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# Prove that a tree with Delta(T)=k (Delta means maximum degree) has at least k vertices of degree 1.

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I don't understand how you count the degree of the vertices.

(See attached file for full problem description)

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2.- Prove that a tree with Delta(T)=k ( Delta means maximum degree) has at least k vertices of degree 1.

Proof. We prove it by contradiction. Suppose that and there are s vertices of degree 1, where s<k. By a theorem , we know that

Note: To count , we know that there is at least one vertex with maximum degree k; and there are s vertices with degree 1, so the rest of |V(T)|-s-1 vertices have degrees at least 2. Hence, we have
(Why happened this I don't understand this inequality can you explain this better, can you make a graph)

So,

As s<k, we know that . Hence,

which is a contradiction, since |E(T)|=|V(T)|-1 for tree T. We complete the proof