 Solution

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.