Final answer:
The branch of the autonomic nervous system that shows greater pre-ganglionic divergence is the sympathetic nervous system, which facilitates a widespread 'fight or flight' response by influencing multiple organs through extensive neural connections.
Step-by-step explanation:
The branch of the autonomic nervous system that shows greater pre-ganglionic divergence is the sympathetic nervous system. This divergence occurs as a single preganglionic sympathetic neuron can innervate multiple postganglionic neurons, making it possible for one impulse to influence a variety of target organs simultaneously. Comparatively, the parasympathetic nervous system is more focused on its neural connections, typically engaging specific targets without such widespread divergence.
The sympathetic nervous system is known for activating the "fight or flight" response, preparing the body to face threats by increasing heart rate, breathing rate, and blood flow to the skeletal muscles, while decreasing blood flow to the digestive system. Such a broad influence on the body is facilitated by the extensive divergence of its preganglionic neurons. The parasympathetic nervous system, on the other hand, facilitates the "rest and digest" functions, with a craniosacral outflow that allows for more precise control over its target organs.