You may be presupposing facts that aren't really true- most of France was not heavily forested either before or during each world war. There were of course regions with more tree cover, most notably the Ardennes (which is mostly in Belgium), but by and large the norm was grassland or farmland.

Not sure if Scientific answers are allowed, but it helps answer the question.

In plants, a major fertilizer are various nitrates. Generally these nitrates come from either fertilizer, or microorganisms breaking down organic matter. The war not only produced a lot of dead bodies (a lot of which were buried or destroyed in the trenches), but also had massive amounts of artillery fire for extended periods of time (gunpowder requires nitrogen).

This new fertilizer, combined with the freshly tilled ground resulted in extra plant growth.

This was noted in William March's Company K

You can always tell an old battlefield where many have lost their lives. The next Spring the grass comes up greener and more luxuriant than on the surrounding countryside; the poppies are redder, the corn-flowers more blue. They grow over the field and down the sides of shell holes and lean, almost touching, across the abandoned trenches in a mass of color that ripples all day in the direction that the wind blows. They take the pits and scars out of the torn land and make it a sweet, sloping surface again. Take a wood, now, or a ravine: In a year’s time you could never guess the things which had taken place there.

I repeated my thoughts to my wife, but she said it was not difficult to understand about battlefields: The blood of the men killed on the field, and the bodies buried there, fertilize the ground and stimulate the growth of vegetation. That was all quite natural she said.