## 3 Results and Discussion
### Simulations for Hierarchical Tree Structured Network
This is the reason to have a lower performance gap between those methods in Figure 12 and a low difference in confidence intervals in Figure 13, when network size is 98. In these conditions EBDKF show highest accuracy since its MSB error percentage is 0.0306. But when the BER is increased to 0.05 by network one having 38 nodes, a very high percentage increment of 4 significant bits of EBDKF is observed compared to BER of 0.005 case making its accuracy least whereas in Average method a relative lower increment in percentage error in significant bits are observed. When comparing Figure 15 with Figure 6, to have a comparison among chain and tree sensor network accuracy wise performance, it can be seen that for Average method percentage increment of errors of significant bits for BER of 0.05 is lower in tree network topology confirming the argument that average fusion's accuracy is higher in tree architecture than chain architecture. For both topologies, the proposed method shows almost similar percentage errors proving that accuracy level of the proposed method remains almost unchanged for chain and tree network topologies. Furthermore, under high communication errors, bit level percentage error values of EBDKF fused results for tree topology is  
Figure 4.1 Network MSE vs. the number of data gathering cycles for BER of 0.0003 and 0.05 among the proposed statistical, AVG and EBDKF information fusion methods in a tree-based IoT platform: (**a**) for first 600 data gathering cycles and (**b**) for first 50 data gathering cycles. higher than chain topology proving that accuracy is worse than chain topology for high BER values. Similarly, the converse of the preceding argument, that is bit level accuracy of chain is worse than tree networks of EBDKF for low BER values. The measurement errors have higher impact on the least significant bits. Therefore, we cannot see considerable improvement in the least significant bits 5, 6, and 7 for any of the fusion techniques considered. Therefore, we omitted them in Figures 6 and 15. It should be investigated how retransmissions affect the information accuracy in a tree structured network that has increased branching with respect to chain structure which does not have any branching. A theoretical upper bound for total data frame/packet transmission per data gathering cycle can be derived similar to Equation (38) derived for chain structure. We follow the same notation used for chain structure here also. We will derive in computation complexity section of fusion techniques for tree structure that 53% of nodes are aggregating nodes in fusion schemes which have unidirectional communication from source nodes to the sink node. Therefore, the total links of a tree structure is \((k-1)\). Therefore, same upper bound in Equation (38) is valid for tree structure too. We extensively simulate the retransmission allowed tree structure for different fusion techniques by varying the maximum number of retransmissions allowed per hop to obtain the result given in Figure 4.2  
Number of nodes of the tree structure was selected as 38 corresponding to an average bit error rate of 0.05 in which the accuracy of the proposed method is highest as it was observed in Figures 12 and 13. Here we have made the assumption that communication channel errors are not affecting on acknowledgment frames/packets so that an acknowledgment sent is always received. It can be clearly observed in Figure 16 that the final results for Average and proposed methods are similar to chain structure. That's because proposed method does not show any significant improvement in accuracy with retransmissions and the AVG method needs maximum of three retransmissions per hop per data gathering cycle to achieve accuracy of the proposed method at a BER of 0.05. But, the EBDKF method needs to set maximum retransmission per hop to a value of 2 to overcome the accuracy of the proposed method. Therefore, \(L=0,2,3\) in proposed method, EBDKF, Average fusion methods respectively in order to achieve a network MSE less than 0.3.