## 5 Simulations
### Performance Tradeoffs  
In this subsection, we look at each of the performance metrics (listed in Section 5.1) in isolation, independent of the trend of values expected of them to minimize the aggregation delay. This way, we are able to identify the desirable trend of values (low, moderate or high) that would be typically expected of these metrics to optimize network measures other than aggregation delay (such as node lifetime and network lifetime) and thereby also identify any tradeoffs. We define node lifetime [34] as the time of first node failure due to energy exhaustion; whereas, network lifetime [34] is the time the network gets disconnected due to the failure of one or more nodes due to energy exhaustion. As noticed in Section 5.3 and Figure 2, to obtain a lower aggregation delay (especially with increase in node density), it would be preferable for the data gathering trees to have a moderate-larger diameter, a smaller fraction of nodes as leaf nodes and a low-moderate number of child nodes per intermediate node. However, one or more of these trend in the values for the above metrics may not be desirable for real-time reporting of events from any node to the leader node and/or for optimizing energy consumption-related performance measures such as the node lifetime and network lifetime. To begin with, we typically desire to have lower values for the tree diameter and hop count per path to facilitate faster real-time reporting of events from an individual node to the leader node of the data gathering tree [12]. However, if we were to incur a smaller hop count per path, then the diameter of the data gathering tree would have to be also smaller [23], thereby increasing the aggregation delay (_diameter-aggregation delay tradeoff_). With respect to the fraction of nodes as leaf nodes, from a network lifetime standpoint, we would typically desire to have a larger number of leaf nodes. This is because the leaf nodes tend to lose relatively less energy compared to the intermediate nodes [16]. For every data aggregation cycle, the leaf nodes lose energy for transmitting the data only once to their upstream intermediate node and there is no energy lost due to reception. On the other hand, the intermediate nodes lose energy for both transmitting (to an upstream node) as well as for receiving and aggregating data (from the downstream child nodes) [22]. Thus, a larger number of leaf nodes in a data gathering tree is desirable to prolong the lifetime of a majority of the nodes in the network so that the time of network disconnection due to node failures (typically referred to as the network lifetime) could be prolonged as much as possible [16,34]. However, a larger fraction of nodes as leaf nodes would more likely increase the number of child nodes per intermediate node (also confirmed through the simulation results in Figure 2) and thereby increase the aggregation delay (_network lifetime-aggregation delay tradeoff_). With respect to the number of child nodes per intermediate node, we notice that a larger value for this metric could increase the aggregation delay as well as trigger premature node failures. As intermediate nodes have to spend energy to aggregate data from their child nodes, a larger number of child nodes per intermediate node could overuse certain nodes (the intermediate nodes) at the cost of the other nodes (the leaf nodes) [24]. A very low value for the number of child nodes per intermediate node would not be desirable either as it would lead to a significantly larger number of nodes functioning as intermediate nodes and aggregating the data (such a scenario could only lower the node lifetime) [6]. Hence, if we are interested in prolonging the lifetime of every node in the network (_i.e._, maximizing the time of first node failure, typically referred to as the node lifetime [34]), it would be desirable to have a moderate number of child nodes per intermediate node. An open question is how to define the range of moderate values for the number of child nodes per intermediate node.