A ternary search tree (TST) is an advanced type of trie data structure specifically designed for the efficient storage and retrieval of strings.

A TST is distinguished from a standard trie (also known as a prefix tree) by its unique structure, which allows each node to have up to three children: a left child, a middle child, and a right child. The left child contains all strings that are lexicographically less than the character in the current node, the right child holds all strings that are greater, and the middle child represents all strings that begin with the character of the current node.

Each node in a TST stores a single character, and a string is represented by traversing a path from the root to a designated node known as the “end of string” node. This characteristic enables a TST to efficiently handle multiple strings that share the same prefix, making it particularly valuable for applications like autocomplete, where the goal is to retrieve all strings starting with a specific prefix.

One of the primary advantages of a TST over traditional tries is its reduced memory usage. In a standard trie, a child node exists for every possible character in the alphabet, which can lead to significant memory overhead. In contrast, a TST only allocates child nodes for characters that are actually present in the stored strings, resulting in improved space efficiency, especially when dealing with large character sets.

Regarding time complexity, searching for a string in a TST typically takes $O(\log n)$ in the average case and $O(n)$ in the worst case, where $n$ represents the length of the string. This efficiency makes TSTs faster than binary search trees for most string-related operations.

In summary, a ternary search tree is a robust data structure that excels in the efficient storage and retrieval of strings. Its design is particularly advantageous for tasks that involve prefix matching, such as autocomplete functionalities.