I will preface my question with the definition of a simple tree that applies to my question:
A simple tree is an undirected and connected graph with no cycles.
I am having difficulty coming up with a compelling argument that a tree of the above definition can be constructed by taking a set of disjoint valid graphs and adding a vertex between each of them. The practice problem states:
Prove that some graph $G$ is a tree if and only if it can be constructed from a set $X_1, X_2, \dots, X_i$ of disjoint valid graphs by adding a new node adjacent to one node from each of $X_1,\dots, X_i$. A single node is a valid graph.
I'm confused whether this question is asking me to prove that two graphs, or trees ("valid"), can be made into one tree by adding a node and adding edges from one preexisting node from each of the trees. If that is how the question should be interpreted, then is it enough to prove that the new tree (or "graph," I guess) has no cycles, and is connected?
I'm new to graph theory and I'm wondering if you can help me understand it.