Shade is caused a number of factors such as buildings, high hedges, fences but more commonly from trees denying light to a lawn.  None of these are helpful for the maintenance of a good quality grass cover.

The denial of light by the tree canopy, the competition for food from the root systems and dehydration through the roots extracting water all combine to impede the healthy growth of grass.  Typical symptoms would be thin bare ground cover, often very dry with moss and lichen in evidence.  Some trees excrete sap through the foliage that can result in a sticky and unpleasant surface.  You may be surprised at the distances roots can travel but as a general rule on deciduous trees, one and a half times the height of the tree is a recognised formula.

It is better to attack the problem at source by thinning out the canopy of the tree removing lower branches and undertaking root pruning, cutting the root system where it enters the lawn area.  None of this will affect a healthy tree or shrub adversely if carried out properly.

A lawn in shade requires regular reseeding with a shade tolerant grass seed mixtures and the use of wetting agents, fertilisers and soil conditioners. The use of wetting agents will assist in moisture retention and penetration on particularly dry areas.  Overseeding in the autumn with a grass seed mixture specifically for shady areas would be beneficial.  In addition aeration and top dressing should improve the situation.

It may be, however, that it is a one sided struggle and in this case it may be better to simply to abandon the area immediately around the tree and allow it to grow wild or putting down a layer of bark mulch.  Planting or sowing wild flowers which will thrive in very dry conditions or planting a range of bulbs to give you colour throughout the year are other alternatives.