Section Recap

You've learned three powerful tree techniques:

$1.$ HLD: Decomposes tree into $O(\log n)$ heavy chains for path queries

$2.$ Centroid Decomposition: Divide-and-conquer with $O(\log n)$ levels for distance queries

$3.$ Treaps: Randomized BST with split/merge for dynamic sequences

$4.$ Euler Tour: Flattens subtrees to ranges for subtree queries

These techniques turn hard tree problems into manageable range queries. Practice combining them, many competition problems require exactly that. You now have a complete toolkit for tree-based data structures. The next step is practice: solve problems, make mistakes, and build intuition.