Mochimochi no Ki (OVA)

Five-year-old Mameta lives with Grandpa. He's a timid boy, who always wakes Grandpa up to accompany him when he has to go to the outhouse in the middle of the night. The shadow of the giant horse-chestnut tree in the yard scares him.

Grandpa tells him that just once a year, on November 20, a single brave child sees a light in the tree, but Mameta is still too scared and goes to sleep. But that very night, Grandpa suffers from intense stomach pain, and Mameta, forgetting his fears, races barefoot down the mountain to summon the doctor. On the way back up the mountain, the doctor carries him piggyback, and from his perch on the doctor's back, Mameta sees a light in the tree.
Even so, the next night and the night after, he continues to ask Grandpa to go with him to the outhouse.

The moral of the story is summarized in Grandpa's words to Mameta: "You should never think of yourself as weak or timid. So long as you have caring in your heart, you'll find that you'll be able to do what needs doing."

(Source: booksfromjapan.jp)