Special case of Minimum spanning tree problem

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I have a initial node called root. In this special case of spanning tree problem I minimize the number of the branch of the tree (not only the cost of the arc) and every branch must observe one rule: it is consider as feasible in the problem if the sum of distance of all nodes in the branch to root is no longer than alpha (this is a parameter) times the distance from last node to root. Is there an algorithm for this case of spanning tree problem?