mod8 - Module 8: Trees and Graphs Theme 1: Basic Properties...

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Module 8: Trees and Graphs Theme 1: Basic Properties of Trees A (rooted) tree is a finite set of nodes such that there is a specially designated node called the root . the remaining nodes are partitioned into disjoint sets such that each of these sets is a tree. The sets are called subtrees ,and the degree of the root. The above is an example of a recursive definition, as we have already seen in previous modules. Example 1 : In Figure 1 we show a tree rooted at with three subtrees , and rooted at , and , respectively. We now introduce some terminology for trees: A tree consists of nodes or vertices that store information and often are labeled by a number or a letter. In Figure 1 the nodes are labeled as . An edge is an unordered pair of nodes (usually denoted as a segment connecting two nodes). For example, is an edge in Figure 1. The number of subtrees of a node is called its degree . For example, node is of degree three, while node is of degree two. The maximum degree of all nodes is called the degree of the tree. A leaf or a terminal node is a node of degree zero. Nodes and are leaves in Figure 1. A node that is not a leaf is called an interior node or an internal node (e.g., see nodes and ). Roots of subtrees of a node are called children of while is known as the parent of its children. For example, and are children of , while is the parent of and . Children of the same parent are called siblings . Thus are siblings as well as and are siblings. The ancestors of a node are all the nodes along the path from the root to that node. For example, ancestors of are and . The descendants of a node are all the nodes along the path from that node to a terminal node. Thus descendants of are and . 1
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C G A E F B L K J I H D M T 1 T 2 T 3 Figure 1: Example of a tree. The level of a node is defined by letting the root to be at level zero 1 , while a node at level has children at level . For example, the root in Figure 1 is at level zero, nodes are at level one, nodes ate level two, and nodes are at level three. The depth of a node is its level number. The height of a tree is the maximum level of any node in this tree. Node is at depth two, while node at depth three. The height of the tree presented in Figure 1 is three. A tree is called a -ary tree if every internal node has no more than children. A tree is called a full -ary tree if every internal node has exactly children. A complete tree isafulltreeup the last but one level, that is, the last level of such a tree is not full. A binary tree is a tree with . The tree in Figure 1 is a -ary tree, which is neither a full tree nor a complete tree. An ordered rooted tree is a rooted tree where the children of each internal node are ordered. We usually order the subtrees from left to right. Therefore, for a binary (ordered) tree the subtrees are called the left subtree and the right subtree .
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mod8 - Module 8: Trees and Graphs Theme 1: Basic Properties...

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