# Is there a tree with 7 vertices: With 5 vertices having degree 1 and 2 vertices having degree 2?

Is there a tree with seven vertices:
a) With five vertices having degree 1 and two vertices having degree 2? b) With vertices having degrees 2,2,2,3,1,1,1?

So I believe the degree is the number of children vertices there are from the vertex? For a) I believe the answer is yes because I was able to draw a tree as such but for b) I believe the answer is no because I could not conceptualize a tree like this.

I may be taking the wrong definition of degree though or might be thinking about "rooted" trees only. Thank you for any help clarifying this concept and if there is a formula/ theorem for this.

• Recheck your answer to (a)... – Casteels Mar 8 '18 at 19:41
• Do you know the handshaking lemma? – Kevin Long Mar 8 '18 at 19:53

HINT

Since every edge will connect exactly two vertices, every edge you add to a graph will increase the sum total of all degrees of all vertices in the graph by $2$. So ... the sum total should always be what kind of number?

So, count a little harder for whatever you drew for a) ...

Also, try a little harder for b) ...

For (b) start drawing from the vertices of degree 1, since the look of the graph at those vertices is unique.

You can probably get to this example:

.-.-.-.-.<: