Let T be a tree, let u be its vertex. The branch weight b(u) of u is the maximum number of vertices of a branch of T at u. The set of vertices u of T in which b(u) attains its minimum is the branch weight centroid B(T) of T. For finite trees the present author proved that B(T) coincides with the median of T, therefore it consists of one vertex or of two adjacent vertices. In this paper we show that for infinite countable trees the situation is quite different.