re: The physics of jumping from a balcony and grabbing on to the branch of a tree.

Yep. What I learned is that once you grab the branch, the momentum of your lower body will continue to swing through. Once your legs swing to the other side and begin to approach being parallel with the ground, your hands have also rotated on the branch. At which point the “C” or hook that is formed by your thumb and fingers on the branch is towards the bottom of the branch and no longer supports your weight at which point gravity directs your body to the ground. Unless you have trained as a gymnast on the high bar and know how to anticipate this, I suspect the first time would result in the same for most people.
I can’t remember what I jumped from, certainty not as high as a balcony but the main tree limb actually bent down as well, so my distance to the ground wasn’t as far. And there was a girl close by I was trying to impress. That was a failure as well.