One of the best, easiest, and cheapest ways of improving tree health is by mulching around the tree. Contrary to popular belief, most of a trees root system is found in the top 6 inches of soil.
Roots need oxygen, nutrients, water, and a good growing medium. These necessities are found at shallow depths. Also, a trees root system extends at least to and many times beyond the drip line (diameter of the tree between the two widest points).
When trees are planted in landscapes containing turf grasses; the grass will out-compete the tree for available nutrients and water. In order to promote tree health one can mulch around the tree, covering the grass.
In the best case scenario the mulch ring should extend to the drip line, but this is not always aesthetically pleasing. Any mulch is better than no mulch though.
One does not need to dig up the grass. A nice three to four inch deep layer will effectively kill the grass and provide more nutrients to the tree. Organic mulches are preferred.
They can increase water retention, decrease soil temperature extremes, increase available nutrients, and also help to aerate the soil over time. They also stop lawn mowers and weed eaters from harming the trunks of trees or exposed roots by killing the grass and thus removing the need for them near the tree.
When mulching trees it is important to keep mulch out of direct contact with the trunk. Keep an inch of two of exposed ground just around the trunk. Then extend your ring as far out as you are willing to.
As mentioned before to the drip line is preferred. In well drained soils, keep the depth of the mulch around three or four inches. In poorly drained soils, keep the depth around two inches. Use fully decomposed organic mulches and replenish when the depth of the mulch decreases.