How did capital punishment start?

You will probably have to go to the library (GASP!) and crack a few books (GASP!) about the history of the old West. It probably goes something like this: Someone robbed someone (bank, cattle rustling, etc.) and were too scared to go after them. But they were just brave enough to go into town and yell "Let's Get Them!" So a bunch of folks on horses (with whatever happened to be attached at the time, ropes etc.) took off and caught the bad guy (groups of people are MUCH braver than individuals). Being poor folk they didn't want to waste expensive bullets, or food & shelter (jail), or time & energy (transporting to jail), so someone generously donated a few feet of rope and the accused was hanged from a convenient tree. The body was strapped to it's own horse and lead to town. The horse and attached belongings (saddle, saddle-bags, etc.) were sold to pay for burial. Any excess money was kept by the town as windfall taxes, and helped to offset paying for law enforcement (Marshall's salary, rewards, etc.). The judicial system probably adopted hangings as a clean (no blood), inexpensive way to dish out capital punishment.