A pseudotree is a connected pseudoforest, i.e., an undirected connected graph that contains at most one graph cycle. Connected acyclic graphs (i.e., trees) are therefore pseudotrees.
Equivalently, a graph 
with edge count 
and vertex count 
is a pseudotree iff
it is connected and satisfies 
. For a connected graph,
the circuit rank is 
, so the bound says its circuit
rank is 0 or 1. A pseudotree is therefore either a tree
or a unicyclic graph. As a subclass of pseudoforests,
pseudotrees are matchstick graphs and hence are
also planar graphs and unit-distance
graphs.
Some care is needed when encountering pseudotrees as some authors use the term to mean "a pseudotree that is not a tree." Such graphs are perhaps better known as unicyclic graphs for clarity.
The numbers of pseudotrees on 1, 2, 3, ... vertices are 1, 1, 2, 4, 8, 19, 44, 112, ... (OEIS A005703), the first few of which are illustrated above.
