18.1. General Trees¶

18.1.1. General Trees¶

Many organizations are hierarchical in nature, such as the military and most businesses. Consider a company with a president and some number of vice presidents who report to the president. Each vice president has some number of direct subordinates, and so on. If we wanted to model this company with a data structure, it would be natural to think of the president in the root node of a tree, the vice presidents at level 1, and their subordinates at lower levels in the tree as we go down the organizational hierarchy.

Because the number of vice presidents is likely to be more than two, this company's organization cannot easily be represented by a binary tree. We need instead to use a tree whose nodes have an arbitrary number of children. Unfortunately, when we permit trees to have nodes with an arbitrary number of children, they become much harder to implement than binary trees. We consider such trees in this chapter. To distinguish them from binary trees, we use the term general tree.

In this module we will examine general tree terminology and define a basic ADT for general trees.

18.1.1.1. General Tree Definitions and Terminology¶

A tree \(\mathbf{T}\) is a finite set of one or more nodes such that there is one designated node \(R\), called the root of \(\mathbf{T}\). If the set \((\mathbf{T} -\{R\})\) is not empty, these nodes are partitioned into \(n > 0\) disjoint sets \(\mathbf{T}_0\), \(\mathbf{T}_1\), ..., \(\mathbf{T}_{n-1}\), each of which is a tree, and whose roots \(R_1, R_2, ..., R_n\), respectively, are children of \(R\). The subsets \(\mathbf{T}_i (0 \leq i < n)\) are said to be subtrees of \(\mathbf{T}\). These subtrees are ordered in that \(\mathbf{T}_i\) is said to come before \(\mathbf{T}_j\) if \(i < j\). By convention, the subtrees are arranged from left to right with subtree \(\mathbf{T}_0\) called the leftmost child of \(R\). A node's out degree is the number of children for that node. A forest is a collection of one or more trees. Figure 18.1.1 presents further tree notation generalized from the notation for binary trees.

Figure 18.1.1: Notation for general trees. Node \(P\) is the parent of nodes \(V\), \(S1\), and \(S2\). Thus, \(V\), \(S1\), and \(S2\) are children of \(P\). Nodes \(R\) and \(P\) are ancestors of \(V\). Nodes \(V\), \(S1\), and \(S2\) are called siblings. The oval surrounds the subtree having \(V\) as its root.

Each node in a tree has precisely one parent, except for the root, which has no parent. From this observation, it immediately follows that a tree with \(n\) nodes must have \(n-1\) edges because each node, aside from the root, has one edge connecting that node to its parent.

18.1.1.2. An ADT for General Tree Nodes¶

Before discussing general tree implementations, we should first make precise what operations such implementations must support. Any implementation must be able to initialize a tree. Given a tree, we need access to the root of that tree. There must be some way to access the children of a node. In the case of the ADT for binary tree nodes, this was done by providing member functions that give explicit access to the left and right child pointers. Unfortunately, because we do not know in advance how many children a given node will have in the general tree, we cannot give explicit functions to access each child. An alternative must be found that works for an unknown number of children.

One choice would be to provide a function that takes as its parameter the index for the desired child. That combined with a function that returns the number of children for a given node would support the ability to access any node or process all children of a node. Unfortunately, this view of access tends to bias the choice for node implementations in favor of an array-based approach, because these functions favor random access to a list of children. In practice, an implementation based on a linked list is often preferred.

An alternative is to provide access to the first (or leftmost) child of a node, and to provide access to the next (or right) sibling of a node. Here are the class declarations for general trees and their nodes. Based on these two access functions, the children of a node can be traversed like a list. Trying to find the next sibling of the rightmost sibling would return null.

// General tree node ADT
interface GTNode {
  Object value();
  boolean isLeaf();
  GTNode parent();
  GTNode leftmostChild();
  GTNode rightSibling();
  void setValue(Object value);
  void setParent(GTNode par);
  void insertFirst(GTNode n);
  void insertNext(GTNode n);
  void removeFirst();
  void removeNext();
}

// General tree ADT
interface GenTree {
  void clear();      // Clear the tree
  GTNode root();     // Return the root
  // Make the tree have a new root, give first child and sib
  void newroot(Object value, GTNode first, GTNode sib);
  void newleftchild(E value); // Add left child
}

18.1.1.3. General Tree Traversals¶

There are three traditional tree traversals for binary trees: preorder, postorder, and inorder. For general trees, preorder and postorder traversals are defined with meanings similar to their binary tree counterparts. Preorder traversal of a general tree first visits the root of the tree, then performs a preorder traversal of each subtree from left to right. A postorder traversal of a general tree performs a postorder traversal of the root's subtrees from left to right, then visits the root. Inorder traversal does not have a natural definition for the general tree, because there is no particular number of children for an internal node. An arbitrary definition—such as visit the leftmost subtree in inorder, then the root, then visit the remaining subtrees in inorder—can be invented. However, inorder traversals are generally not useful with general trees.

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To perform a preorder traversal, it is necessary to visit each of the children for a given node (say \(R\)) from left to right. This is accomplished by starting at R's leftmost child (call it \(T\)). From \(T\), we can move to \(T\)'s right sibling, and then to that node's right sibling, and so on.