Give a tree with n vertices, the vertices are randomly colored with two equiprobable colors. Thus, connected hoods of the same color are formed.
1) Which is the average size of such a neighborhood?
2) Which is the average size of such a neighborhood that contains a special vertex V ?
If the tree is a linear one, the size of a neighborhood is almost 2. A pointed neighborhood has a size of almost 3.
While trying to solve it, I had the impression that the same sizes are valid for any tree.