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

We like watching American Ninja Warrior and some of the obstacles are similar to this. Its definitely doable for someone with the right training but a lot of things matter including the distance of the jump.

Most important factor is probably size of branch, for it to work the branch has to be strong enough to hold bodyweight +, and also small enough to get a grip. Unless someone has Michael Jordan sized hands there aren't too many branches that fit both criteria.

Also learned from ninja warrior, mixed grip (one had grabs facing forward, the other grabs facing towards body) is your best chance by far.