What would a modern landscape look like with the absence of lignin decomposing fungi, bacteria, and etc.
What I mean is...would a forested area just become filled with fallen trees instead of a more "open" forest floor.
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This condition actually existed on Earth over 300 million years ago. Read here
Effectively, the trees would grow, live, die and they fall. They would pile upon each other. They would pile so high that their corpses would compress into coal. The trees had to struggle to find any bit of earth they can root to and grow high to compete for sunlight.
In time, the wind would blow dust over these piled up trees and bury them.
This condition won't last long (time scale of millions of years) as something will eventually evolve to fill this niche, in which there is an abundance of food to be had.
According to Willk in the absence of natural decomposers, such as fungi or bacteria, dead organic material would be broken down by the UV rays from the sun. Willk referenced dry valleys in Antartica where seals are mummified and stated that:
The mummies weather in the elements. UV radiation breaks down tissue. Wind and windblown grit wear it away. It is much like what happens to wood left outside in a dry environment.
In terms of your question, if a tree fell over, it would not decompose. Instead, UV rays from the sun would start to break down the tree and wind, rain and frost would break apart the wood, essentially causing it to weather. As it is no longer alive, it can’t repair or protect its self from the damage. Without natural decomposers, the process would take a lot longer to fully break down a tree in this way, possibly leading to a reduced number of ‘open floor’ spaces, as you pointed out in your question.
Acid rain may help to break down these fallen trees much faster than they otherwise would be. Much in the same way limestone structures are broken down slowly by ‘normal’ rain but far faster in acidic rain.
If you wanted to have more open spaces in a forrest without natural decomposers, you could have ‘territorial’ trees which purposely undercut other trees. By this i mean they grow their roots rapidly, taking water and nutrients from the imediate area around a ‘rival’ tree. The result would be large sections of forrest which contain just a single tree isolated in an open area.
To further reinforce this idea of ‘territorial’ trees, you could have them release a toxin under the roots of other trees or saplings, killing them off and preventing them from growing. To avoid large areas of dead land, perhaps the grass around the tree is not seen as a rival or has a symbiotic relationship with it.