(data structure)

**Definition:**
(1) A data structure accessed beginning at the *root* node. Each *node* is either a *leaf* or an *internal node*. An internal node has one or more *child* nodes and is called the *parent* of its child nodes. All children of the same node are *siblings*. Contrary to a physical tree, the root is usually depicted at the top of the structure, and the leaves are depicted at the bottom. (2) A *connected*, *undirected*, *acyclic graph*. It is *rooted* and *ordered* unless otherwise specified.

Thanks to Joshua O'Madadhain (jmadden@ics.uci.edu) for the figure, 6 October 2005.

**Formal Definition:** (1) A tree is either

- empty (no nodes), or
- a root and zero or more subtrees.

**Specialization** (... is a kind of me.)

*heap*, *B-tree*, *binary tree*, *balanced tree*, *multiway tree*, *complete tree*, *search tree*, *digital tree*, *Merkle tree*.

**See also**
other vocabulary: *descendant*, *ancestor*, *tree traversal*, *height*, *depth*, *degree* (3), technical terms: *ordered tree*, *rooted tree*, *free tree*, *arborescence*.

*Note:
Appearing in the Dictionary of Computer Science, Engineering and Technology, Copyright © 2000 CRC Press LLC. *

A tree in the data structure sense (1) is not the same as a tree in the graph sense (2). Consider possible *binary trees* with two nodes. There are two possible data structures: a root and a left subtree or a root and a right subtree. However there is only one possible graph: a root and a subtree. The graph definition doesn't allow for "the subtree is the right subtree and the left subtree is empty". Also there is no "empty" graph tree.

Thanks to Sharat Chandran (sharat@cs.umd.edu) for clarifying the difference between these two senses.

* The formal definition is after [CLR90, page 94].*

Go to the Dictionary of Algorithms and Data Structures home page.

If you have suggestions, corrections, or comments, please get in touch with Paul Black.

Entry modified 15 December 2017.

HTML page formatted Fri Feb 23 10:06:08 2018.

Cite this as:

Paul E. Black and Algorithms and Theory of Computation Handbook, CRC Press LLC, 1999, "tree", in
*Dictionary of Algorithms and Data Structures* [online], Vreda Pieterse and Paul E. Black, eds. 15 December 2017. (accessed TODAY)
Available from: https://www.nist.gov/dads/HTML/tree.html