Are you wanting a flowering shrub, an evergreen or what. In my experience it isn`t that competive so that about anything will do well near it. River Birch likes plenty of moisture so I would plant shrubs that like plenty of moisture as well but are short enough to not cover the trunk of the birch.

I think I would like to use evergreen shrubs...got any suggestions ore experience with them?
Thanks for your help!
Pam

'Sunset Western Garden Book 1997' suggests these plants for Under Oaks. They also say this: "...The following list will help you select plants that don't deman lots of water, so are suitable for planting beneath these grand trees...A few cautions: Do not plant, irrigate, or disturb the soil within 10 feet of the trunk, and avoid injuring the roots. Keep lawn outside the dripline. Plant sparingly; use drip irrigation or soaker hoses, not sprinklers. Avoid planting under declinng (oaks)."

In my opinion the same would apply for any (mature specimen). Whether or not your Birch is mature or not, still follow these guidelines while you plant.

Shrubs:
Berberis darwinii
Carpenteria californica
Ceanothus ovatus
Erigonum
Holodiscus discolor
Mahonia
Rhamnus (most)
Ribes (drout-tolerant) "Currant, Gooseberry"

Ground Covers:
Cotoneaster
Juniperus
Mahonia repens

Perennials:
Achillea tomentosa
Artemisia
Hemerocallis
Heuchera
Kniphofia uvaria
Nepeta faassenii
Penstemon (some)
Salvia (most)
Santolina chamaecyparissus
Sedum (most)
Thymus

I'm sure Spirea would be another good one to add. As would Lavender.

I believe all the plants listed above will be hardy in zone 5. It would be best that you doublechecked though as I'm just thumbing through their selections and since the book was written for the West Coast and has their own zonal system.

As Jeff mentions above, if your tree will be in a heavily irrigated space, go with shrubs that like water.

Dax