As someone else mentioned, it used to be wolves, lynxes, bears etc. But we killed them all so that humans could eat the deer. Then hunting deer became the sport of the king and associated royalty. This is why eating venison became associated with wealth and only eaten on special occasions. When it used to be available for all.

We now have a problem as there is no natural predator to keep the numbers down.

Wolves, lynxes, and (used to be) bears