NP-hardness of existence of spanning tree with given maximal degree I am trying to solve the following exercise:

Let $G = (V,E)$ be a graph. Show that the following two problems are NP-hard:

$G$ has a spanning tree where every node has at most $k$ neighbors, and $k$ is part of the input.

$G$ has a spanning tree where every node has at most 5 neighbors.



It's written that 1 is supposed to be a hint for 2. You can of course choose to ignore it and solve 2 directly.
I tried to reduce $k$-MST to Steiner tree, but I don't know if this is a right way. I read that can use a Hamiltonian circuit, but I don't understand how. Can anybody help me?
Hint for part 1: What does it mean when $G$ has a spanning tree where every node has at most 2 neighbors?

Hint for part 2: Suppose you know that it is NP-hard to show that $G$ has a spanning tree where every node has at most 2 neighbors. Can you reduce this problem of at most 2 neighbors to the current problem of at most 5 neighbors?

This post is made out of user53923's comment as I have verified it is useful.