Posted: 23/09/2014 at 10:34
I suspect that what is wanted is a tree in shape and form so the short answer is no, neither of these trees is suitable for growing in pots as they get far too big and coppicing would make it just a big shrub. Eucalyptus are far too vigorous and birch can bleed to death if pruned at the wrong time or too heavily.
Since your garden is small, you definitey don't want tree roots in the soil where they will expand to seek nutrients and moisture and eventually disrupt either foundations or drain and swer pipes or all of the above. Search instead for a small tree which can be container grown, something like an apple or pear on dwarf rooting stock will fit the bill and give you fruit too if there's a pollinator nearby. The RHS suggests these trees - https://www.rhs.org.uk/advice/profile?PID=274
You will be entirley responsible for all its needs in terms of good soil, nutrients and watering so use the biggest container you can find (minimum 60cms deep and wide) and use top quality John Innes no 3 compost in which to anchor its roots. Give a top dressing of pelleted chicken manure or blood, fish and bone every spring and regumar feeds of liquid tomato food through the growing season.