Recently, multihop wireless sensor networks (WSNs) are widely developed and applied to energy efficient data collections from environments by establishing reliable transmission radio links and employing data aggregation algorithms, which can eliminate redundant transmissions and provide fusion information. In this paper, energy efficiency which consists of not only energy consumptions but also the amount of received data by the base station, as the performance metric to evaluate network utilities is presented for achieving energy efficient data collections. In order to optimize energy efficiency for improvements of network utilization, we firstly establish a graphical game theoretic model for energy efficiency in multihop WSNs, considering message length, practical energy consumptions and packet success probabilities. Afterwards, we propose a graphical protocol for performance optimization from Nash equilibrium of the graphical game theory. The approach also consists of the distributed protocol for generating optimum tree networks in practical WSNs. The experimental results show energy efficient multihop communications can be achieved by optimum tree networks of the approach. The quantitative evaluation and comparisons with related work are presented for the metric with respect to network energy consumptions and the amount of received data by the base station. The performances of our proposal are improved in all experiments. As an example, our proposal can achieve up to about 52% energy efficiency more than collection tree protocol (CTP). The corresponding tree structure is provided for the experiment.