Levelsum

Levelsum is a concept used in graph theory and data structures to describe the distribution of node values across depths in a rooted structure. It focuses on grouping nodes by their distance from a designated root and summing their assigned weights within each group.

Formal definition: Let T be a rooted tree with root r. Each node v has a nonnegative weight w(v). For each level k, define L_k = { v : distance from r to v equals k }. The level sum at level k is S_k = sum_{v in L_k} w(v). The level-sum sequence with respect to r is the sequence {S_0, S_1, S_2, ...}. The total weight of the tree is the sum of all level sums, which equals sum_{v} w(v), since every node belongs to exactly one level. The level-sum sequence provides a view of how weight or value is distributed across depths.

Computation: A level-order traversal (breadth-first search) yields the levels directly. As nodes are dequeued, their weights are added to the current level’s total, and the process continues to the next level. The time complexity is O(n), where n is the number of nodes.

Generalizations: The concept extends to graphs that are not trees by using the distance from the root as the level, with unreachable nodes treated as absent or assigned an infinite level. In directed acyclic graphs, levels can be defined via shortest-path distances or by a layering scheme derived from a topological ordering.

Applications: Level sums can help analyze load distribution in hierarchical data, balance in organizational structures, and performance characteristics of level-wise algorithms. They also serve as a simple metric for visualizing how a quantity is spread across different depths in a tree or network. Example: a small tree with a root of weight 3, two children with weights 2 and 1, and a grandchild with weight 4 yields level sums [3, 3, 4].