Question

Consider the vertex set V = {1, 2, . . . , n}. Think about Cayley’s...

Consider the vertex set V = {1, 2, . . . , n}. Think about Cayley’s Theorem and, perhaps, about Pr ̈ufer codes to make and prove the claim:

Claim 1. There are exactly ___________ trees on the vertex set V for which the vertex 1 is a leaf (i.e. has degree 1).

Write a Proof.

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Answer #1

By Cayley's Theorem number of trees with n vertices is . Let the first vertex be 'v'. Now consider the rest of (n-1) vertices, with these (n-1) vertex we could construct trees. So now we just need to connect the vertex v with any of the (n-1) vertices of the trees. This make the vertex v , a leaf in every tree and all possible such trees with n vertices are now constructed. The total number of such trees = .

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