To the best of my knowledge I believe all the tree is good, but the best part is the sapwood layer just below the bark, it will have the most nutrients. See if you can get some mulch off one of those chippers when you see crews trimming hardwoods. Depending on what you want to grow, an outdoor patch of mulch may work, otherwise sterilized sawdust blocks can be used. Another option is to get small logs (3-4 feet by 4-6 inch diameter) & innoculate them with spawn plugs. there are a few pics of the logs I knocked up with shitake on this page.
fungusfun.com/mushrooms/modules.php?...clude=view_album.php
Cheers
C1