## 1 Introduction  
Guided by the development trend of the \"sensor city\", wireless sensor networks (WSN) have been widely used in the field of information, from the field of military and national defense to the fields of medical care, industry and agriculture, urban management, environmental monitoring [1], and smart homes [2] that are closely related to people. The WSN is an ad hoc network composed of randomly distributed sensor nodes [3]. The nodes perceive the environment to enable data collection, processing and transmission. However, the sensor nodes in WSN use limited energy (such as batteries), and in some complex working environments, it is difficult to supply power in time, which will lead to the unreliable lifetime and quality of the network. In addition, for scenarios with high real-time requirements, such as disaster monitoring, military supervision, medical inspection, etc., it is even more necessary to consider how to balance the energy consumption of nodes [4]. Therefore, how to balance the energy consumption and extend the lifetime of network is the research focus in WSN. The selection of cluster head (CH) nodes in routing protocols is a key to efficient communication in WSN. The CH nodes undertake the tasks of information collection,data fusion and data transmission in the cluster, and their energy consumption is faster than other nodes. By designing an effective clustering protocol, each sensor node can use the limited energy more reasonably. The clustering routing protocol is divided into four processes: cluster head election, cluster formation, data fusion, and data transmission. It divides the entire network into multiple clusters. In each cluster, select one node as the cluster head (CH) node and the other nodes as the cluster member (CM) nodes. The CM nodes communicate with the CH node in each cluster and forward the data to the CH node. The CH nodes integrate the data and send it to the sink node, and then the sink node transmits the data to the network for communication management between users [5], as shown in Figure 1. In addition, the performance of routing clustering protocols in homogeneous networks is not good in heterogeneous networks, so the research based on clustering routing protocols is one of the research hotspots in heterogeneous networks at present. Using the metaheuristic algorithm to design the cluster routing protocol has always been a popular direction of industry research [6]. The metaheuristic algorithm is a powerful tool to solve complex optimization problems [7]. It can obtain the best approximate solution for more complex NP-hard problems in polynomial time [8]. As the research on bionics becomes more and more mature, metaheuristic algorithms are proposed one after another; for example, the particle swarm optimization algorithm (PSO) [9; 10], genetic algorithm (GA) [11; 12], bat algorithm (BA) [13], seagull optimization algorithm (SOA) [14], and the grey wolf optimizer (GWO) [15], etc. The formulas for individual movements of many metaheuristic algorithms are based on operations such as addition, subtraction, multiplication, and division. The mathematical model of the algorithm is not closely related to the essence of things, and there is no specific scientific theory support. In order to come up with a metaheuristic algorithm with good performance and a close connection between the mathematical model and the essence of things, we searched for formula derivation about the growth principle of bamboo forest in biology. Based on the differential model of the bamboo growth and the Gaussian mixture models [16], this study proposes a new metaheuristic optimization algorithm named the bamboo forest growth optimizer (BFGO) and demonstrates the effectiveness of the algorithm's optimization ability is proved on the CEC test sets and engineering optimization problems. In addition, for the problem of energy consumption in WSN, based on the BFGO algorithm, this paper proposes an energy-efficient clustering mechanism of protocol (BFGO-C) for two-level heterogeneous WSN is proposed.