How big are the root systems of a beech and an oak?

Answer

In general you can say that the root system of a tree extends at least as far as its crown. Once you’re under the branches, you’re standing on the roots. However, this is just a rough rule of thumb.
The majority of the roots of most plants are in the upper layers of the soil. Nutrient roots are formed here that are responsible for the absorption of mineral nutrients and water.
These nutrient roots consist of large networks of very finely branched roots that find their way between the soil particles. Each root branches into several smaller roots, which in turn branch out and so on. So many roots are formed on a fairly limited volume of soil. Root hairs develop at the end of the finest roots. These are outgrowths of the outer cell layer of the fine roots. The plant absorbs substances from the soil via the surface of these root hairs.
Every root, every fine root and every root hair naturally has a length and a volume. The surface area of ​​the root hairs can also be calculated. In theory, you could measure all of this to determine the “size” of a root system. In practice this is of course quite difficult, especially for mature trees such as oak or beech.
For comparison; the entire root system of a four-month-old grain plant has a length of many hundreds of kilometers. The total surface with which this plant absorbs substances from the soil is more than 400 m². Thus, how root system of an oak several centuries old will be impressively large.