Maybe a Cornus, dogwood.

It is indeed Amelanchier, a common shrub or small tree in my area (western NC, USA) called Sarvice (Service Berry), and other varieties and species of it are called "Juneberry" or "Shadbush" outside of the mountains, as it blooms when the Shad run in the spring (early!) and fruits in June.

I too have a tree with red berries on it. It looks like a small blackberry, but is red. The leaves are almost heart-shaped, small with ridges along the sides of the leaf. It is more like an oak leaf but smaller. The berries are clustered on a pod like a blackberry and there is no core in the middle, just like a blackberry. What is it??

There are at least three possibilities, all in the holly family: Ilex ambigua (Carolina Holly), Ilex decidua (Possomhaw), and Ilex montana (Mountain Winterberry) If the trees still have leaves, try to get a good closeup picture of them and the berries and post them here. It will help in identifying it. You might try calling your local cooperative extension office and asking them. Somebody there may be familiar with the specific trees and know what they are. If no answer is forthcoming, wait till spring and get a sample of one and take it to a GOOD full service nursery or to the extension office. That should get an answer for you. Sandy