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

Best chance is if the branch is near the height of the balcony, in which case you need some horizontal thrust in the branch's direction. As per Galileo and Newton, if the branch is significantly lower than your starting point, your exponential acceleration will magnify the forces on the branch, and likely tear it from the tree or your arms from your torso. Be sure and say "splat" when some of your parts hit the ground.