Growing lawn & turf under trees can always be a problem. However there are ways that you can grow lawn under trees in most situations. It comes down to a number of factors mainly to do with soil type, lawn type and soil moisture.
Firstly have a look at the variety of grass you presently have, is it a variety that will grow under trees or take shade ? Most varieties of Couch will not grow under trees. If you have Kikuyu it will but only in a open, semi shade situation. Kikuyu will still grow in about 50% shade to some extent. If you have buffalo some of the newer varieties such as Sir Walter and Shade Master will also grow to some extent under trees to around 50% shade.
Next check your soil around the tree, is it dry or not draining. You may be able to improve the soil by adding organic matter (cow or sheep manure) or remove the soil and add and improved soil mix. You normally do not have to remove much around 10 to 15cm of top soil. Trees tend to draw up mositure around their root zone so making the soil dry, however most of this occurs deeper in the soil. By improving the very top of the soil zone you can increase the mositure holding capacity of the soil. Most lawn types only draw mositure from the top 15 cm of the soil. Plant some runners from the lawn and water well.
If your lawn is still not growing the next option is to use a different grass around the tree. There are may seed grown varieties which will grow well under trees where running varieties won't. But be aware that they will look different to the lawn you already have. This will mean a different leaf and possibly colour of grass. You can purchase packs of seed call "shade mixture" which have a combination of grass seeds suitable for growing under trees or in shady spots. Most varieties which will grow under trees are Rye grass or Fescues which will take a good amount of shade. They are not running varieties, so be aware that a good spread of seed is required to get coverage.