## 3 Results and Discussion
### Simulations for Hierarchical Tree Structured Network
In chain topology, the average number of child nodes is always equal to one and in tree it is almost two such that mean values fused in tree is more accurate than chain for average fusion. When comparing the accuracy of the average method for the tree network between its sink node at network, it can be identified that the accuracy of the sink node for low BER is higher than its network and vice versa. The performance of sink node and network has been similar when the number of nodes is around 55, that is BER of 0.0214. This phenomenon can also be explained as follows. The increment of BER does not affect the accuracy of the measurement in non-aggregating nodes for average method. Therefore, at low BER values, the accuracy of the network is low as the sensor noise of non-aggregating nodes have a significant impact. However, the accuracy of sink node is always affected by change in BER values as it fuses two already fused results in average for tree network. Therefore, under high communication errors, the sink node's accuracy goes lower and the network accuracy is maintained higher by the non-aggregating sink nodes whose accuracy does not depend on BER s.  
Further, there is no significant difference in network MSE of the proposed method between chain and tree sensor networks when comparing Figure 4 with Figure 12. However, for very low BER values (<10\({}^{-5}\)) in chain topology, the network MSE had been reduced to the level of EBDKF, whereas in tree topology, there is no significant improvement in the accuracy of the proposed method for such BER (\(220<nodes<240\)). The network MSE value of proposed method remain less than 0.31 for different number of sensor node combinations (all combinations of average BER values) in the considered range from 20 to 240 nodes. In comparison to these results, the MSE performance of EBDKF for very low BER values is higher for tree topology than chain topology and vice versa. This can be also explained as a result of increased branching in tree resulting higher number of neighbors for diffusion than chain topology. In chain, there are maximum of only two neighbors to exchange local and individual estimates. However, in tree topology as it will be proved in following section on complexity, there are in average three neighbors per non-leaf nodes for diffusion to take place So, under low BER values the diffusion process has increased the accuracy of network than chain by increased number of neighbors and at the same time it has been worsen under high BER values. These arguments can be further verified by analyzing the bit level accuracy under high and low communication errors as analyzed later in this section. To justify the network MSE values obtained for three different data fusion techniques in Figure 12, and to visually represent the mean estimate, we plot the mean estimate and 95% confidence limits for tree network as shown in Figure 13.  
When analyzing the results obtained in Figure 13, it can be seen that for low BER values the network mean estimate is very close to the system state of 1.8 for all data fusion techniques. In these conditions, the 95% confidence interval is also narrow and its width is least in EBDKF and overlaps for Average and proposed method proving the previous result obtained for network MSE for low probability of communication errors. This is because we can say that when the confidence interval for the results is higher, so is the network mean square error. The mean estimate of the proposed method decreases slightly and remains very close to the real value of 1.8 when BER increases. But the mean value tends to deviate more from real value of 1.8 when BER increases for other two data fusion techniques where the EBDKF method having a moderate deviation and Average method having the highest deviation.