In computer science

Computer science

Computer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...

, a ternary tree is a tree data structure in which each node has at most three child nodes

Node (computer science)

A node is a record consisting of one or more fields that are links to other nodes, and a data field. The link and data fields are often implemented by pointers or references although it is also quite common for the data to be embedded directly in the node. Nodes are used to build linked, often...

, usually distinguished as "left", “mid” and "right". Nodes with children are parent nodes, and child nodes may contain references to their parents. Outside the tree, there is often a reference to the "root" node (the ancestor of all nodes), if it exists. Any node in the data structure can be reached by starting at root node and repeatedly following references to either the left, mid or right child.

Ternary trees are used to implement Ternary search tree

Ternary search tree

In computer science, a ternary search tree is a special trie data structure where the child nodes of a standard trie are ordered as a binary search tree. Searching for a string in a ternary search tree consists of a series of binary searches, one for each character in the string...

s and Ternary heaps.

Definition

• Directed Edge - The link from the parent to the child.
• Root - The node with no parents. There is at most one root node in a rooted tree.
• Leaf Node - The node which has no children.
• Child Node - The nodes that are located underneath of a node.
• Depth - Length of the path from the root to the node. The set of all nodes at a given depth is sometimes called a level of the tree. The root node is at depth zero.
• Height - Length of the path from the root to the deepest node in the tree. A (rooted) tree with only one node (the root) has a height of zero. In the example diagram, the tree has height of 2.
• Sibling - Nodes that share the same parent node.

- A node p is an ancestor of a node q if it exists on the path from q to the root. The node q is then termed a descendant of p.

- A size of a node is the number of descendants it has including itself.

Properties of ternary trees

• Maximum number of nodes

– Let be height of a ternary tree.

– Let be the maximum number of nodes in a ternary tree of height h

h	M(h)
0	1
1	4
2	13
3	40

–

– Every tree of height h has at most nodes.

• If a node occupies TREE [k}, then its Left Child is stored in tree .
• Mid Child is stored in TREE .
• Right Child is stored in TREE .

Insertion

Nodes can be inserted into ternary trees in between three other nodes or added after an external node. In Ternary trees, a node that is inserted is specified as to which child it is.

External nodes

Say that the external node being added on to is node A. To add a new node after node A, A assigns the new node as one of its children and the new node assigns node A as its parent.

Internal nodes

Insertion on internal nodes is more complex than on external nodes. Say that the internal node is node A and that node B is the child of A. (If the insertion is to insert a right child, then B is the right child of A, and similarly with a left child insertion or mid child.) A assigns its child to the new node and the new node assigns its parent to A. Then the new node assigns its child to B and B assigns its parent as the new node.

Deletion

Deletion is the process whereby a node is removed from the tree. Only certain nodes in a binary tree can be removed unambiguously.

Node with zero or one child

Say that the node to delete is node A. If a node has no children (external node), deletion is accomplished by setting the child of A's parent to null and A's parent to null. If it has one child, set the parent of A's child to A's parent and set the child of A's parent to A's child.

Comparison with other trees

The picture below is a binary search tree that represents 12 two-letter words. All nodes on the left child have smaller values, while all nodes on the right child have greater values for all nodes. A search starts from the root. To find the word "ON", we compare it to "IN" and take the right branch. Every comparison could access each character of both words.

in
/ \
be of
/ \ / \
as by is or
\ \ \ / \
at he it on to

Digital search tries to store strings character by character. The next picture is a tree that represents the same set of 12 words;

_ _ _ _ _ _ _ _ _ _ _ _ _
/ / / \ \ \
/ / / \ \ \
a b h i o t
/ \ / \ | / | \ /|\ |
s t e y e n s t f n r o
as at be by he in is it of on or to

each input word is shown beneath the node that represents it. In a tree representing words of lower case letters, each node has 26-way branching. Searches are very fast: A search for "IS" starts at the root, takes the "I" branch, then the "S" branch, and ends at the desired node. At every node, one accesses an array element, tests for null, and takes a branch.

i
/ | \
/ | \
b s o
/ | \ / \ | \
a e h n t n t
| \ | / \ |
s y e f r o
\
t

The above picture is a balanced ternary search tree for the same set of 12 words. The low and high pointers are shown as solid lines, while equal pointers are shown as dashed lines. Each input word is shown beneath its terminal node. A search for the word "IS" starts at the root, proceeds down the equal child to the node with value "S", and stops there after two comparisons. A search for "AX" makes three comparisons to the first letter "A" and two comparisons to the second letter "X" before reporting that the word is not in the tree.

Examples of ternary trees

• Ternary search tree
  Ternary search tree
  In computer science, a ternary search tree is a special trie data structure where the child nodes of a standard trie are ordered as a binary search tree. Searching for a string in a ternary search tree consists of a series of binary searches, one for each character in the string...
  
  
• Ternary heap
• An infinite ternary tree containing all primitive Pythagorean triples is described in Tree of primitive Pythagorean triples
  Tree of primitive Pythagorean triples
  In mathematics, a Pythagorean triple is a set of three positive integers a, b, and c having the property that they can be respectively the two legs and the hypotenuse of a right triangle, thus satisfying the equation a^2+b^2=c^2; the triple is said to be primitive if and only if a, b, and c share...
  
   and in Formulas for generating Pythagorean triples. The root node contains triple [3,4,5].