mikevan

Endomycorrhizal fungi is important for most plants and critical for a few. Others depend on Ectomycorrhizal fungi that form sheaths on the outside of the roots but those are mostly relegated to the likes of conifers and oaks and similar.

Compost is irrelevant to the fungi - it resides on plant roots. Compost is beneficial to the overall soil health tho. It's best to put it on top of the soil and let the soil life bring it down into the soil. As for mulch, I mulch with the purpose of making on-location compost, or sheet-composting. So I prefer smaller granulation that decomposes faster and just keep piling it on as it decomposes and is taken into the soil.

If your soil is a typical suburban yard, you may or may not have a decent colonization of endomycorrhizal fungi. Most suburbs nowadays is developed from old farmlands that have been blasted with phosphorous rich fertilizers. These fertilizers shocks mycorrhizal fungus to death so these soils can be very much devoid of this fungus and you may want to consider inoculating. The typical over-fertilized lawn can be similarly devoid of mycorrhizal fungi. If you're living on fairly virgin meadows and forest land, inoculating may not be necessary.

Inoculation involves exposing the roots to spores and hyphal fragments. The best way to inoculate for certain is to use prepared inoculas like what can be obtained from Fungi Perfecti or TandJEnterprises. Direct application via holes poked in around the plant is an option. Or vectoring it in with another plant can work too - clover and alfalfa can be inoculated and planted around the tree and their roots will form a network that will facilitate hyphal strands infecting the target roots. Both methods can be used simultaniously too. Soil composition does have an effect on how quickly this happens. Thick clay soil is slower to be colonized than loose, loamy soil.