Too Young to Remember
Laboratory rats demonstrate this effect pretty clearly. Like human toddlers, 18-day-old rats soon forget many of the memories they've laid down (such as the association between a sound and an electric shock). The rate of forgetting is quicker in rats than in humans, but the mechanism is thought to be the same.
If a rat can't remember being 18 days old, they certainly can't remember being 14 days old or younger. But when rat pups are removed from the nest for 3 hours each day (a stressful experience for a rat pup) from 2-14 days old, they show long-lasting changes in physiology and behavior. Even though they presumably have no memory of being removed from the nest, it still makes them anxious, hypersensitive to stress, and prone to alcoholism. (Little know fact: rats like alcoholic drinks about as much as humans do, with the same range of individual variation in voluntary drinking.)
The same is true of humans. Studies of children adopted from institutions* between around six months and three years old (when most children will have little or no memory of their life pre-adoption) has shown that these children nevertheless tend to function poorer than children adopted at younger ages or non-adopted children, with different ages being crucial for different specific symptoms. In this study of children from Russian orphanages, for example, the latest age at adoption was 27 months old (a little over 2 years), but even so, later-adopted children had poorer self-control than earlier-adopted children. And this study found that children adopted from well-run but emotionally deprived institutions between 13-24 months old had significantly more behavior problems than children adopted from similar institutions before 13 months. (Interestingly, though, few studies have shown any lasting effect of trauma occurring before 6 months, even though some studies show short-term effects. It seems that experiences in later infancy and toddlerhood can completely reverse the effects of experiences before 6 months of age.)
In both humans and rats, conscious memories of trauma are only one part of the impact that trauma can have on an individual. Far more significant is the impact that extreme stress can have on brain structure and function - and since young children's brains are growing and changing much more dramatically than older children or adults, the effects of trauma on the brain can be even greater** in these children.
So don't discount the effect of a trauma the child was too young to remember. It could have reshaped the child's brain, changing the way xe thinks and feels in ways xe can't tie to any conscious memories.
* Adopted children are the best to study the effects of early childhood trauma, because the change in home environment usually means the child suffers no trauma in later childhood that could confound the results.
** The study linked to, though old, is the only one I know of that directly compares adopted children who experienced good infant care and trauma in later childhood to children experiencing early trauma. Unfortunately, it's only available through institutional access. If you'd like to read it, let me know your e-mail address in the comments and I'll mail it to you.