Even though the bounty is gone, since the accepted answer gives the extremely-false impression that binary-trees are not very useful, I will post another answer.
To squabble about the performance of binary-trees is meaningless - they are not a data structure, but a family of data structures, all with different performance characteristics. While it is true that unbalanced binary trees perform much worse than self-balancing binary trees for searching, there are many binary trees (such as binary tries) for which "balancing" has no meaning.
The reason that binary trees are used more often than n-ary trees for searching is that with every comparison in a (balanced) binary tree, you eliminate about half the tree from consideration. In contrast, with an n-ary tree, you can eliminate (n-1)/n of the tree using log(n) comparisons (using a binary search).
Applications of binary trees- Binary Search Tree - Used in many search applications where data is constantly entering/leaving, such as the map and set objects in many languages' libraries.
- Binary Space Partition - Used in almost every 3D video game to determine what objects need to be rendered.
- Binary Tries - Used in almost every high-bandwidth router for storing router-tables.
- Hash Trees - used in p2p programs and specialized image-signatures in which a hash needs to be verified, but the whole file is not available.
- Heaps - Used in heap-sort; fast implementations of Dijkstra's algorithm; and in implementing efficient priority-queues, which are used in scheduling processes in many operating systems, Quality-of-Service in routers, and A* (path-finding algorithm used in AI applications, including video games).
- Huffman Coding Tree (Chip Uni) - used in compression algorithms, such as those used by the .jpeg and .mp3 file-formats.
- GGM Trees - Used in cryptographic applications to generate a tree of pseudo-random numbers.
- Syntax Tree - Constructed by compilers and (implicitly) calculators to parse expressions.
- Treap - Randomized data structure used in wireless networking and memory allocation.
- T-tree - Though most databases use some form of B-tree to store data on the drive, databases which keep all (most) their data in memory often use T-trees to do so.
