М-арные деревья: структура, свойства и применение.

Tree data structure in which each node has at most m children. In graph theory, an m ary tree (for nonnegative integers m) (also known as n ary, k ary or k way tree) is an arborescence (or, for some authors, an ordered tree) in which each node has no more than m children. A binary tree is an important case where m = 2; similarly, a ternary tree is one where m = 3.

A full m ary tree is an m ary tree where within each level every node has 0 or m children. A complete m ary tree (or, less commonly, a perfect m ary tree) is a full m ary tree in which all leaf nodes are at the same depth.

Traversing a m ary tree is very similar to traversing a binary tree. The pre order traversal goes to parent, left subtree and the right subtree, and for traversing post order it goes by left subtree, right subtree, and parent node. For traversing in order, since there are more than two children per node for m > 2, one must define the notion of left and right subtrees. One common method to establish left/right subtrees is to divide the list of children nodes into two groups. By defining an order on the m children of a node, the first nodes would constitute the left subtree and nodes would constitute the right subtree.

Using an array for representing a m ary tree is inefficient, because most of the nodes in practical applications contain less than m children. As a result, this fact leads to a sparse array with large unused space in the memory. Converting an arbitrary m ary tree to a binary tree would only increase the height of the tree by a constant factor and would not affect the overall worst case time complexity. In other words, since
First, we link all the immediate children nodes of a given parent node together in order to form a link list. Then, we keep the link from the parent to the first (i. e., the leftmost) child and remove all the other links to the rest of the children. We repeat this process for all the children (if they have any children) until we have processed all the internal nodes and rotate the tree by 45 degrees clockwise. The tree obtained is the desired binary tree obtained from the given m ary tree.

m ary trees can also be stored in breadth first order as an implicit data structure in arrays, and if the tree is a complete m ary tree, this method wastes no space. In this compact arrangement, if a node has an index i, its c th child in range {1, ,m} is found at index , while its parent (if any) is found at index (assuming the root has index zero, meaning a 0 based array). This method benefits from more compact storage and better locality of reference, particularly during a preorder traversal. The space complexity of this method is .

One of the applications of m ary tree is creating a dictionary for validation of acceptable strings. In order to do that, let m be equal to the number of valid alphabets (e. g., number of letters of the English alphabet) with the root of the tree representing the starting point. Similarly, each of the children can have up to m children representing the next possible character in the string. Thus, characters along the paths can represent valid keys by marking the end character of the keys as "terminal node". For example, in the example below "at" and "and" are valid key strings with "t" and "d" marked as terminal nodes. Terminal nodes can store extra information to be associated with a given key. There are similar ways to building such a dictionary using B tree, Octree and/or trie.