So you get heavy rain on a wooded hillside with little or no run off. Heavy rain on a grassed hillside (or even worse, bare earth), streams of muddy water running into whatever stream or river sits at the bottom of the hill. I have seen a veritable waterfall running off our local fields into the stream.
Why? Let's think about it. Trees, large green things, lots of branches and leaves, large roots leading deep into the soil, leaf mulch underfoot. What happens when rain hits it? Well, we have a Leylandii in our garden, quite a big one. Only in the heaviest rain do you get wet beneath its branches. There is a huge surface area of leaves and branches that needs to get wetted before water starts to hit the ground beneath. Otherwise it just drips gently down to water the tree. The effect is the same in any woodland. A large surface area of plant matter to hold water before it hits the ground, slowing run off. When it does hit the ground we have deep soft leaf mulch to absorb water, and roots going deep into the earth, channelling that water back into the water table to feed springs (and tree roots) when rain is sparse.
What happens when heavy rain hits grass, or other crop land? Well, we have a small amount of surface area in the vegetation, so water is almost instantly hitting the earth. The earth is often relatively hard packed from the weight of vehicles and animals, and it is earth, not leaf mulch, so slower at absorbing water. No deep roots to channel the water downwards, so once the first inch or two of surface earth is soaked, we start to get run off. Clearly, on sloping ground, this can happen fairly quickly, and if the ground is already wet, even more so.
So trees reduce run off, farmed land increases run off. Fast run off moves water downstream more rapidly. So let's pay our hill farmers to clear land and breed sheep, and farm the hills as intensively as they can, causing faster run off, and sending the water downstream more rapidly, and in greater quantity.
So we push the water down into the lowland more rapidly, causing flooding on the river's flood plain (the clue is in the name). But here we find more farmers paid to farm the land as intensively as they can. Having all that land underwater can't be good, so they insist on the dredging of the river to move water through even faster. And now what? Towns and cities further downstream are flooded and cry out for improved flood defences. Other tax payers in areas that don't flood shout about not spending their taxes to support the few flood victims (wonder if they would feel the same about other tax payers not paying for policing of high crime areas) and more and more money gets spent.
So, we are spending money to clear trees and encourage intensive farming, and then spending money on dredging and flood defences to cope with the problem we have caused.
Simple answer? Recognise the true value of trees to our whole economy and ecology. Pay farmers to keep them rather than get rid of them. Apply some logic and common sense to the "Common Agricultural Policy". Will it happen? Not a chance. We don't have a politician in this country that can see beyond the next headline. "More money on flood defences" and "more money for hill farmers" make better headlines than "plant more trees". Sad.