i think i tried responding to this once before but this web site was experiencing difficulties and wouldn't let me post it!!!

Hostas..... reqiure light to full shade and can grow in zones 3 to 7. Hostas are adaptable to a wide range of sites and soils. Hostas are fairly drought tolerant, although they will not attain full size without even moisture and will not survive constanly in dry soil. Hostas will grow in heavy clay as well as constantly moist conditions.
U mentioned planting underneath a large tree. The one thought that would come to my mind would be competition between the tree and whatever u put under it. Generally, if u look at how far the tree limbs branch out, that is basically the spread of the tree's root system. I don't know anything about the tree u have.. maybe it's not as demanding. I know with my oaks in my front yard, the grass has a much tougher time underneath the tree. Those sections always dry out first. Not only is there competition for water, but also for nutrients. Let me give u an example.. (this is no lie) My wife's dad put a raised flower bed around a large oak tree. Would u believe the roots of that tree grew right up into that raised bed. He tore the raised bed out and it was a solid matt of tree roots!
One potential problem with hostas is slugs and snails. They chew holes in the leaves... It's like caviar to them! They do apparently have slug resistant varieties... can't name them though.
Also, voles, and moles relish hosta crowns and roots... but your cats ought to be able to handle that problem... ;-)

If u have good grass growing under that tree--maybe hostas would do just fine. These are just a few thoughts--hope i was of some help!