What are the applications of binary trees?

Question

I am wondering what the particular applications of binary trees are. Could you give some real examples?

Answer 1

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.

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 implementing efficient priority-queues, which in turn are used for scheduling processes in many operating systems, Quality-of-Service in routers, and A* (path-finding algorithm used in AI applications, including robotics and video games). Also used in heap-sort.
• 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.

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The reason that binary trees are used more often than n-ary trees for searching is that n-ary trees are more complex, but usually provide no real speed advantage.

In a (balanced) binary tree with m nodes, moving from one level to the next requires one comparison, and there are log_2(m) levels, for a total of log_2(m) comparisons.

In contrast, an n-ary tree will require log_2(n) comparisons (using a binary search) to move to the next level. Since there are log_n(m) total levels, the search will require log_2(n)*log_n(m) = log_2(m) comparisons total. So, though n-ary trees are more complex, they provide no advantage in terms of total comparisons necessary.

(However, n-ary trees are still useful in niche-situations. The examples that come immediately to mind are quad-trees and other space-partitioning trees, where divisioning space using only two nodes per level would make the logic unnecessarily complex; and B-trees used in many databases, where the limiting factor is not how many comparisons are done at each level but how many nodes can be loaded from the hard-drive at once)