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N-ary Trees

In the preceding section we considered trees in which the nodes can have arbitrary degrees. In particular, the general case allows each of the nodes of a tree to have a different degree. In this section we consider a variation in which all of the nodes of the tree are required to have exactly the same degree.

Unfortunately, simply adding to Definition gif the additional requirement that all of the nodes of the tree have the same degree does not work. It is not possible to construct a tree which has a finite number of nodes all of which have the same degree N in any case except the trivial case of N=0. In order to make it work, we need to introduce the notion of an empty tree as follows:

Definition (N-ary Tree)  An N-ary tree   T is a finite set of nodes  with the following properties:
  1. Either the set is empty, tex2html_wrap_inline62586; or
  2. The set consists of a root, R, and exactly N distinct N-ary trees. That is, the remaining nodes are partitioned into tex2html_wrap_inline62594 subsets, tex2html_wrap_inline62596, tex2html_wrap_inline62362, ..., tex2html_wrap_inline62600, each of which is an N-ary tree such that tex2html_wrap_inline62604.

According to Definition gif, an N-ary tree is either the empty tree, tex2html_wrap_inline62608, or it is a non-empty set of nodes which consists of a root and exactly N subtrees. Clearly, the empty set contains neither a root, nor any subtrees. Therefore, the degree of each node of an N-ary tree is either zero or N.

There is subtle, yet extremely important consequence of Definition gif that often goes unrecognized. The empty tree, tex2html_wrap_inline62586, is a tree. That is, it is an object of the same type as a non-empty tree. Therefore, from the perspective of object-oriented program design, an empty tree must be an instance of some object class. It is inappropriate to use the null reference to represent an empty tree, since the null reference refers to nothing at all!

The empty trees are called external nodes  because they have no subtrees and therefore appear at the extremities of the tree. Conversely, the non-empty trees are called internal nodes .

Figure gif shows the following tertiary   (N=3) trees:

eqnarray14569

In the figure, square boxes denote the empty trees and circles denote non-empty nodes. Except for the empty trees, the tertiary trees shown in the figure contain the same sets of nodes as the corresponding trees shown in Figure gif.

   figure14572
Figure: Examples of N-ary trees.

Definitions gif and gif both define trees in terms of sets. In mathematics, elements of a set are normally unordered. Therefore, we might conclude that that relative ordering of the subtrees is not important. However, most practical implementations of trees define an implicit ordering of the subtrees. Consequently, it is usual to assume that the subtrees are ordered. As a result, the two tertiary trees, tex2html_wrap_inline62628 and tex2html_wrap_inline62630, are considered to be distinct unequal trees. Trees in which the subtrees are ordered are called ordered trees  . On the other hand, trees in which the order does not matter are called oriented trees  . In this book, we shall assume that all trees are ordered unless otherwise specified.

Figure gif suggests that every N-ary tree contains a significant number of external nodes. The following theorem tells us precisely how many external nodes we can expect:

Theorem  An N-ary tree with tex2html_wrap_inline58076 internal nodes contains (N-1)n+1 external nodes.

extbfProof Let the number of external nodes be l. Since every node except the root (empty or not) has a parent, there must be (n+l-1)/N parents in the tree since every parent has N children. Therefore, n=(n+l-1)/N. Rearranging this gives l=(N-1)n+1.

Since the external nodes have no subtrees, it is tempting to consider them to be the leaves of the tree. However, in the context of N-ary trees, it is customary to define a leaf node  as an internal node which has only external subtrees. According to this definition, the trees shown in Figure gif have exactly the same sets of leaves as the corresponding general trees shown in Figure gif.

Furthermore, since height is defined with respect to the leaves, by having the leaves the same for both kinds of trees, the heights are also the same. The following theorem tells us something about the maximum size of a tree of a given height h:

Theorem  Consider an N-ary tree T of height tex2html_wrap_inline62658. The maximum number of internal nodes in T is given by

displaymath62564

extbfProof (By induction).

Base Case Consider an N-ary tree of height zero. It consists of exactly one internal node and N empty subtrees. Clearly the theorem holds for h=0 since

displaymath62565

Inductive Hypothesis Suppose the theorem holds for tex2html_wrap_inline62668, for some tex2html_wrap_inline60478. Consider a tree of height k+1. Such a tree consists of a root and N subtrees each of which contains at most tex2html_wrap_inline62676 nodes. Therefore, altogether the number of nodes is at most

equation14778

That is, the theorem holds for k+1. Therefore, by induction on k, the theorem is true for all values of h.

An interesting consequence of Theorems gif and gif is that the maximum number of external nodes in an N-ary tree of height h is given by

displaymath62566

The final theorem of this section addresses the maximum number of leaves in an N-ary tree of height h:

Theorem  Consider an N-ary tree T of height tex2html_wrap_inline62658. The maximum number of leaf nodes in T is tex2html_wrap_inline62700.

extbfProof (By induction).

Base Case Consider an N-ary tree of height zero. It consists of exactly one internal node which has N empty subtrees. Therefore, the one node is a leaf. Clearly the theorem holds for h=0 since tex2html_wrap_inline62708.

Inductive Hypothesis Suppose the theorem holds for tex2html_wrap_inline62668, for some tex2html_wrap_inline60478. Consider a tree of height k+1. Such a tree consists of a root and N subtrees each of which contains at most tex2html_wrap_inline62718 leaf nodes. Therefore, altogether the number of leaves is at most tex2html_wrap_inline62720. That is, the theorem holds for k+1. Therefore, by induction on k, the theorem is true for all values of h.


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