Tree (data structure)
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Mathematically, it is an ordered directed tree, more specifically an arborescence: an acyclic connected graph where each node has zero or more children nodes and at most one parent node. Furthermore, the children of each node have a specific order.
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[edit] Terminology
A node is a structure which may contain a value, a condition, or represent a separate data structure (which could be a tree of its own). Each node in a tree has zero or more child nodes, which are below it in the tree (by convention, trees are drawn growing downwards). A node that has a child is called the child's parent node (or ancestor node, or superior). A node has at most one parent.An internal node (also known as an inner node or branch node[1]) is any node of a tree that has child nodes. Similarly, an external node (also known as an outer node, leaf node, or terminal node) is any node that does not have child nodes.
The topmost node in a tree is called the root node. Being the topmost node, the root node will not have a parent. It is the node at which operations on the tree commonly begin (although some algorithms begin with the leaf nodes and work up ending at the root). All other nodes can be reached from it by following edges or links. (In the formal definition, each such path is also unique). In diagrams, it is typically drawn at the top. In some trees, such as heaps, the root node has special properties. Every node in a tree can be seen as the root node of the subtree rooted at that node. A free tree is a tree that is not rooted.
The height of a node is the length of the longest downward path to a leaf from that node. The height of the root is the height of the tree. The depth of a node is the length of the path to its root (i.e., its root path). This is commonly needed in the manipulation of the various self balancing trees, AVL Trees in particular. Conventionally, the value −1 corresponds to a subtree with no nodes, whereas zero corresponds to a subtree with one node.
A subtree of a tree T is a tree consisting of a node in T and all of its descendants in T. (This is different from the formal definition of subtree used in graph theory.[2]) The subtree corresponding to the root node is the entire tree; the subtree corresponding to any other node is called a proper subtree (in analogy to the term proper subset).
Representations
There are many different ways to represent trees; common representations represent the nodes as dynamically allocated records with pointers to their children, their parents, or both, or as items in an array, with relationships between them determined by their positions in the array (e.g., binary heap).Trees and graphs
The tree data structure can be generalized to represent directed graphs by removing the constraints that a node may have at most one parent, and that no cycles are allowed. Edges are still abstractly considered as pairs of nodes, however, the terms parent and child are usually replaced by different terminology (for example, source and target). Different implementation strategies exist, for example adjacency lists.In graph theory, a tree is a connected acyclic graph; unless stated otherwise, trees and graphs are undirected. There is no one-to-one correspondence between such trees and trees as data structure. We can take an arbitrary undirected tree, arbitrarily pick one of its vertices as the root, make all its edges directed by making them point away from the root node - producing an arborescence - and assign an order to all the nodes. The result corresponds to a tree data structure. Picking a different root or different ordering produces a different one.
Traversal methods
Main article: Tree traversal
Stepping through the items of a tree, by means of the connections between parents and children, is called walking the tree, and the action is a walk
of the tree. Often, an operation might be performed when a pointer
arrives at a particular node. A walk in which each parent node is
traversed before its children is called a pre-order walk; a walk in which the children are traversed before their respective parents are traversed is called a post-order
walk; a walk in which a node's left subtree, then the node itself, and
then finally its right sub tree are traversed is called an in-order traversal. (This last scenario, referring to exactly two subtrees, a left subtree and a right subtree, assumes specifically a binary tree.) A level-order walk effectively performs a breadth-first search
search over the entirety of a tree; nodes are traversed level by level,
where the root node is visited first, followed by its direct child
nodes and their siblings, followed by its grandchild nodes and their
siblings, etc., until all nodes in the tree have been traversed.Common operations
- Enumerating all the items
- Enumerating a section of a tree
- Searching for an item
- Adding a new item at a certain position on the tree
- Deleting an item
- Pruning: Removing a whole section of a tree
- Grafting: Adding a whole section to a tree
- Finding the root for any node
Common uses
- Representing hierarchical data
- Storing data in a way that makes it easily searchable (see binary search tree and tree traversal)
- Representing sorted lists of data
- As a workflow for compositing digital images for visual effects
- Routing algorithms
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