Source: patkgreen fom a reddit post 14 years ago answering the question "do trees die of old age?"

The answer, of course, is sort of. Like another poster said, old age is technically a secondary cause. What happens when a tree gets old is that it can't protect most of its inner heartwood or exterior. It certainly does not fight damage as well.

Woody plants heal through CODIT (http://en.wikipedia.org/wiki/Compartmentalization_of_decay_in_trees) in four steps, called walls. This is originally from a study where a guy named Shigo axed a bunch of trees and watched them heal.

The four walls are explained in the wiki, but I'll simplify it.

Wall 1: the tree prevents the wound/infection from spreading up and down the tree by blocking the transport tubes on either side of the wound.

Wall 2: the tree prevents the wound/infection from spreading inward by building up a ring to act as a barrier

Wall 3: the tree prevents the wound/infection from spreading circularly by sectioning the wound laterally. This basically means the wound/infection can't spread left or right. After wall 3, the would should look like a piece of pie.

Wall 4: the new growth rings seal off the wound/infection so it doesn't spread outward, as the trunk grows.

So in old age, the trees lose the ability to compartmentalize their wounds and eventually die. When wall 4 fails, you'll find a standing tree with a hollow inside- it's not uncommon to see them with leaves still. But unless wall 4 has succeeded just in the nick of time, the tree will die soon.

Another way trees die is they girdle themselves because they get too big or something that is sucked up by the roots get caught in the xylem and phloem. Think of it as getting a piece of food caught in your throat. It slowly chokes and dies.

Girdling is also a cause of death from insects and beavers. Most animals and insects only eat the newest part of the tree, where the xylem and phloem channels are active. This area, just under the bark, is called the cambium. When this cambium is eaten, the tree has no way to pass water/food/nutrients up and down the tree, essentially starving.

When a tree is old enough for the bark to start falling off simple weathering is enough to damage the flow of food/water/nutrients and that is as close to dying of old age as a tree can get, in my experience.

Lastly, some trees literally have no ceiling of growth (redwood, white pine) except for when the atmospheric pressure prevents the tree from getting its food/nutrients/water to the roots or leaves. In this case, they eventually die because they weaken and can no longer transport at a functional rate.

As an edit: when trees get unhealthy, you'll see weird sucker sprouts along a normally bare stem, or lots of leaves in clusters in weird spots.

source: professional ecologist