Section 9.1 Introduction to Trees
Tree terminology • Tree: a connected, undirected graph that contains no simple circuits • must be a simple graph: no multiple edges or loops • an undirected graph is a tree if and only if there is a unique simple path between any 2 of its vertices
Example 1 The graph at left is a tree; it is connected, and contains no simple circuits The graph at right is not a tree because it is not connected
Forest • A forest is a graph containing no simple circuits, not necessarily connected • Each connected component of a forest is a tree • For example, the second graph on the previous slide is a forest consisting of the following trees:
Rooted tree • A rooted tree is the directed graph that results from choosing a root vertex in a tree and directed each edge away from root • The parent of vertex x is the vertex y such that there is a directed edge from y to x • If y is the parent of x, then x is a child of y • Vertices with the same parents are siblings
More rooted tree terminology • Ancestors: all vertices in the path from root to a particular vertex; root is the ancestor of all vertices • Descendants: all vertices that share an ancestor • Leaf: vertex with no children • Internal vertex: non-leaf vertex; root is always internal, unless it’s the only vertex in the tree
Example 2 The tree at left can be converted to a rooted tree by choosing any vertex as root; two examples are shown below
M-ary tree • A rooted tree in which every internal vertex has at most m children is called an m-ary tree • A full m-ary tree is an m-ary tree in which every internal vertex has exactly m children • If m = 2, we call it a binary tree
Example 3 The tree at left is a full 5-ary tree The tree at right is not a full m-ary tree; m can’t be 2, since some vertices have 3 children. If m is 3, the tree isn’t full.
Ordered rooted tree • A rooted tree in which the children of each internal vertex are ordered is an ordered rooted tree • in an ordered binary tree, we refer to the first child as the left child, and the second child as the right child • the trees rooted at a vertex’s left and right child are the left and right subtrees of that vertex
Trees as Models • Chemistry: graphs can represent molecules, with atoms as the vertices, and bonds between them as the edges • Business: trees are often used to represent chain of command within organizations • Computer Science: File systems are typically organized into directories containing files; the entire file system is represented as a tree with a root directory containing subdirectories, each of which may be a subtree
Properties of Trees • A tree with n vertices has n-1 edges • A full m-ary tree with n vertices has: • v = (n-1)/m internal vertices and • e = [n(m-1) + 1]/m leaves • A full m-ary tree with v internal vertices has: • n = mv + 1 vertices and • e = v(m - 1) + 1 leaves • A full m-ary tree with e leaves has • n = (me - 1) / (m - 1) vertices and • v = (e - 1) / (m - 1) internal vertices
Example 4: using tree properties to solve problems • Suppose somebody starts a chain letter, asking each recipient to forward the message to four other people • Some people send it on, others do not • How many people have seen the message, including the original sender, if: • no one receives it more than once • the letter ends when 100 people have read it but didn’t forward it • How many people did forward the message?
Example 4 solution • The chain letter can be represented as a 4-ary tree, with internal vertices representing people who forwarded the message, and leaves representing people who didn’t • So we know that the number of leaves e = 100
Example 4 solution • Using the formula from 3 slides back, the number of people who saw the letter = the number of vertices n = (me - 1) / (m - 1) or (4 * 100 - 1) / (4 - 1) = 133 • The number of internal vertices represents the number of people who forwarded the message; since we know there are 100 leaves out of 133 vertices, there are 33 internal vertices
More tree properties • Level of a vertex: length of the unique path from root to that vertex - the level of root is 0 • Height of a rooted tree is the maximum of levels of all vertices - in other words, the length of the longest path from root to any vertex • A rooted m-ary tree of height h is balanced if all leaves are at levels h or h-1
Example 5 The tree at left is a balanced binary tree; its height is 3 and all leaves are at levels 2 and 3 The tree at right is an unbalanced binary tree; its height is 3 and there are leaves at levels 1, 2 and 3
More tree properties There are at most mh leaves in any m-ary tree of height h If an m-ary tree of height h has e leaves, then h If the m-ary tree is full and balanced, then h =
Section 9.1 Introduction to Trees
