New research shows, although they don't talk like we do, with audible noises, they do communicate, underground, via root networks that actually transmit electrical signals to other trees. So when one tree comes under attack from a tree disease, or damaging insect, it sends out signals to other trees, both to let them know that it is under attack from either a  tree disease or insects, and to get ready to defend themselves. Also they may send out signals that they may need help from their neighbors to fight off the attackers. They send these signals via a network of filament thin roots that travel at a rate of a third of inch per minute, compared to human pain signals that travel in milliseconds.

Scientists have found 500 year old tree stumps that are actually being kept alive by their neighboring trees, which are pumping nutrients (sugars, chorophylls) via fungal networks around root tips that help facilitate nutrient exchange between trees.  Stumps that would normally have completely rotted away to hummus, have been show to still have living cambium (the arterial network of the tree that moves nutrients up and down the tree).  So next time you are driving by a forest and you see a sick or dying tree, know that its neighbors are actively trying to help it recover from some kind of tree disease or insect attack,  or possibly drought stress.
There are other ways that trees communicate; emitting scents (gasses). and pheromones. In Africa, scientists have recently discovered that Acacia trees that are being eaten by giraffes emit a warning gas called ethylene, to warn nearby trees that they are under attack. Those trees immediately begin pumping toxins into their leaves that taste terrible to the giraffes.

There is similar tree behavior here in the US and in Europe. When under attack by some tree diseases or insects, many trees such Beech, Spruce and Oak,  produce compounds to help fend off the attacks, either by directly attacking the invader, or by attracting beneficial insects that will kill and eat the insects that are attacking the tree.

Incredibly, the trees are able to accurately determine which insect or tree disease is attacking it, by the  saliva of the insect. It can then emit the proper chemical pheromone to attract the correct beneficial insect.
So what do you think, do trees talk?