The Eccentric Empire

In Sheldor's empire, the cities form a tree with cities as vertices and roads as edges. He wishes to choose a capital among his cities and to do that, his ministers come up with the following definition:

  • The eccentricity of a city \(v\) is the maximum distance from \(v\) to any vertex.

Sheldor wishes to choose a city which is the least eccentric. What is the maximum number of such cities that could exist?

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