In the picture below taken after a rain, the beech shows a pattern of moisture about 3 feet up the tree. None of my other trees, including beeches (you can see another one on the very left of the picture) is showing the same pattern. I have recently had two trees fall in that area because the bottom rotted out. I'm wondering if this moisture pattern is indicating that this tree is also rotting at the bottom.
First, from a distance, I would look at the canopy (top) of the tree and see if it is dying or thinning. If the tree is rotting at the base then it will surely show signs at the top as the tree is deprived of nourishment. If you see dramatic thinning then time to remove the tree. End of story.
If you do not see a lot of thinning, then approach the tree with care and see what the soil around the tree is like. If it is squishy and persistently wet It is possible the tree is sitting in more water than it would like, and the priority is to find out why so much water and take action to get the water table down where it normally would be.
Take a pointed tool like a pocket knife or screwdriver and push the tip gently into the dry bark first to get a feel for how resistant the normal wood is, then compare that to the same action on the wet bark. See if the wet areas shed bark easily - if it does then this is definitely not normal for beech trees. Compare the dry areas to the wet to see if the basic woody stuff is the same, as it should be.
Edit: pay particular attention to the apex of the triangular stains. It may be that sap is escaping from the tree and flowing down in a spreading pattern due to bark irregularities. Beech can suffer from bacterial wetwood.
If you see anything odd that is not easily explained, take some supplemental photos of the issue and update the question, if you please.
