We consider the problem of optimal communication tree construction in a given undirected weighted graph. Such a problem occurs while minimizing the power consumption of data transmission in different distributed networks in the case when network elements are able to adjust their transmission ranges. In this paper, the most general strongly NP-hard formulation, when edge weights have arbitrary non-negative values, is considered. We propose new heuristics, mostly based on variable neighborhood search, for getting an approximate solution of the problem. Extensive comparative analysis between the proposed methods was performed. Numerical experiments demonstrated the high efficiency of the proposed heuristics.