I'll be sure to tell Baxter (the dog) that he can't climb a tree. He'll be devasted, lol, since it is a favorite pastime.
What I have observed is he "clutches" the tree trunk with his front legs and boosts himself upward with his back paws/claws. I'm told this is not unusual behavior for the breed. 
@berdie That might work with a short,thin tree,but not with something like a grown pine.
The "trick" isn't so much the climbing as it is the "getting back down. Even cats have a problem with this,and they are much better equipped when it comes to claws,and much lighter.
I think we have all seen cats "back down" a tree (their claws won't work if they try to come down head first),and the look of terror in their eyes as they come down.