Tries

A trie stores a set of sequences as one tree where every edge carries a symbol and every path from the root spells a prefix. Words that begin the same way share nodes, so a lookup costs the length of the word, not the size of the set.

Node holds	C++	Python	Java
Fixed alphabet	array<int, 26>	list of 26	TrieNode[]
Sparse alphabet	unordered_map	dict	HashMap
Word ends here	bool	bool	boolean

No standard library ships a trie, so the node is the whole design. A fixed array per node is fastest on 26 letters or 2 bits, a map per node when the alphabet is wide or sparse. Keep the end-of-word flag separate, because a stored word is also a prefix of longer ones and a trie without the flag quietly accepts strings nobody inserted. The alphabet need not be letters — feed it the bits of an integer, most significant first, and a walk preferring the opposite bit at every level maximises an XOR.

Operation	Trie	Hash set
Insert or find a word of length $L$	$O(L)$	$O(L)$
Every word under a prefix	$O(L + \text{hits})$	$O(nL)$
Memory	one node per distinct prefix	one entry per word

Use one when the question is about prefixes rather than whole words — the shortest stored prefix of a query, a word built a letter at a time from other words, or the best XOR partner, the same question over bits.

The problems below start with the shared-prefix idea on its own and two character tries, then switch substrate to Binary Tries for the maximum XOR pair and the same walk under a limit.