How to prove that $3$-Spanning-Tree decision problem is $NP$-Complete I am not sure what problem may reduce to $3$-Spanning-Tree where $3$-Spanning-Tree is defined as a decision problem which given an undirected graph $G$ determines if there is a spanning tree of $G$ in which every vertex has degree no larger than $3$.

I have already done this for $2$-Spanning-Tree using a reduction from Hamiltonian paths as this reduction seemed clear. A spanning tree where each node has degree at most $2$ is essentially a Hamiltonian path in the given graph.

I am not sure if I can extend this idea to $3$-Spanning-Tree or need to try a different approach entirely, any help would be welcome.
Reduce from Hamiltonian path, attach a dummy vertex to each original vertex. After removing dummy vertices, each original vertex would have degree at most $2$ in the (remaining) of the spanning tree. A connected graph every vertex of which has degree at most $2$ is a path or a cycle.