O. Henry takes a very interesting tone; it is almost condescending and sarcastic. He seems to be making fun of all of his characters, deriding them for shallowness. They are flippant and ineffectual. He makes fun of the entire neighborhood of Greenwich Village, and creates several characters of the sort he seems to encounter there, only to mock them. But then the very end takes all of that and slaps it with a very serious, dreadful ending, which seems to indicate that careless conversation and attitudes can have devastating effects.
The climax of the story would be when Johnsy and Susie are watching and waiting for that last leaf to fall. When it doesn't, and Johnsy decides to live, we have the resolution. However, the dramatic ending is a very intense and powerful finale, one that seems to mirror a climax in its importance.