The central image of A Tree Grows in Brooklyn is the "Tree of Heaven," which will grow anywhere its seeds fall but is only found in Brooklyn's poor neighborhoods. Francie is compared to the Tree of Heaven several times throughout the course of the novel; like the tree, which grows gradually, Francie's understanding of her own power to change her circumstances grows gradually as well.
We can find a "seed" of this understanding in chapter 12, when Francie's mother Katie moves herself and her two children to a house where they can live rent-free as long as Katie does the cleaning. Katie makes the move because she realizes that she cannot rely on her husband Johnny (Francie's father) to support the family due to his drinking problem and flighty ways.
While it doesn't immediately occur to Francie that she can change her own fate, Katie sets an example for her here by changing the family's living circumstances rather than letting Johnny's drinking and unemployment drag them down further. Chapter 12 is the first time we see Francie exposed to the idea that someone living in her circumstances can work to make them better.
In chapter 17, Katie again sets an example for Francie by trading housecleaning for piano lessons for Francie and Neeley. This is how Francie meets Miss Tynmore, who plants in Francie's head the idea that Francie could be a writer some day.
In chapter 23, Francie transfers to a better school after Johnny helps her lie about her home address. Francie's storytelling skills also begin to flourish, and she starts to understand that she is receiving opportunities her parents never had—opportunities she can use to better her situation.
In chapter 30, Francie sees her name in print for the first time in the school magazine. On her way home, however, she refuses to help an unmarried neighbor named Joanna, resulting in Joanna and her baby being injured. Francie leaves her one copy of the magazine for Joanna and hides in her family's cellar, wondering how women can be so cruel to one another.
Although she doesn't fully realize it yet, in chapter 30, Francie's behavior is already changing. While in previous chapters Francie had lied to get coveted material goods like dolls and pies, in chapter 30, Francie gives up a coveted object (the magazine) to make amends for refusing to help Joanna.
In chapter 39, Francie takes control of her own writing by burning all her papers that are about topics she hasn't experienced. She keeps only the papers that earned her poor grades, even though they are about "ugly" subjects, because those are the subjects she knows. Here, Francie again takes control of her circumstances: she decides what kind of writer she will be, rather than letting others (here, her teacher) tell her how to write.
In chapter 43, Francie suggests to Neeley that they start a new tin-can bank, this time without telling their mother. Having just cashed her first paycheck, Francie is realizing the power money has to help her change her circumstances and those of her family.
In chapter 48, Francie signs up for three summer school courses, taught at the local college. For the first time, she realizes just how much she has already changed her own circumstances: her parents had no education, yet she is taking college classes.
In chapter 56, Francie takes several steps to change what to her seemed great injustices in childhood. She buys a toy at the candy store, then tells the owner to give it to a poor child. She goes to the library and realizes the librarian has never even looked at the flowers that used to give Francie so much hope and joy. She says goodbye to Neeley, and she sees a little girl watching her from the fire escape, just as Francie herself used to watch her neighbors.
By the end of the book, Francie's transition from a child to an adult is complete. Like the Tree of Heaven, she has conquered harsh living conditions, creating her own ability to thrive. Also like the tree, this growth is gradual; an argument can be made for it happening at several different points in the book, particularly in chapter 30 and later chapters.