If you faced with poorly draining clay soil, you may have a difficult time growing trees.  Soil that doesn't drain well and has periods of standing water isn't very accommodating to most trees.  But there are varieties that can handle these types of conditions. Here are just a few that you might want to consider.

Birch trees would be a good choice for these types of soil conditions.  There are several species of birch to choose from, all that like moist soils.  The River Birch (Betula nigra), Heritage Birch (Betula nigra 'Heritage'), Paper Birch (Betula papyrifera), and Japanese White Birch (Betula platyphylla) are great, just to name a few.

Cypress trees are another type of tree that grow well in moist conditions.  They can even grow in swamp like conditions.  Generally listed as fast growers (depending on the growing conditions), there are a few choices from this genus.  The Bald Cypress (Taxodium distichum) and the Pond Cypress (Taxodium ascendens T. dustichum var. nutans) are two popular choices.

If you are still looking for more variety, take a look at willows, poplars, and cottonwoods.  For willow trees, you could go with the Weeping Willow (Salix babylonica), the Corkscrew Willow (Salix matsudana 'Tortuosa'), and the Navajo Peking Willow (Salix matsudana 'Navajo').  As far as poplar trees go, the Lombardy Poplar (Populus nigra 'Italica') and Theves Poplar (Populus nigra 'Afghanica') are popular.  For cottonwood trees, you may want to check out the Cottonwood Siouxland (Populus deltoides 'Siouxland').

So if you are faced with poor draining soil conditions, there are still several tree options to choose from.  Here are just a few.  You might have to be a little more particular about which varieties that you select, but there are several that can still thrive.