Is connected forest a spanning tree?
A connected forest is a tree (as well as a forest).

The terms spanning forest and spanning tree refer to an ambient graph. Given a connected graph $G$, a spanning tree is a subset of the edges of $G$ (spanning all vertices) which forms a tree. Similarly, if $G$ is a graph with $c$ connected components, then a spanning forest is a subset of the edges of $G$ (spanning all vertices) which is a disjoint union of $c$ trees.