Prove that a graph $G$ is a forest if and only if every induced subgraph of $G$ contain a vertex of degree at most $1$
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Let $G$ be a forest. Then $g$ is the collection of a bunch of trees. For the tree there at least one vertex of degree $0$ or $1$, so every induced subgraph of $G$ contain a vertex of degree at most $1$
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Assume that every induced subgraph of $G$ contain a vertex of degree at most $1$. I want to show that these subgraph are tree, meaning I want to show that non of them contain any cycle and all of them are connected graph. But I'm not sure how.