ICAR-DMR says shiitake will grow on almost any size and age of log, then gives the working specification: 9 to 18 cm diameter, from trees 15 to 20 years old. The species named are broad leaved hardwoods, including oak, Castanopsis, Elaeocarpus, Lithocarpus, birch and Carpinus.
Age matters because a young tree has thin sapwood and an old one is often already rotting. Diameter matters because a thin log dries out and a thick one takes longer to colonise.