If the tree needs supplemental elements add those by deep root fertilization. Inject the fertilization below the competing turf plants directly to the feeder roots of the tree in a grid pattern from the drip line to the trunk. Deep root fertilization helps keep phosphorus runoff out of our lakes and streams. Remember phosphorus is reletively immobile once bound
in the soil.
for more info check out
http://www.extension.umn.edu/info-u/environment/BD288.html
I strongly disagree with your assessment.
When grass is growing in a tree's root zone, the two plants share the exact same soil.
You may be thinking grass roots grow from the surface to 3" down, and then tree roots grow from 3" down to 6'. That is just not how it works!
Most tree roots grow up! The main root grows outward, and at each fork the root tip takes the path of least resistance. If that path leads to soils that are low in oxygen, fertility, or water, it grows slowly or not at all. Those roots that hit soils with good growing conditions thrive.
Oxygen levels at the surface are the highest, and depending on the type of soil you have, after about a foot of depth, the O2 is just too low to support vigorous growth.
Because of drying cycles at the surface, those main roots tend to be down a few inches and then send up smaller roots toward the surface.
Almost all the fine feeder roots are very close to the surface. Interestingly, some grow up to, and even through the surface, only to do their thing and die back, all in one day.
Where are the grass roots? Unlike tree roots, grass roots mostly grow down! They too will grow toward favorable conditions, but are a bit more aggressive than tree roots. If you think for one minute that you can fertilize grass without fertilizing the tree, or the other way around, you have another think coming.
If you read your link, it is not the liquid fertilizer that is running into the lake, it's the erroded soil, plant parts, leaves and what not. Where you get liquid phosphorus into the lake water, is pumping it below the soil, into the loose gravel sub soil where it can't chemically bind.
Go to construction sites and study soil profiles, you'll see.