most every trip i fly through denver and often misconnect. if everything is on time, 50 min is plenty of time to connect. if there is even a slight delay, you are cutting it very close. if the delay is mechanical, crew, or another united issue, they will automatically rebook you on the next available flight. depending on loads, that may be the next flight or it may be in (pick a time, the worst i was offered was 3 days later). some is this is based on your mileage status with united, at least in my experience.
if the delay is air traffic, weather or something outside of united's control, they will also rebook you. if the delay is their fault, they will place you (at your request, if available) on a different airline. if it is not, they will not.