A central issue in A Tree Grows in Brooklyn is Johnny Nolan's alcoholism. At the time the novel takes place, over a hundred years ago, substance abuse was considered shameful and was not generally thought of in the clinical terms in which we tend to see it today. Alcoholism was regarded primarily as a moral failing, not as an illness.
In this context, it's obvious why Katie doesn't want it to be known that her husband was "a drunk." With the death certificate whitewashing his problem, no evidence exists about his alcoholism after he's gone. In a sense, this is a kind of justice, because everything in the way Johnny is presented shows us that he's basically a good man. Nevertheless, though, Johnny is the proverbial ne'er-do-well who has never been able to provide adequately for his family.
Despite still being a child, Francie knows about his problem and is protective of him; she idolizes her father and seems to forgive him everything. In a remarkable scene, Johnny opens up to his daughter and tries to explain why he drinks. There is a fatalism in him, a belief that it's impossible for him to succeed in life. "I drink because I haven't got a chance," he tells Francie. To everything in his confession, his revelatory monologue to her, she patiently answers, "Yes, Papa," knowing that any other reaction will embarrass him and ruin the father-daughter relationship both of them cherish. The scene is almost unbearably sad, as is the overall story of seeming hopelessness in working-class Brooklyn—despite Francie's knowledge that some day she will break out of the situation, growing and thriving against the odds like the eponymous tree.