Besides what the person said above, when you are learning English, it's best to be as literal or obvious with your meaning as you can be. :) Otherwise, the meaning of verbs and phrases can get very confusing. In the example you give, I would say that the best and most accurate way to say that would be: I climbed up the tree. Try to use a verb that is as close to your meaning as possible! With that said, walking and running can be used interchangeably a lot of the time, and don't always mean that you literally walked or ran. Example: I ran to the store for eggs and milk. That doesn't necessarily mean I actually RAN to the store. It just means I took a quick trip to the store. Or if I'm "running errands" it just means that I'm out performing those tasks.

The most commonly used one is "I climbed up a tree." The example of a tree doesn't suit "ran up" or "walk up", though they both are almost the same thing. The only difference is that "ran up" implies more urgency. "go up" is the most universal one among them.

Actually, of the four, the one that I would say is wrong is "I walked up a tree." Generally that usage isn't done. "I went up a tree," "I climbed up a tree", and "I ran up a tree" are all valid usages, although "I ran up a tree" is only really appropriate if you want to emphasize a particularly fast manner of climbing and is sort of metaphorical, since it implies you are running up on two feet only if you take it literally.