While exact figures are not listed, black bears can climb trees, young grizzlies too, can climb – perhaps not as quickly, but older ones usually don't. Bears can run more than 60 kilometers an hour, and they can do it up hills, down hills or along a slope. To put that in perspective, that’s 15 m/sec or 50 ft/sec – more than twice as fast as we can run.
Black bears spend a lot of time in trees – their first instinct when threatened is to retreat, and often climb trees and send them cubs climbing up trees when threatened by other predators such as wolves, grizzles or rival black bears. Black bears have claws that have specifically evolved to climb trees, and even if you have enough speed to get into a tree, there’s absolutely nothing stopping that black bear from climbing up there with you. Not only is that bear a faster runner than you, it’s also a better climber.
Now, because grizzly bears evolved in coastal and mountain forests without dense tree canopies, they do not climb nearly as well, and more often than black bears use their size as an offensive strategy when threatened. So perhaps with a grizzly you’d have a slightly better chance once you’re in the tree, but grizzlies can also climb when they need to.
Check out these Bear Myths Debunked at National Geographic.