There is, in fact, an entire ANSI book on playgrounds.
ANSI-614, 100+ pages covering construction, installation,
inspection and maintenance. Materials, dimensions, (both
inclusionary and exclusionary), inspection requirements,
signage, it's all there with the requisite "should's" and
"shalls".
As it relates to trees, they are not mentioned specifically,
but would fall under the exclusionary dimensions for any
particular apparatus.
Trees and playgrounds can be a good mix if they are kept to
the perimeter as the depth of the playground surfacing must
be from 10" to 14" deep depending on the material, too deep
for the continued health of most trees. Alternately tree
wells could be created.
Northwind, CPRA CCPI