Prove that removing k − 1 edges from a tree T results in a graph which has k components, by induction

Question

T is a tree with an amount of components, I need to prove this by induction. The base case $k=1$ works as removing $1-1=0$ edges keeps the number of components the same.

Answer 1

Since you want to do it with induction:

Base Case: k = 1. Clear.

Step: k-2 -> k-1

Remove k-2 edges from the Tree, by Induction hypothesis, you get k-1 components. Now each of this components is a tree itself. Now remove one edge from one of those trees. Again by induction hypothesis, you get 2 components, so all in all you get k components if you remove k-1 edges.