Most of the cultivated species we grow are saprophytic, eating dead wood, like the Oyster Mushroom. If one were to inoculate a living tree with these, they might act as parasites, slowly eating the tree. That's why we typically inoculate cut logs and stumps with these kinds of mushrooms.
However, most edible mushrooms out there are not cultivated because they are symbiotic - that is, their mycelium lives in tree roots, and furnishes the trees with nutrients in return for sugars. This makes cultivation too difficult and unpredictable for commercial growers.
Some species of mushrooms will only pair up with trees of a certain kind (A Birch Bolete is typically only found under Birch trees, for instance). Some trees will support multiple species of mushrooms. (Douglas Firs are a good example - the older the tree, the more kinds of mushrooms one will find growing under them.)
Most mushroom field guides will have observations of pairings found in nature. When you find mushrooms in the wild, note which kinds of trees they are growing with, and note the other plants growing nearby. Grass? Other green plants? Is there a lot of leaf litter? Woody debris? Make a note of the surroundings, and think how you can duplicate the circumstances near your home. Mushrooms that like leaf litter may not grow well in grass (or the reverse may be true).
While there are no proven methods, it is believed that simply laying the ripe caps of compatible species of mushrooms under the proper host tree so they may release their spores there may be enough to establish a fungal symbiosis. It is certainly so simple a technique that there is no danger or harm or even much effort involved in trying it. Another thing to do would be to collect spores and make a spore slurry, in which the roots of tree seedlings/saplings would be dipped.
In any case, establishing symbiotic mushrooms will take several years to bearfruit, just as with planting a tree it will be several years before you get fruit. But the results would certainly be well worth it.
I read somewhere about how some researchers were trying to re-establish native prairie species in lands damaged by heavy farming over the past century. They weren't having much luck until they had the brainstorm to transplant soil from other healthy prairie ecosystems to their test plot. The transplanted soil was full of mycelium and other soil biota that proved essential to the health of the native plants.
Alder trees are great for most species. Paul Stamets says these are his tree of choice for cultivation of wood loving species.
There are symbiotic species like Morels that do well with Box Elder and Elm but there again, they are somewhat unpredictable for fruiting and require time to establish themselves.
If you want to knock up trees that are dead or dying, you can almost guarantee good results unless they are pine or other sap-filled trees. Oyster mushrooms fly in colonizing almost anything you put them in.
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