Data structures & algorithms.

Hash a key to a bucket and you jump straight to the value — no scanning. That's the O(1) magic behind every dictionary, set and cache. Understand hashing, collisions and load factor and a huge swath of "how is this so fast?" stops being mysterious.

Big-O is how you talk about scaling: throw away the constants and ask what happens as n gets huge. It's the difference between code that survives production and code that falls over the first time the data gets big — and it's simpler than it looks.

A binary search tree keeps data ordered so you can find, insert and delete in O(log n) — as long as it stays balanced. See exactly how a naive tree degrades into a slow list, and how self-balancing trees keep the guarantee.

A heap is a tree that always keeps the smallest (or largest) item at the top, so you can pull the next-best in O(log n). It's the engine inside priority queues, task schedulers and Dijkstra's shortest path — anywhere "what's next?" has to be fast.

A trie is a tree keyed by the characters of a string, so words that share a prefix share a path. That structure is what makes autocomplete, spellcheck and IP routing fast — anything that needs to search by prefix instead of by whole key.

Dynamic programming is the technique that turns exponential brute force into polynomial time by remembering sub-answers instead of recomputing them. The moment you learn to spot overlapping subproblems, DP stops being the scary interview topic and becomes mechanical.

The finale ties the arc together: a tree, a heap, a trie are all just graphs with rules. BFS and DFS are how you walk them, and shortest-path algorithms are how you find your way — the tools underneath maps, social networks and dependency resolution.