Prove the number of vertices of degree 1 in an tree must be greater than or equal to the maximum degree in the tree. Prove the number of vertices of degree 1 in an tree must be greater than or equal to the maximum degree in the tree.
-My professor stated this as a fact in class, and said that we could find the accompanying proof online. I've tried to find it but was unsuccessful, I was wondering if anyone knows where I could find it or even how to do the proof itself, so that I could better understand why this fact is true.

Take a spanning tree for the tree. Contract all edges that join two vertices of degree > 1 to a single point, resulting in one 'main' vertex and a whole lot of edges sticking out from it to all the verts of degree 1. 
That main vertex must have degree at least as large* as the greatest degree of any node in the original tree, but in the reduced tree, its degree is exactly the sum of the degrees of the degree-1 vertices. 


*

*proof of this: consider an edge-by-edge contraction. If we contract an edge PQ, the degrees at P and Q, $d_P$ and $d_Q$, and both at least 2, but the degree of the merged vertex is $d_P + d_Q - 2$ (the "2" being for the lost edge). Since $d_Q - 2 \ge 0$, this is at least as large as $d_P$; a similar argument holds for $d_Q$. 

