Final answer:
A tree contains at least Δ(T) leaves
Step-by-step explanation:
A tree is a connected graph with no cycles. The degree of a vertex in a tree represents the number of edges incident to that vertex. The maximum degree of any vertex in a tree is denoted by Δ(T).
To show that if T is a tree, then T contains at least Δ(T) leaves, we can use the fact that in any connected graph, the number of edges is one less than the number of vertices.
Let's assume that T contains less than Δ(T) leaves. In that case, the sum of the degrees of all vertices in T would be less than 2*(Δ(T)). However, the sum of the degrees of all vertices in T is always equal to 2 times the number of edges in T. Since T is a tree, it has (n-1) edges, where n is the number of vertices. Therefore, we have a contradiction. Hence, T must contain at least Δ(T) leaves.