Enumeration
A graph is simple if it has no self-loops and no multiple edges with the same endpoints. The number of simple 1-trees with n labelled vertices is
The values for n up to 18 can be found in sequence A057500 of the On-Line Encyclopedia of Integer Sequences.
The number of maximal directed pseudoforests on n vertices, allowing self-loops, is nn, because for each vertex there are n possible endpoints for the outgoing edge. André Joyal used this fact to provide a bijective proof of Cayley's formula, that the number of undirected trees on n nodes is nn − 2, by finding a bijection between maximal directed pseudoforests and undirected trees with two distinguished nodes. If self-loops are not allowed, the number of maximal directed pseudoforests is instead (n − 1)n.
Read more about this topic: Pseudoforest