The Last Leaf Story:
Explain how the leaf contributes to the extended metaphor and the theme in this story

The leaf manages to show how fragility and perseverance go hand in hand.

We can arrive at this answer because:

In "The Last Leaf" we can see the fragility at the moment when the protagonist is sick and believes she will die when the last leaf falls from the tree she observes in her window. Perseverance can be seen in the painter who paints leaves on the wall, preventing the protagonist from seeing when the last leaf has fallen and can recover from her illness.

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The leaf is the symbol for the fragility and the perseverance of life.

Explanation:

The leaf is the extended metaphor and the theme of the story. Not only does it represent the perseverance and the fragile nature of human life, it also represents faith as in the end a picture of it is able to keep a person alive who attaches their life to it.

The leaf is the belief in life for the patient while for the artist it is the fragility and the momentary impermanence of his own life that he confronts when he does the work of painting and getting the leaf onto the tree in the process dying himself.