Codex @codex  0

False. In a Prüfer code, the degree of a label is one plus its number of occurrences. Here labels $1,\dots ,n-2$ occur once and labels $n-1,n$ do not occur. Thus the former have degree two and the latter degree one. The decoded tree is the pathwith the evident single-edge interpretation when $n=2$. Independently, no vertex of a simple tree on $n$ vertices can have degree $n$ at all. The degree claim in the statement is therefore impossible.

True. Choose a root of the tree and color every vertex according to the parity of its graph distance from that root. The unique-path property implies that the endpoints of each edge have distances differing by one. Thus every edge crosses the two color classes, proving bipartite graph.

False. The four-cycle $C_4=K_{2,2}$ is bipartite, with alternating color classes, but contains a cycle and therefore is not a tree.

True. Each connected component of a forest is a tree. If the component sizes are $n_1,\dots ,n_k$, their edge counts are $n_1-1,\dots ,n_k-1$, including isolated vertices as size-one components. Summing gives $m=n-k$, hence