Basic implications and equivalences. Additional implications and equivalences. D. Necessity of large cardinals. We begin with the concrete representation of finite trees that is used throughout the paper. A tree is said to be trivial if and only if it has exactly one vertex, which must be its root.
Every vertex has at most one parent.
However, vertices may have zero or more children. This is a partial ordering on trees. Let N be the set of all nonnegative integers. Thus there is a special treatment of roots in k-trees. There is exactly one trivial k-tree, which consists solely of the root infinity.