## 1 Introduction  
Multiple sensor nodes make up a wireless sensor network (WSN), where these nodes can sense, measure, gather, and transmit information from the environment via radio. In a wide range of applications, such as healthcare, industrial control, public security, commercial, military, and environmental monitoring, WSNs are widely used [1, 2, 3]. A WSN has limited storage, battery, and processing capacity, which affects the network lifetime, quality of service, and cost [4, 5]. Furthermore, data collection and transmission in WSNs may be affected by attacks or interference in harsh environments. Therefore, a WSN's reliability is an important issue, to be sure that the WSN works properly. It is essential to calculate WSN reliability, to minimize the cost, minimize the node power consumption, and maximize the network lifetime, as unreliable WSNs may fail to accomplish the tasks, resulting in inefficient use of sensor resources [6]. The reliability of the WSN can be evaluated using various methods such as Markov chain theory, universal generating function (UGF), a Monte Carlo (MC) simulation approach, a reliability block diagram (RBD), fault tree (FT) [7], and a signal strength and trust model [8]. The routing issue in transferring data from the source node to the target node is one of the most significant challenges facing WSNs. Existing routing protocols can be classified into single-hop and multi-hop, where in single-hop, the data is directly sent from the node to the base station (BS), while the data is sent to the BS through several nodes in multi-hop [5]. Multi-hop routing protocols fall into two main categories based on the network structure: hierarchical and flat [9; 10; 11]. In the flat routing protocol, all nodes perform the same task and function. Whereas in hierarchical routing protocols, different tasks and functions are assigned to the nodes [10; 12; 13; 14]. The three main types of the flat routing protocol are: proactive (table-driven), reactive (source-initiated), and hybrid [10]. RPL, Hydro, CPT, and Zigbee are the most popular proactive routing protocols [15; 16]. LOAD, LOADng, AODVbis, and Tiny AODv are some examples of reactive routing protocols [15; 16; 17]. Moreover, zone routing is an example of hybrid routing protocols. Figure 1 illustrates the three main types of hierarchical routing protocols: cluster-based, tree-based, and chain-based [2; 5; 9; 13; 18]. In cluster-based routing [19; 20; 21; 22], each cluster has at most one cluster head (CH) that manages the cluster, while other nodes are called cluster members (CMs), as illustrated in Figure 2, where the CMs send the collected data to the CH and do not directly communicate with the BS. In the WSN, the BS serves as both the central data-collecting node and the common destination for data gathered from the nodes. There are usually no power restrictions on the BS, which acts as a bridge between WSN and the end user (via communication). The most popular cluster-based routing protocol is the low-energy adaptive clustering hierarchy (LEACH) [13]. As shown in Figure 3, the tree-based topology deploys nodes in a logical tree configuration, with all sensor data being transferred from leaf nodes (children) to their respective parents [12; 18]. Figure 1: Routing protocols in WSN. Figure 2: Cluster-based WSN (each cluster is depicted in a distinct color). An appropriate neighbor node is selected as the parent node for every node in a tree-based approach. A routing tree is constructed with a sink node that acts as a root node, and all nodes are connected to the root through the most energy-saving path [23]. The sensor node's data is transmitted along the tree towards the root node (sink) and is fused between nodes. There are different tree-based routing protocols such as energy-aware data aggregation tree (EADAT), balanced aggregation tree routing (BATR), power-efficient data gathering and aggregation protocol (PEDAP), and enhanced tree routing (ETR) [18].