That assassinations occur after a planning process like this is an insight from the U.S. Secret Service, Ash said. If agents come across someone who has been tracking the president's movements for six months, that's a much higher-risk situation than someone who just has an idea about attacking.
In the April 20, 1999, Columbine High School shootings, Harris and Klebold had clearly done significant planning. Officials found a piece of notebook paper "showing a diagram of the Columbine High School cafeteria with two X's next to the pillars," and a list of how many people were there before and after the first lunch period, the sheriff's department said. Both boys had written an itinerary for the day of the shooting.
The largest single school shooting in U.S. history, which occurred at Virginia Tech in 2007, was perpetrated by Seung-Hui Cho. Cho had been declared mentally ill and "an imminent danger" to himself by a Virginia special justice, CNN reported in 2007.
The massacre, in which Cho killed 32 people before turning the gun on himself, was not spur-of-the-moment, either. During the two hours between the two shootings he committed at different dormitories on April 16, 2007, Cho mailed a package with 27 videotaped messages and an 1,800-word statement to NBC News. "You forced me into a corner and gave me only one option," he said in one of the videos.
Schizophrenia is a diagnosis that has come up in some instances of mass killings -- for instance, Jared Loughner, who killed six people and wounded former U.S. Rep. Gabrielle Giffords outside an Arizona supermarket in 2011.
Ted Kaczynski, the "Unabomber," received a diagnosis of schizophrenia from court-appointed psychiatrists. This is an especially likely diagnosis in cases when the shooter is in his late teens or early 20s, because symptoms often do not develop until that age, Levin said.
But in other cases, the shooters' "pathology" is more clearly tied to their situations, Levin said. They have given up hope for the future, and they seek revenge. They believe they are the victims, and the people they are shooting are villains.
School shooters specifically tend to come from middle- or upper-class families, Garbarino said. In poorer areas already rife with violence, kids with the same vulnerabilities may drop out of school or end up in the criminal justice system much earlier than in a resource-rich community, so school itself might hold less meaning for them.
"For school shooters, they're sort of emotionally living and dying in high school," Garbarino said.
Kipland P. Kinkel, a freshman at Thurston High School in Springfield, Oregon, killed his parents and two students in his school's cafeteria in 1998. His parents were both teachers. His family had taken him to therapy, concerned about his obsession with guns and explosives.
"They got every possible resource, and it still didn't work," Garbarino said.
Garbarino interviewed a young man who had brought guns and bombs to school on Valentine's Day of his senior year of high school. The boy had studied the Columbine shooting and contemplated killing himself or his tormenters, who had been bullying him. Before he acted, a couple of girls saw him and found his behavior odd; one of them went to get a police officer. He surrendered and went to jail. He said in retrospect that he didn't want the girls to get hit if he started shooting.
School shootings by students dropped significantly after Columbine because schools took action to encourage everyone in their communities to bring threats to school officials' attention, Ash said.
"We've averted a number of mass killings at schools around the country in recent years because young people are beginning to inform when they hear one of their peers threatened in the hallway," Levin said. "They may inform a resource officer, a parent, a teacher, a school psychologist."
In the past, "It wasn't cool for teenagers to inform on their peers, and they didn't," he said. "More and more young people now recognize the seriousness of these threatening utterances at school, and they're much more likely to inform someone."
What have emerged instead are instances where members of the general public have taken up arms and killed innocent people.
Garbarino also points to a culture of gun violence, where killing people may seem like a viable and available solution to life's problems.
The good news is that most people who have these risk factors will never take the next step, Garbarino said. They may play violent video games, talk about violence with friends and research it on the Internet, but they would never implement a full plan.
A major problem is that often, interventions happen when a person becomes dangerous or threatening. A young person should receive help when he or she is merely "troubled," Levin said -- for instance, if he or she is being bullied and harassed, and feels a profound sense of powerlessness.
Anti-bullying laws can help, he said. Parents, teachers, principals and school psychologists should step in when they realize a young person feels terrorized in school, Levin said. Most school shooters had been bullied chronically; bullying is not necessarily just a part of growing up.
"When we see those red flags in the life of a youngster, we should intervene," Levin said. "Not to prevent a murder, but to do it because it's the right thing to do.
"We would improve the quality of life for lots of people," he added, "and in the process we probably would prevent a murder or two."