One easy way to get the city's attention is to write to one of the city council members and tell them that the trees pose a legal liability for the city since limbs are breaking off and falling in your yard. Emphasize that they need to be taken down so they don't fall on a home, or a limb fall on a child and ask them who to contact. Changes are they'll forward your email to the correct department [parks most likely] who will then send someone out to see what needs to e done.

I did this when I was almost hit head-on at an intersection and again when I had some concerns about drug dealing in my neighborhood. I was amazed at how helpful the city council was--of course the term 'legal liability' didn't hurt either.....