Although the book talks about many different issues--poverty, alcoholism, lying, etc.--one of the main themes is hope and the determination to rise above difficult circumstances. There are naturalistic elements in the book, but it is not naturalistic. The Nolans are restricted materialistically by poverty, but still find ways to enjoy life and follow their own desires. Francie talks about how she can get drunk simply by looking at flowers, and Katie tells Francie that she can go without an education for the moment because Francie will fight for it later. This theme claims at least part of the meaning of the title; a tree growing in Brooklyn can survive the tough life that might prevent it from growing, just like the people in Brooklyn can and must fight to obtain their sun and air.
Another major theme is the question of morality. Johnny helps Francie lie in order for her to attend a better school, but this enriches Francie's life and presents opportunities that might not have been available to her otherwise. Sissy helps Johnny recover from alcohol withdrawals with questionable methods including some elements of sexuality, but it helps Katie and Johnny stay together in their rocky relationship. Katie also explains to Francie about love and sexuality from two somewhat clashing points of view: as a mother and as a woman. This book examines the gray shades that exist around traditional notions of what is right, and asks the reader what he or she might consider "right" in a different situation.
Gender roles are also a prominent theme throughout the book. Some of these roles are dated as the book takes place before women's suffrage in the United States is achieved, but many of the gender conflicts examined are still pertinent today. While Katie is giving birth, she tells Francie her views on how much involvement men should have with such intense sorrows only women experience. Katie also works multiple jobs to help sustain her family, yet Johnny receives the luxuries of nicer clothing articles for his part-time work. As Francie grows up and begins to discover her desire for companionship, she begins to understand the complexities and injustices surrounding the difficult circumstances women were often forced when pregnant out-of-wedlock.
Though there are many themes, here is a list of some others:

Pop Culture References    Wikipedia

The title of the 1947 Bugs Bunny cartoon A Hare Grows in Manhattan is based on the title of the book A Tree Grows in Brooklyn. At the end of the cartoon Bugs Bunny shows the book to a pack of menacing dogs and they turn away from him and run to Brooklyn.
The third episode in season 4 of Daria is named A Tree Grows In Lawndale.
Season 1 episode 22 of Ugly Betty is named A Tree Grows in Guadalajara
The book also makes an appearance in the HBO series Band of Brothers, in episode 9.
Super Grover segment in Sesame Street #4224 takes place "where two trees grow in Brooklyn."
This is the favorite book of the character Katerina, in the short story Lermontov by Amanda Michalopoulou
The opening lines of "Interlude" on Jay-Z's "The Black Album" refer to "...a tree that grows in Brooklyn"
The Lifetime album "Jersey's Best Dancers" includes a song titled "Francie Nolan".
Rapper Talib Kweli refers to A Tree Grows in Brooklyn in Blackstar's track Respiration
In an episode of I Love Lucy, Lucy schemes to get her husband, Ricky, into a play she wrote called "A Tree Grows in Havana", in reference to his Cuban homeland.