The exposition (or background information) of the story is that there is a very wealthy cattle rancher who has a daughter, Josefa, who is very skilled at riding and shooting.
Rising action occurs when Ripley, foreman of one of the cattle outfits owned by Josefa's dad, decides to propose to her. H e makes the long journey to her house. On the way, he hears a mountain lion, and the inciting incident is when he sees Josefa at the watering hole about to be pounced on by the lion, and jumps in-between the two. Josefa shoots it mid-air, and the dead lion lands on top of him in an embarrassing scuffle. Given's pride is wounded; more rising action occurs when he lies and says the lion was "Bill", his pet of 2 years. She indicates that she believes him, and is impressed that he would have risked his life for a pet. He offers to ride her home, then leaves. In the story's climax, when she gets home she tells her father that she had just shot the infamous "Gotch-eared devil", a mean mountain lion known for a patch of ear missing. So, she had lied to Givens so as to not wound his pride.
With O. Henry, the climax and denouement (or falling action, or resolution) is always tricky, because he usually has a surprise ending that is the most climatic moment of the story. So, with the climax being at the end, there really is no denouement. If you want to shift the climax to the mountain lion pounce, then everything after that would be the denouement.