It's a general trend, one that I believe can be vouched for by both statistics and scientific study. Of course, you will have outliers and exceptions. But people who've endured abuse/trauma as a child are much more likely to turn into troubled adults themselves. I agree with Syntax. Childhood plays a crucial role in shaping a person's morals, ethics, character, personality, etc. I would go so far as to say a childhood defines that person, and can explain their actions, their beliefs, their reactions to certain situations, etc.
I think there are generally two ways children react to abuse. The first case is when the child understands what is normal. The child knows that the abuse is wrong, that the parent is in the wrong for doing whatever they are doing. The child will probably experience hatred, which he/she will keep deep in their heart and harbor resentment to their parents long after they grow up. This kind of child, despite living in a substandard environment, whether because of outside influences or what, knows what is right and what is wrong. Therefore, even as the child goes through a troubled childhood, he/she can vow to himself "never to be like his parents."
That's case A. More commonly though, we see cases where the kid just never understands what is "normal." The child is essentially isolated from the outside world, from "normal people", basically raised in a substandard childhood with no absolute way to discern what is right, what is wrong, more importantly, what life *should* be like. In this case, the kid will take after the parent, as there is no alternative to take after. Contrary to logic, the child will not harbor any resentment to the adult or authority figure because that child has been raised believing all along that being abused was "normal." In the most severe of cases, where the parent is absolutely abusive, and the kid is continually abused, the child will completely lack any understanding of how children, people, spouses, etc. are supposed to be treated. Here's where you tend to see severely troubled people, where you get psychopaths, where you get people with no empathy, no sympathy, etc.
In most cases, though, parents will just neglect the child, and so the child grows up to be a bum who was never disciplined as a kid.
The key difference in between these two types of children is exposure to the outside world. Does the kid know what a happy family feels like? Does the kid have the opportunity to explore, in school, how other families are like, etc.? If the kid has trouble communicating socially, which is highly likely among substandard childhoods, then the kid could very well live his entire elementary school life without knowing "normalcy."
Well, I've had my fair share of experience with a crappy childhood. And in my experiences, a childhood is a kind of past that you sometimes just can't run away from. It'll get to you. Most people with substandard childhoods, will turn into substandard people. It's just because they had that big of a setback they had to overcome just to get even with people who had normal lives. If you compare life to a race, it's basically like being set fifty meters behind the starting line. It's not that you can't win. It's that you're just going have to run 50 meters just to break even, and then you've still got the rest of the track to finish.
Wellp, I'm rambling. Just my two cents.