On trees older than 10,000 years...
It depends what you mean by "tree". The bristlecone pines are often cited as the oldest living trees. They are up to 4,700 years old.
On the other hand, some trees can be much older than this, in cases where the organism sends out repeated shoots, rather than living with a single trunk for all its life. Whether this represents one individual or not is debatable. Scientists mostly seem to regard these examples as legitimately one individual, from one continuously living root system.
In any case the age itself is not disputed by such questions.
For a long time (from about 1980 until fairly recently) the oldest individual was considered to be "King Clone", a creosote bush in the Mojave Desert, which is around 11,700 years old. See
Creosote Bush (Larrea divaricata or tridentata) and many other similar pages on-line. This picture (from an amateur
photo tour) may not be the King Clone itself (which is not marked, for conservation reasons I guess) but it gives an idea of what they look like.
More recently, a new record breaker is an individual of "King's lomatia" in Tasmania, which is around 43,000 years old. See
King's lomatia - the Oldest Plant Clone in the World?, and many similar pages on the web.
Cheers -- Sylas