What happens to the man's second fire in To Build a Fire?

During the course of this short story, there are three fires—or rather, there are two fires and a third attempt at making a fire.

The first fire is at the creek's divide, where the man and his dog stop to eat lunch and warm themselves up. After continuing to walk through the treacherous weather, the man falls through ice and accidentally soaks his lower legs. In order to get dry and prevent frostbite, he lights another fire, which is the second fire in the story.

A sudden movement, however, causes a branch above his head to drop a load of snow atop his second fire, extinguishing it at once. The sight of his second fire going out causes the protagonist to berate himself for his stupidity in having built his fire directly under a tree. He chose the location for its convenience, but it made the fire vulnerable to the branches directly above it. This moment of extinguishment marks a turning point, as the man grows increasingly desperate due to the severity of his situation.

The man tries to light a third fire in a new spot, but by this time, frostbite has taken control of his hands, and he is unable to light the third fire. After trying everything in his power to get some warmth back into his hands, including a cruel attempt to kill the dog, he surrenders to his fate and waits for death.

What occurs when the man builds his fire under the tree in "To Build a Fire"?

When the man builds his fire under the tree, snow falls on it from the boughs.

The man is really struggling because he does not seem to have outdoorsman skills.  He stumbles his way into the limited shelter of the tree, and that is where he chooses to build his fire.  After it begins to crackle, he looks up and realizes his mistake.

Now the tree under which he had done this carried a weight of snow on its boughs. ... Each time he had pulled a twig he had communicated a slight agitation to the tree—… an agitation sufficient to bring about the disaster. 

The snow falls, inevitably, and lands on the man and the fire.  Of course, that’s the end of the fire.  The man is “shocked” and feels like this is his death sentence.  He can’t survive out here in the wilderness without a fire.

London depicts one man’s struggle for life against the perils of nature in stark detail.  One man, alone in the snow, trying to do nothing more than build a fire.  The man’s ignorance is his death sentence.  At every turn, he becomes more and more endangered by the weather and lack of protection from it.