We know that trees with at least two vertices have at least...

We know that trees with at least two vertices have at least two leaves. But typically trees have more leaves than that. (a) Show that if a tree has a vertex of degree k, then it has at least k leaves. (b) Let T be a tree with n vertices, k leaves, and a vertex with degree k, where k>=2. Suppose that n>k+1. Prove that T has a vertex of degree two.

Transcript

00:01 Hey there, in this video will prove that a tree with no vertices of degree 2 must have at least half of its vertices be leaves.

00:07 To prove this, we'll use two facts.

00:09 First of all, if n is the number of edges, or sorry, number of vertices, and e is the number of edges, then in a tree, the number of edges has to be equal to the number of vertices minus 1.

00:28 The second fact is true for any graph.

00:31 Let's let v be the set of vertices.

00:34 Then the sum over all vertices and v of the degree of the vertex is going to be equal to twice the number of edges.

00:48 Where we remember the degree of a vertex is exactly the number of edges that are coming out of that particular vertex.

00:54 Putting these together, we can say that for a tree, the sum over all the vertices of the degree of the vertex has to be equal to 2 times n minus 2, where n here is the total number of vertices.

01:09 We can rewrite the term on the right hand side, actually, as the sum over all the vertices of 2 minus 2.

01:20 Because it just add 2 to itself the number of times i have vertices, i'm going to get 2 times the number of vertices.

01:27 So what i've got is the sum of the degrees is equal to the sum over 2 minus 2.

01:35 And we can rewrite this a little bit.

01:38 Further, just by bringing all the sums to the left -hand side, and i can say that the sum over all the vertices of the degree of the vertex minus 2 is equal to negative 2.

01:48 Now let's think about that sum on the left -hand side.

01:53 When v is a leaf, the degree of v is going to be exactly 1, because leaves are the terminal parts of the tree.

02:01 The degree of v can never be 2 by assumption.

02:05 So the terms on the right -hand side are the left -hand side of the sum are always going to negative 1 if i've got a leaf or greater than 1 if i don't have a leaf.

02:16 So if i let n l be the number of leaves, i can re -express the sum as the sum over all vertices, which are leaves, plus the sum over all vertices which are not leaves...

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