Question
We have an older Japanese Maple tree that gets full sun yet for this year and last, has a green mold growth on the bark of the tree including the branches.  I used a hose to get rid of most of it.

We just brought home two new trees (a Cleveland ornamental pear and a Japanese maple(Tamuke)and when planting them, I noticed that they both also have the mold on their bark but on a much more limited basis.

Is there some way I can remove this mold without harming the trees?  Thank you.

Answer
These sounds like lichens and are not a health problem for the tree.
Lichens are unusual creatures. A lichen is not a single organism the way most other living things are, but rather it is a combination of two organisms which live together intimately. Most of the lichen is composed of fungal filaments, but living among the filaments are algal cells, usually from a green alga or a cyanobacterium.

When lichen is wet from rain or dew, it grows actively. It also likes sunshine, and will grow well in winter after the leaves have fallen from deciduous trees and no longer block the light, or on trees with badly thinned canopies.

If you have lichen growing on the trunks or branches of your trees or shrubs, don't panic. Lichen will not actually harm your trees. That's because it takes its nourishment from the air rather than from its host. Lichens are often blamed for the decline and death of shrubs and trees because they are commonly found on dead branches and limbs. In actual fact, exposed limbs on damaged plants simply give lichens access to the sun they need for growth with little competition. There is no need to do anything, as lichen is not detrimental to the health of your plants.
The lichen will not effect the health of the tree.
If you feel you must do something and do not want the lichens on the trunk you might try this. No products are labeled for lichen control, but copper fungicides labeled for ball moss, or the home remedy of baking soda (sodium bicarbonate) for ball moss control (label pending) are detrimental. In pecan orchards, the fungicides used for scab and downy spot control reduce lichen numbers due to their effects on the fungal partners in lichens. However I would leave the lichens be.