Fortunately, the tree should not need any fertilizer, as mature trees typically have all the nutrients they need in the root zone. You can have a laboratory soil test performed for the soil in its root zone if you have cause for concern about nutrient deficiencies or improper pH (neither of which appears to be the case in the photos), and we can help with results interpretation, but that is probably not necessary. Plus, if any of the tree's roots infiltrate a lawn (either yours or a neighbor's) which is fertilized yearly, they are also able to access some of those nutrients.

Miri

Whether you overseed the lawn (recommended each autumn) or lay sod on bare areas (the area around the tree looks quite bare), you can use the guidance on our Lawn Fertilizer Schedule chart to determine what to apply, as it will depend on lawn type (tall fescue, zoysia, etc.) and the time of year. Autumn is when cool-season turf like fescues are seeded, since zoysia is planted much earlier in the year and is cared for differently. Fertilizer choice also depends on the results of a laboratory soil test, which is used to determine if the soil is deficient in any particular nutrients, such as phosphorus (P). Maryland's lawn fertilizer law prohibits the use of a fertilizer formulation that contains P unless the lawn is being planted new (renovated) or unless a soil test shows a P deficiency.

Be aware that most turfgrasses do not fare well under the canopy of a mature tree, especially when they also compete with its roots. If you intend to grow lawn in the area pictured, expect that is might start out lush but will decline in the coming years due to those stress factors, even with fertilization. Tall fescue and fine fescue are the most shade-tolerant of the turf types used in Maryland, but they are not able to thrive long-term in semi-shade, so consider using other types of groundcovers instead in areas close to a tree, or just a layer of mulch.

Miri