Have you ever noticed how the grass growing underneath trees in a savanna remains green while the grass growing out in the open turns brown in a mild drought. It's not just because the grass is growing in the shade of the trees. Tree roots pumpwater up from deep underground 24 hours a day. During the day the tree's leaves are using this water, but at night the leaves don't need this water so the roots release the water pumped up from deep below into the top soil around the tree. The next day the tree's roots would reabsorb some of this water for use in its leaves, but this water is also available for any grass growing in the tree's root zone to use. So a pasture set up as a savanna uses water from a wider range of soil depths and is more drought resistant than a pasture with no trees growing in it.
Grass may not be the best thing to design under a tree, but trees definitely can add to the moisture content of soil simply by condensing moisture out of the air and dripping onto the surface. There are some places on earth where it never rains, but still have sufficient moisture due to this continuous condensation. Some of the Canary Islands fall into this category and the natives there dug trenches filled with stones to collect the condensation from their "rain tree".
(Ocotea foetens, commonly called til, tilo, stinkwood, or rain tree of Hierro island, is a species of tree in the family Lauraceae. It is evergreen and grows up to 40 m tall. It is a common constituent of the laurisilva forests of Madeira and the Canary Islands.)
I believe average contribution of moisture to the soil from condensation is set at 40% of total available moisture, although this can be quite variable
I think there could be a couple more variables at play here. Perhaps years and even decades of leaf fall from this tree has increased the organic matter in a localized area of soil. The more organic matter in a soil, the more water a soil can absorb and hold onto. Maybe the symbiotic relationship between the soil fungi and the tree roots are also aiding the grass roots to reach water in the soil that is not in immediate contact with the grass roots.
I have clusters of trees amid the pastures on my farm, and I see something similar to what Mike is talking about in the opening post. Spreading out about 15 or 20 feet from the tree clusters is much more grass than compared to the dominant broomsage on my pasture at the time this photo was taken earlier this year.
"Study books and observe nature; if they do not agree, throw away the books." ~ William A. Albrecht
It's not just trees. The grass seems to thrive in the light/shade/light cycle occurring all day, instead of one long dark cycle and one light cycle. You'll see it under signs or other overhangs as well. Though I do admit, the tree as water source and humus supplier probably intensifies this effect. That is why silvopasturing works well, as the grass benefits from this booster effect, and the animals benefit from a ready shade source and supplemental forage.
Always look on the bright side of life. At least this ad is really tiny: