## 3 Results and Discussion
### Simulations for Hierarchical Tree Structured Network
#### 3.2.2 Evaluation of Complexity of Information Fusion  
In tree-based network, multiple nodes can be connected to a fusing node. If the fusion node act as a small clusters head, the cluster head might have high computational complexity. Therefore, we analyze total number of fusion nodes in a tree network with minimum distance neighbor connection method that was specified earlier as depicted in Figure 17.  
It can be observed in Figure 17 that approximately 53% of nodes act as aggregating nodes in networks using average and proposed method while all the nodes act as aggregating nodes in Diffusion Kalman Filtering techniques. As it can be observed in simulation results, the total number of child  
Figure 16: Network MSE comparison among the proposed statistical, AVG, and EBDKF information fusion methods in a retransmission allowed tree structured network when BER is 0.05. Figure 17: Average number of fusion nodes and total child nodes in tree-based IoT platform using maximum reliable connection method for different fusion techniques. nodes in networks using average and proposed method for fusion has \(N-1\) number of total child nodes, and a network using EBDKF will have \(2*(N-1)\) total child nodes where N is the total number of nodes. Therefore, the average number of child nodes per aggregating nodes of the network represented by \(n_{child}\) for proposed and average fusion techniques will be as depicted in Equation (40). \[n_{child} =\frac{(N-1)}{0.53N} \tag{40}\] \[=\approxeq 1.87\]  
when \(N\) is high, \(n_{child}\) value given in Equation (40) is close to 2. The leaf nodes for these fusion schemes have no child nodes. Now we will derive the \(n_{child}\) for EBDKF as shown in Equation (41). \[n_{child} =\frac{2*(N-1)}{N} \tag{41}\] \[=\approxeq 2\]  
As it is evident from the Equation (41), a network employing EBDKF will also have average number of child nodes per aggregating nodes of the network almost equal to 2. Therefore, the average computational complexity of an aggregating node will be as if each aggregating node has two children for all of the data fusion techniques considered in this context. The number of leaf nodes for EBDKF are still \(0.47(N-1)\) having one child per each node. Therefore, the total number of child nodes remaining for intermediate nodes will be as given in Equation (42). \[2*(N-1)-0.47*(N-1)=1.53(N-1) \tag{42}\]  
As there are \(0.53(N-1)\) number of non-leaf nodes for EBDKF fusion, the average number of child nodes per non-leaf nodes (\(n_{child_{non-leaf}}\)) is given by Equation (43) as,  
\[n_{child_{non-leaf}} =\frac{1.53(N-1)}{0.53(N-1)} \tag{43}\] \[=2.87\]  
Therefore, as seen from Equation (43), the non-leaf nodes will have in average close to 3 child nodes. The root node is also a non-leaf node. Therefore, the root (sink) node will have in average 1.87 nodes that is approximately two nodes which is one lesser than a typical non-leaf node. So, the average number of child nodes for both the sink node and the network for a tree structured diffusion scheme is equivalent to two. Therefore, the average fusion complexity of the network and the fusion center (sink node) must be equivalents. In chain-based WSNs, each node has only one child. Thus, each node needs to combine only two data frames/packets during the fusion process except the first node for AVG and proposed statistical method. But when it comes to the tree topology, an aggregating node needs to combine in average three data frames/packets in average considering the network. Two of them are information sent from the child nodes and the other is the self-measurement. Now we will consider about the complexity of diffusion technique. The average number of neighbors of EBDKF for both chain and tree topology were proved to be two nodes in previous sections. Therefore, the fusion complexity of chain and tree topology for diffusion technique must be equivalent.