How is the effectiveness of search in trees, which have maintained a balance between nodes, determined?

The efficiency of search in balanced trees is determined by the fact that the height of the tree is minimal and proportional to the logarithm of the number of nodes (O(log n)). This ensures quick access to elements, as approximately half of the remaining elements are discarded at each level of the search.

A balanced tree maintains an even distribution of nodes, preventing the structure from degenerating into a list, which would significantly degrade performance.

For example, in red-black trees or AVL trees, the balance is automatically maintained after insertion and deletion operations, guaranteeing logarithmic time for search, insertion, and deletion.

Thus, the efficiency of search is determined by the height of the tree and the balancing algorithms that keep this height minimal.