Watering Trees

My lawn is dying in areas where I have trees. What should I do differently as far as watering in those areas?

The grass and the trees are competing for all resources. 

What kinds of trees are we talking about?

2 young trees, 3-5 years old: one is a red mulberry, one is a flowering dogwood. 1 mature tree - another flowering dogwood.

What do you do with the leaves?

Are the roots of the trees showing?

I don't do anything with the leaves.
The roots are not showing.

As the leaves fall, they can smother the grass.  If you remove the leaves, the soil slowly turns into a dirt that is not hospitable to the trees or the grass.  The trick is to find a way to convert the leaves to mulch that does not smother the grass.  A mulching mower will often do this. 

That is true. I use a mulching mower so it does get the leaves that stick around. It's pretty windy here though.