The sharp and sudden crack! broke the summer calm.

Alexis Handwerker had been sitting on a bench beneath a towering elm in Stuyvesant Square Park in Manhattan — now she was pinned to the ground, bleeding, disoriented and smothered by leaves. One arm was rammed back unnaturally, broken. Panicked parkgoers struggled to free her from a huge tree limb that had plummeted 30 feet.

“I don’t want to die,” she screamed. “I don’t want to die.”

Ms. Handwerker, a 29-year-old social worker, survived the July 2007 accident with grievous injuries, and sued. Her lawyers pieced together evidence that untrained parks workers had missed signs that the elm was rotting — even though the 80-foot tree, one of the biggest in New York City, had sent limbs crashing down before. The city settled in February, paying $4 million.

Ms. Handwerker’s suit is just one of at least 10 stemming from deaths or injuries caused by falling limbs and branches in New York City that were quietly resolved over the last 10 years, or are now winding their way through the courts. The city has paid millions of dollars in damage claims, with far more expected. It all comes at a time of steep cutbacks in the amount of money the city dedicates to tree care and safety.

Lawyers and investigators hired by the victims have gathered parks records, taken sworn testimony from city officials and parks workers, and hired tree-care experts to review city procedures. The collected evidence, taken together with public records and interviews with outside experts and parks officials, depicts an overstretched and haphazard system of tree inspections and care, one in which the crucial job of spotting dangers can be left to untrained workers, and repairs and pruning are delayed to save money.