"Split Cherry Tree" is about a son and his father and the two worlds they live in: the working-class rural world and the more educated and refined new class. Dave Sexton, the boy, is in between the two worlds, working hard at home but also educating himself at high school so that he can have a better life. Luster, his father, is very old fashioned, focused completely on work. He doesn't believe in the existence of things he can't see, like germs, and he is skeptical of the changes in education, with things like field trips and boys and girls being taught together.
The story finds Luster getting upset with the school for making Dave stay late to repay a debt to someone whose tree he broke. Luster goes into school to reprimand the teacher (bringing his gun with him), but eventually learns to understand that things have changed and that he must accept that. After spending the day with Dave's teacher, Luster says, "School has changed from my day and time. I'm a dead leaf, Dave. I'm behind. I don't belong here."
To me, there are a couple messages to be taken from the story. The first is to not judge a book by its cover. Luster is visually poor and ragged:
Pa's overalls legs were baggy and wrinkled between his coat and boot tops. His blue work shirt showed at the collar. His big black hat showed his gray-streaked black hair. His face was hard and weather-tanned to the color of a ripe fodder blade. His hands were big and gnarled like the roots of the elm tree he stood beside.
It is also implied that Dave's teacher is the opposite. However, they are both able to get along and find a mutual understanding, even though they judged each other negatively initially. I think this also extends to a great message of general understanding between people; despite Luster's anger towards the teacher, they end up friends and learning from each other, showing that if you can just sit down and talk with someone, you can resolve problems.
Luster ends the story with a message for his son that encapsulates his old-fashioned view of the world.
You'll allus look cleaner than your old Pap. Jist remember, Dave, to pay your debts and be honest. Jist be kind to animals and don't bother th' snakes. That's all I got agin th' school. Puttin' black snakes to sleep and cuttin' 'em open.
So, another message is to be honest and repay debts. Finally, I would say that the story emphasizes an ability to understand change and difference between generations and between cultures, as exemplified by Luster and his son's differences. Good luck!