Concepts4

An Euler Tour Tree represents each rooted tree as a DFS open/close sequence so that every subtree is a single contiguous interval.

Persistence lets you keep every past version of a data structure while making O(log n) updates and queries on any version.

An implicit treap is a randomized balanced binary tree that treats array positions as keys without storing them explicitly.

A treap is a binary search tree on keys combined with a heap on random priorities, which keeps the tree balanced in expectation.