I saw this video, and I'm curious what color bark would be if the sun was in one position constantly.
The video explains that birch trees are white because of stuff in their bark. The reason is because the sun where birches are usually is only in one direction, but makes a small arc in the sky. The white color helps the tree avoid temperature differences across its surface.
The bark would be any color. If the sun was always in the same position, plants would evolve to catch all the delicious light for photosynthesis with their leaf analogues so no significant radiation would hit the trunk. Or, maybe simpler way to think about it, if there was significant radiation hitting the trunk, the trees would be under pressure to evolve a way to catch it for photosynthesis.
The difference with birch is that the heat is transient. It lasts long enough to cause damage but would have little value as extra energy due to its transient nature and low angle.
With a tidally locked planet the radiation you get is the radiation you get. There is nothing transient about it and the angle it has is the angle it has. You either use it effectively or you give up on photosynthesis and become a parasitic plant (aka animal?) instead.
An interesting question is what the leaf analogues would like. For radiation a flatter surface would be nice but you also need to think about gas exchange and cooling. Maybe different climates would drive different solutions? Well, it is a separate question, so I'll not speculate.
If the sun doesn't move in the sky, then there's no danger of temperature gradients like the ones shown in the video -- there's no warming and cooling every day. Trees on a stuck-sun world would face a constant heat that maybe damped slightly by a passing cloud from time-to-time. That indicates to me that the bark could be just about any color depending upon what it wants to do. If it wants to be highly heat-repellant so as to not over heat, it will get whiter bark. If it wants to absorb energy in its branches, it will have blacker bark (the tree might use heat in its branches to warm its roots far underground).
Usually the darker shades get more attraction and as a result more energy because they reflect less solar energy, so the side of the planet that gets more fun-in-the-sun will probably have a lighter shade. This is due to them not needing to evolve a sun catching color averse to the side of the planet that gets less sun time; the side that would have a darker shade of tree bark.
Earth has many different shades of trees due to how populated the forest are and because the Earth has a slow rotation on its axis. Thus less populated forest would have a lighter shade of bark and more populated forest would have a darker shade of bark.
