tree TREES
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Trees
A is a restricted form of graph. It has a nite set of nodes and edges. But the edges have a direction (from parent node to child node), and a non-empty tree must satisfy the following rules: 1. One of the nodes is special the root. Every non-empty tree has a unique root. 2. Each node has exactly one parent, except the root, which has none. 3. Each node is connected to the root by a unique path from the root to that node. 4. There are no cycles, that is no paths that form loops through two or more nodes. Also, the children in our trees are ordered; some people call such trees ordered trees.
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About the diagram: One draws trees upside-down. The arrows indicate the parent-child relationship: they arent necessarily object references. Crunchy nugget: The rst two properties are sucient to dene a tree, since together they imply the third. Questions: Can you prove this? Are any other subsets of these properties sucient?
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Denitions
A node with no children is called a leaf. A node that is not a leaf is an internal node. Thus, the root is internal unless it is the only node, and a node with just one child is internal. Nodes with the same parent are siblings. The tree formed from a node together with all its descendants and the edges among them is a subtree. If m1 is the parent of m2 , m2 is the parent of m3 , . . ., and mk1 is the parent of mk , then the sequence of nodes m1 , m2 , . . . , mk is a path in the tree. The length of such a path is k 1, i.e., the number of edges on the path. The height of a tree is the length of its longest path from the root to a leaf. The depth of a node is its distance from the root: the number of edges on the path from the root to that node. The depth of the root is 0.
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Special Kinds of Trees A tree may have a xed branching factor: a maximum number of children for any node. A binary tree has branching factor 2. Operations on trees include inserting a new node removing a node traversing a tree: visiting the nodes in some order and doing something at each, e.g., printing its label attaching a new subtree at a given node removing a subtree A tree can be implemented many ways, not necessarily with node objects and references. For now, think of it abstractly.
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Uses of Trees
Trees can be used to represent things that are in some hierarchical relationship: some family relationships, e.g. father-of. Question: what about parent-of? The class inheritance hierarchy in Java. Which directories occur inside which others in a computer le system. The relationship between method calls and their arguments. Example: a( b(c()), d(), g(h(), i()) ) The relationship between operations and operands in an arithmetic expression. Example: 3 + (4 (6 8)) + (25/2).
+ + 3 4 6 * 8 25 / 2
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Binary Search Trees
A binary search tree (or BST) is a binary tree where the nodes are labelled, and each nodes label is: greater than the labels of all the nodes in its left subtree, and less than the labels of all the nodes in its right subtree. Any subtree of a binary search tree is a binary search tree. Question: Is it sucent to only require that a nodes label is greater than the left childs and less than the right childs?
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An Example of a BST
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78
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Question: Is the above BST the only way to store those elements in a BST? Question: How would you search in a BST?
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A BST combines the best features of an array and a linked list: Because it is ordered, one can use an ecient search strategy analogous to binary search in a sorted array. But it is dynamic like a linked list, so it can grow and shrink in size easily. Question: Why not just use a sorted linked list?
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Eciency of BST Operations
BST search seems to be ecient because, like binary search, it allows us to disregard a lot more of the problem every time we go down a level in the tree. But we may get to disregard a lot or a little, depending on the shape of the tree. Think of some examples.
In fact, the eciency of the operations depends on the height of the tree. What do we know about a trees height? . . .
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Height Binary of Trees What is the maximum height of a binary tree with n nodes? What is the minimum? What kind of tree has the minimum height?
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Height maximum number of nodes Let M (h) represent the maximum number of nodes in a binary tree of height h. Prove: For all h 0, M (h) = 2h+1 1. Base Case: Prove that M (0) = 20+1 1. A binary tree with height 0 has one node, so M (0) = 1. And 20+1 1 = 2 1 = 1.
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Let k 0 be an arbitrary integer. Induction Hypothesis: Assume that for all 0 i k, M (i) = 2i+1 1. Induction Step: Prove that M (k + 1) = 2k+2 1. Let T be any binary tree of height k + 1. Let L and R be the two (possibly empty) subtrees of the root. Clearly, the maximum number of nodes in T , namely M (k + 1), is just 1 (for the root) plus the maximum number of nodes in L and R. But T has height k + 1, so L and R have height k. (The proof of this is left to reader.) So the IH applies to both L and R. As a result: M (k + 1) = 1 + 2 M (k) = 1 + 2 (2k+1 1), by the IH = 1 + 2k+2 2 = 2k+2 1. Conclusion: For all h 0, M (h) = 2h+1 1.
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Number of nodes minimum height Let n be the number of nodes in some binary tree of height h. We proved that the maximum number of nodes in such a tree is 2h+1 1. So: n 2h+1 1 n + 1 2h+1 log2(n + 1) h + 1 h log2(n + 1) 1 The minimum height of a binary tree is then O(log n).
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Eciency of Search in a BST The height of a BST containing n nodes varies from log2(n + 1) 1 to n 1. So the number of steps required to search a BST with n nodes varies from log2(n + 1) like binary search to n like linear search, depending on the shape of the tree! We can keep a BSTs height on the order of log2 n by careful rearrangement during insertion and deletion. And there are ways to do this without making insertion and deletion take much longer. However, we dont cover this in our course.
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Abstraction again Weve gured out a lot about trees without ever seeing an implementation. Weve talked completely in terms of the abstract notion of a tree. Question: Why is this a good thing?
Now on to implementations . . .
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Implementing Trees
Using References Intuitive implementation: each node is an object, and each edge is a reference. (There are still other decisions, such as how to represent the list of edges from a node to its children.) Using Arrays Question: How can you represent a tree using an array?
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ADT or Data Structure
The tree can be viewed as either an ADT, or a data structure. Example: A tree can be no more tha...