I'm trying to build roads between flags, but something, possibly a tree or two, is preventing them from being laid down.

Do I have to cut down trees before roads can be laid, or is there some way to build roads through forests and other areas without taking them down first?

After playing for awhile myself, it looks like you can't build roads over trees. The trees can be cut with woodcutters though like you mentioned.

I learned this the hard way when I had a couple Foresters Huts too close to my town and consequently couldn't properly build a road network.