@Prickly Pear You'll be happy to know that while the adventure makes a big show of telling us DMs about the consequences of falling off the big tree, it also makes it clear that there's no need to ask for skill checks to get across (it's six feet wide and solidly immobile). Chrysagon will slip and fall, giving everyone a fright, but he won't fall off into the water. I guess the designers gave the falling-off info in case a DM wanted to run a random encounter there (and have whatever push you off?) or it was an oversight. Hmm.
Oh, don't get me wrong. I love volunteer skill-checks. The thought crossed my mind to ask for them, just to put a scare into you all. Your description was apt - the bark is slippery, it's surprisingly nerve-wracking.