That's what I believed too, that trees would help breaking your fall, right ? Until, one fine day, I was construction workers emptying a 20 floor building for demolition, basically throwing furniture out the window. Uhmm, thick wood doors get impaled on tree branches.
Also, parachutists know to avoid landing in trees at all costs. So nope, trees don't help at all.
Maybe the type of tree makes a difference? A lot of trees we plant in North American cities have fairly rigid branches all the way to the top, and the foliage is relatively thin. I don't know what kind of trees grow in Peruvian rainforests, but I wouldn't be surprised if the branches were softer and covered with much thicker foliage, like human-sized leaves you could just slide off of.
Also, parachutists know to avoid landing in trees at all costs. So nope, trees don't help at all.