Question
I have an 40year old linden that has developed small holes at the bottom of its trunk under which small mounds of sawdust are accumelating.I also noticed small insects coming in and out of those small holes. Any ideas? Thank you.

Answer
Sounds like Linden borer. Linden borer, Saperda vestita.
Small-sized linden trees are often injured near the base by the larva of this insect. It eats out quite a large cavity, decay follows, and the tree soon breaks over. The larva is white and at maturity is about an inch long. It usually burrows near the ground, sometimes going into the roots below the surface. There is usually a vertical gallery next to the bark and a cavity toward the center of the tree. The adult is a longhorned beetle about 3/4" long, black, and densely covered with flattened hairs that are light olive-green. Usually four small black spots are present just below the center of the wing covers. The beetles emerge in late summer and feed on the leaves and tender shoots before laying eggs in the bark.
Spray the area of the trunk with an insecticide called Onyx or Merit. These products will soak into the trunk and kill the insects in the galleries under the bark and in the wood.  Check with your local garden type store for these products. Other insecticides will also work but not as well as the Onyx or Merit. here is a web link to more information on linden borer. http://www.entomology.umn.edu/cues/Web/164LindenBorer.pdf