Hope in urban life
The main idea in the novel is clear from the novel's title. The Tree of Heaven grows among the tenements in Brooklyn. The idea is simply that the tree, having sprout up randomly between the cement, is now a testament to the idea of hope, which breaks through and infects urban life, like a tree springing up from cement instead of dirt. At the end of the book, after a whole life cycle after which, Francie is a full adult, in college, we learn the tree is still alive and well.
This is about the ironic persistence of hope in the face of certain disaster. In the end, everything works out, although Johnny dies. But his death came as the direct result of his suicidal drinking habit. And, although both Katie and Francie regret his death tremendously, their fate is ultimately better with Katie's new husband because he is a source of hope. He is like the tree, plugged into the ground by roots, whereas Johnny was wayward and blown about by circumstance.
Breadwinning and gender roles
The book plays a lot on gender. For instance, instead of highlighting Neeley's story, we learn mostly about Francie and the development of her sense for what life is like as a woman, shaped by her horrifying encounter with a would-be pedophile (her mom saves her by shooting the man and killing him), but also by her watching her mother.
In the novel, Katie represents the foil of Johnny's failure. A hard-working person, Katie doesn't mind doing hard or dirty jobs, even when her husband "should" be making the money in the family. Katie steps in willingly and takes that job on. In other words, Johnny has abdicated his role as head of the household, but Katie's grace, resilience, and strength allow her to fill that role better than her spouse could have. The house is not a patriarchy, but a generous matriarchy.
Tragedy and redemption
It's important to honor the women in the novel by not reducing them to their relationships to men. For Katie this doubly true, because Katie never complains or hesitates to step in when she realizes that her husband, Johnny, is becoming a homebody and an alcoholic.
That being said, one common theme in the book is that tragic situations aren't always permanent. For instance, when Francie realizes that at her school she is judged and bullied (likely for being Irish, although she doesn't understand that exactly), she is allowed to switch to a new school, where she doesn't have friends, but she isn't bullied. Also, the novel ends with a reversal of fortune and a wedding, making the story a comedy. The redemption comes when Katie finally gets the husband she deserved, an honorable man with integrity who also cares about the fate of the children, sending them off to college.
