  Tree sets are abstract structures that can be used to model various
tree-shaped objects in combinatorics. Finite tree sets can be represented by
finite graph-theoretical trees. We extend this representation theory to
infinite tree sets.
  First we characterise those tree sets that can be represented by tree sets
arising from infinite trees; these are precisely those tree sets without a
chain of order type ${\omega+1}$. Then we introduce and study a topological
generalisation of infinite trees which can have limit edges, and show that
every infinite tree set can be represented by the tree set admitted by a
suitable such tree-like space.
