By “sick tree,” I mean one that has something affecting it; a pathogen, either a disease or an insect. Diseases come in two varieties here: fungus and bacteria.
A tree with a pathogen may be perfectly healthy, but there is a good chance that the tree’s natural defenses have been weakened by water stress from too little water. See my blog post Symptoms of a dry tree. When a tree mounts a defense to a pathogen, it concentrates chemicals at the wound site. This takes energy and water, because the chemicals will have to be moved inside the trunk.
The sick tree will not look right. Perhaps a branch in a healthy section of the tree suddenly dies; the leaves turn brown for no apparent reason. Other symptoms might include a change in the colour of the leaves or the development of brown patches, leaving some of the leaf looking normal.
There are many changes pathogens bring to the tree. Usually each one has its own signature. Once you can recognize Fireblight on your apple tree, for example, it will look the same if you ever get it again. It is a really good idea to inspect your tree regularly throughout the growing season. Walk around, look up. How is it doing? You don’t have to know what is wrong with your tree; that’s an arborist’s job. What is paramount is that your inspection find the problem, the abnormality. Then make the call.