And since trees are the biggest plants we have, they tend to last quite a long time after they die. In fact, they are an integral part of their ecosystems, even after they stop photosynthesizing.
They provide food directly for decomposers, an indirectly for birds and other creatures that eat the decomposers. As the parts rot away, the nutrients held within those parts go back into the food cycle, feeding young plants - maybe even a tree's own seed.
Dead trees also provide habitat for insects, birds and mammals. Insects burrow under the bark, or even into the decaying wood. Birds can still build nests in the branches until they break off, and cavity nesters (birds that nest in holes) excavate nests, or take over nests excavated by others. Squirrels and blue jays, among others, will store nuts in holes in trees. Possums will sleep away chilly winter days in large tree holes stuffed with leaves and other warmth-conserving materials. Fishers will raise families in tree cavities.
And dead trees can give us interesting images to consider as well. On my walk this afternoon, I noticed a tree that was clearly dead, and rotted around the bottom. It looks like any slight breeze could knock it over. I went over and gently pushed on the trunk. It is still pretty stable - I'm sure a strong wind could take it down, but my nudging barely made it move.
After that, I kept noticing dead trees. Here are pics of several that were interesting in one way or another....
too decayed to identify
same pine, up close
At this time of year, many trees look "dead" with all their leaves gone, but if you look closely at the twig tips you will see the buds of this coming year's growth. Those buds were formed as last summer was ending, and they sit there all winter, waiting for warmth and light to release them. The flowers and leaves will emerge and the living tree will continue to grow.