The long island battle was in New York. It was a British win and the America was forced to retreat. America lost over 20% of there men in that battle. They lost New York because of that battle.
Brandywine
The Trenton battle was in New Jersey. America won the battle and it gave them a high moral. Washington crossed the Delaware river and snuck up on the British.
Saratoga
America won the battle of Princeton because the British had to retreat. The Washington leading his Troops.
Summary
Washington attempts to prevent the patriot capital at Philadelphia from being captured, but failed. this long and bloody conflict ended the patriots winning streak.
A major victory for George Washington and the patriot troops. This battle was the turning point in the war. This patriot win persuaded the French the war on there side.