A (possibly infinite) graph is called a tree if it is connected and acyclic. For k >= 1, a tree is c ...

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A (possibly infinite) graph is called a tree if it is connected and acyclic. For k >= 1, a tree is called k-regular if every vertex has exactly k neighbors. Prove that, given a vertex v in a (q + 1)-regular tree and a positive integer i, there are exactly (q + 1) q^(i-1) vertices at distance i from v.