With climax being the point of greatest emotional intensity, interest, of suspense in a narrative, the plot of O. Henry's "The Last Leaf" has suspense after Sue asks Johnsy to not look out the window by her bed until Sue's art work is completed.  Johnsy asks her to go in another room, but Sue refuses, so Johnsy instructs Sue,

"Tell me as soon as you have finished...because I want to see the last one [leaf] fall.  I'm tired of waiting.  I tired of thinking.  I want to turn loose my hold on everything, and go sailing down, down, just like one of those poor, tired leaves." 

At this point, the reader wonders if the leaf will fall, and Johnsy will lose her will to live, as she has declared she will.  Here, O. Henry leaves his reader at this point of suspense, having Sue talk with old Mr. Behrman downstairs. Then, when Sue tells him how sick Johnsy is, Behrman becomes angry, saying that he will paint his masterpiece and they will all go away.  However, as the little curmudgeon finally agrees to come upstairs and pose for Sue, she shows him the window and they "looked at each other without speaking." 

This sentence is the climax, for it indicates that the leaf has fallen off.  Behrman and Sue know that Johnsy will soon see that the vine is empty, and she may give up her spirit completely.  Of course, Behrman takes action for that when Johnsy does awaken, she will not find an empty vine.