I have a tree on the side walk of the city of reading and the tree limb fell into my backyard and the neighbors. The neighbor said we were negligent because we didn't trim the tree and now I have a letter from the city theat I have to keep the tree trimmed or will be charged for them to trim it is this true I thought if the tree was in my property it is my concern but the tree is on the street my property stops at my gate

Technically anything that hangs into your yard is yours to maintain or trim. If the limb was over your property line, the onus is on you to trim back. The same thing would apply if this was your direct nextdoor neighbor's tree growing over your property line. You could trim it back to your property line and the neighbor could do nothing about it as it became your tree limb the second it grew onto your property.