## 2 Routing Protocol Design
### TSBR Design  
To balance the network burden and increase lifetime, we propose an equal probability method. In the route tree establishment process, the network adjusts the number of each route tree's sub-nodes according to the equal probability method to equilibrate the probability for each route tree. Therefore, the network can consume the energy of each node equally and avoid some nodes exhausting their energy prematurely. Network lifetime would be prolonged through this protocol. In the following content, we will present some concepts about equilibrium:  
**Definition 1: equilibrium**  
If route tree \(T\) is in equilibrium, then \(T\) must satisfy: \(T\) is null or there is no more than \(I\) different values of neighbor nodes in the same layer. **Definition 2: equilibrium factor**  
If one node's low layer has several nodes, \(i^{\prime}\) is the node who has the least sub-layers. Its sub-node number is \(k_{i^{\prime}}\), equilibrium factor is \(x_{i^{\prime}}=I\). The sub-node number of another node \(i\) is \(k_{i}\). The different value between each node and the node has least sub-node is \(\alpha=k_{i}+k_{i^{\prime}}\). The equilibrium factor is:  
\[x_{i}=1/\left(\alpha+1\right)\qquad\qquad 0<x_{i}\leq 1 \tag{1}\]  
This paper use equilibrium degree to measure whether a route is equal. The more the equilibrium degree, the more equal the route, the more equal the consumption of energy and the more prolonged the network lifetime. Equilibrium degree is defined in Definition 3. **Definition 3:**  
Equilibrium degree is a linear set of its sub-nodes, formatted as follows:\[\gamma=\left\{ \begin{array}{ll}1&i=0,1\\ \dfrac{1}{\alpha}\sum_{i=1}^{n}x_{i}\gamma_{i}&i=2,3,\ldots\end{array} \right. \tag{2}\]  
where \(\gamma\) is equilibrium degree, \(n\) is sub-node number of the node, \(\gamma_{i}\) is the \(i_{th}\) sub-node of the node, \(x_{i}\) is the equilibrium factor. The following sections show the equal route tree algorithm is just an approximate probability equilibrium which cannot achieve non-conditional equilibrium. The main idea of this algorithm is to add a numeric _SubNodeNum_ field into a broadcast-route packet and feedback-route packet to give the sub-node number of the next layer whose default value is 0. The _SubNodeNum_ will be plus 1 if its node adds a new sub-node. Node \(A\) has more sub-nodes than its neighbor node \(B\) and the different value is \(a\). Meanwhile, node \(C\) is in the communication range of \(A\) as well as \(B\) as is shown in Figure 5, where \(C\) is registered as a sub-node of \(A\) after receiving a broadcast-route packet which come from A, then receiving a broadcast-route packet from \(B\). If \(C\) knows \(A\) has more sub-nodes than \(B\) and the value difference is more than 1, it will quit from \(A\) and then register as a sub-node of B. Therefore, the difference between the sub-node numbers of \(A\) and \(B\) has decreased. In such way, we obtain an approximately equal route tree. In Figure 5, after a route tree has been established, \(A\) has four sub-nodes and \(B\) has one sub-node. \(A\) will consume more energy than \(B\) in the same situation. In such a case, \(A\) would die earlier than \(B\). We can conclude from the equilibrium definition that the equilibrium factor of node \(S\) in Figure 5(a) is 1/4 and the equilibrium degree is 5/9. Therefore, the network is unequal. After using TSBR, the sub-node number of \(A\) decreases to three, and that of \(B\) increases to two. The equilibrium of node \(S\) in Figure 5(b) is 1/2 and the equilibrium degree is 5/6. Due to this equilibrium is the most important factor to measure if a network is equal. We can confirm the network would become more equal by using TSBR and thus more suitable for underwater sensor networks. Figure 5: Balancing process. (**a**) No balancing method is applied. (**b**) Balancing method is applied. According the basic idea of TSBR, we deduce the probability of a node \(A\) changing to become \(B\)'s sub-node.