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Silvopasture is an ancient practice that integrates trees and pasture into a single system for raising livestock. Pastures with trees sequester five to 10 times as much carbon as those of the same size that are treeless while maintaining or increasing productivity and providing a suite of additional benefits. Livestock continue to emit the greenhouse gases methane and nitrous oxide, but these are more than offset by carbon sequestration, at least until soil carbon saturation is achieved.

Silvopasture also offer financial benefits for farmers and ranchers. Livestock, trees, and other forest products, such as nuts, fruit, and mushrooms, generate income on different time horizons. And help protect farmers from risk. The health and productivity of both animals and the land improve.

https://drawdown.org/solutions/silvopasture

Trees in silvopasture systems provide livestock with protection from sun and wind, which can increase animal comfort and improve production. Trees can provide shade in the summer and windbreaks in the winter, allowing livestock to moderate their own temperature. Heat stress in livestock has been associated with decreased feed intake, increased water intake, and negative effects on production, reproductive health, milk yields, fitness, and longevity.[4][5]

Certain tree types can also serve as fodder for livestock. Trees may produce fruit or nuts that can be eaten by livestock while still on the tree or after they have fallen. The leaves of trees may serve as forage as well, and silvopasture managers can utilize trees as forage by felling the tree so that it can be eaten by livestock, or by using coppicing or pollarding to encourage leaf growth where it is accessible to livestock.[1]

Well-managed silvopasture systems can produce as much forage as open-pasture systems under favorable circumstances. Silvopasture systems have also been observed to produce forage of higher nutritive quality than non-silvopasture forage under certain conditions. Increased forage availability has been observed in silvopasture systems compared to open-pasture systems under drought conditions, where the combination of shade from trees and water uptake from tree roots may reduce drought impacts.[1]

https://en.m.wikipedia.org/wiki/Silvopasture

I can tell you from my lawn that the grass under my Texas trees stays green longer and grows faster than the direct-sun grass. I think your hypothesis is worth exploring.