How many edges has a tree with n vertices Provide an argumen

How many edges has a tree with n vertices? Provide an argument for your claim.

Solution

A gruop of n vertices has n-1 edges. This can be proved by induction.

Base case: For n=1, if there is one vertex, so there are no edges. Which is same as n-1.

Let us assume it is true for n vertices. So it has n-1 edges. Now we have n+1 vertices, as we already assumed for n vertices there are n-1 edges. If we have to add one more vertex we need to add atleast an edge with one of the already existing n vertices. We cannot create another edge with the new vertex as it would create a cylce. So now in the tree with n+1 vertices there are n edges. So the assertion is also true for n+1.

Hence the assertion is proved.