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

Assuming it was decently close and the angle was right, I would think you’d want to kind of hit the branch with your gut and sort of wrap up around it.

I doubt many people could Tarzan it and actually jump and grab it and have any control over what happened next. Your only hope is that it’s close enough to do a reach/lean/fall to it and grab the branch. Any more momentum than that and your smart arse friends are posting it on YouTube.