What does "Tree someone's or their game" means?

As contextually hinted here in the sentence "1838 – "Instead of having treed their game....", the meaning of possibly rarely-known phrase "tree his/her/someone's/their game" has some positive meaning like "mastering the game", which if not realized, it suggests futile and ignorant efforts or course of action(barking up wrong tree).

Now this usage of the word "tree", I haven't come across even after much efforted searching of it's phrases' list or origins.

Can someone confirm whether the contextually hinted meaning is correct and accurate and also explain when and by what origin, the word "tree" was used to suggest positive meaning hinting "mastered/well-understood their game" ?

Both expressions likely come from English fox hunting. The hunter (usually on a horse) would have several dogs which would seek out and pursue a fox, with the purpose of tiring it or cornering it so that the hunter could shoot it. (Note that I may be a little off on the details of fox hunting here.)

If the fox chose to scamper up a tree it was said to be "treed". Usually the dogs would then circle the bottom of the tree, barking at it. But sometimes the dogs would mistake which tree the fox went up, or the fox would leap from the first tree to another, leaving the dogs "barking up the wrong tree".

While these terms are used metaphorically in many different circumstances, the origins have nothing to do with "games" that are "played" (eg, cricket). Rather, "game" refers to animals that are being hunted.