discrete mathematics

Question

Let T be an n - vertex tree having one vertex of  degree i for i=2,3,…,k and the remaining n−k+1 vertices are of degree 1 each. Determine n in terms of k.

Comments

n = $\frac{k^{2}-k+4}{2}$ ??

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@joshi_nitish how you are getting this answer?

As if k = 3 then n = 5, we have vertex {A,B,C,D,E} now A is adj to 3 vertex as k = 3 A-B, A-C, A-D now how we will have E as degree 1?