasp it using the locking carabiner.
Prepare two slings made out of the utility cord, tying them with a secure knot such as a double fisherman's bend.
Throw the rope over a sturdy branch. Using one of the slings, tie a Prusik around both strands, put it into the locking carabiner on your harness, and lock the carabiner. Tie another Prusik lower down. Carefully dress and test the Prusiks, since you're going to trust them to hold your weight. Put your feet into the bottom Prusik. Ascend the rope by alternately putting your weight on one Prusik and sliding the other up the rope.
To descend, you can either reverse the process or rappel on a Munter hitch. If you're going to rappel, read the chapter in Freedom of the Hills on rappelling, and set up a Prusik backup as described there, as well as following the other safety precautions described, such as tying backup knots on the ends of the rope.
If you mess up any of the techniques, e.g., by not tying one of the knots correctly, then you could fall and get hurt or killed. Ditto if you rappel without a backup and your hand goes off of the rope. For these reasons, you would be well advised to practice at home with a friend who can provide you with a belay on a second rope. While you're at the top of the climb, you may want to use a separate sling to anchor yourself, so that if you slip, you can't fall to the ground.
If space in your pack is really at a premium, then you could reduce the length or diameter of the rope. A shorter rope will reduce the height to which you can climb, and may also make you more vulnerable to rappelling off the end of your rope, which can be a deadly mistake. A smaller diameter rope may be more easily damaged, and there is probably a limit on how small you can go and still have the Prusiks and munter work.