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Binary Trees

Binary Trees. Tree Example. Tree Structures. A tree is a hierarchical structure that places elements in nodes along branches that originate from a root . Nodes in a tree are subdivided into levels in which the topmost level holds the root node.

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Binary Trees

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  1. Binary Trees

  2. Tree Example

  3. Tree Structures • A tree is a hierarchical structure that places elements in nodes along branches that originate from a root. • Nodes in a tree are subdivided into levels in which the topmost level holds the root node. • Any node in a tree can have multiple successors at the next level • Therefore a tree is a nonlinear structure.

  4. Tree Structures (continued) • Operating systems use a general tree to maintain file structures.

  5. Tree Terminology • Tree structure • Collection of nodesthat originate from a unique starting node called the root. • Each node consists of a value and a set of zero or more links to successor nodes. • The terms parent and child describe the relationship between a node and any of its successor nodes.

  6. Tree Terminology (continued) s

  7. Tree Terminology (continued) • Consists of nodes connected by edges • A tree is an instance of a more general category called a graph(a later slideshow discusses graphs) Nodes Edges Nodes connected by edges (a graph – but not a tree)

  8. Tree Terminology • Tree – recursively defined as empty or a root node with zero or more sub-trees • Node – a holder for data plus edges to children • Edge – connects a parent node to a child node • Root – a pointer to the first node, if it exists, or NULL • Leaf node – a node with no children • Internal node – a node with one or two children • Path – sequence of edges between two nodes • Height – longest path in a tree from root to any other node • Depth – number of edges from root to a node

  9. Tree Terminology (continued) • A path between a parent node P and any node N in its subtree is a sequence of nodes • P=X0, X1, . . ., Xk = N • k is the length of the path • Each node Xi in the sequence is the parent of Xi+1 for 0  i  k-1. • Depth (also called Level) of a node • Length of the path from root to the node. • Equivalent - number of edges from root to the node. • Viewing a node as a root of its subtree, the height of a node is the length of the longest path from the node to a leaf in the subtree. • The height of a tree is the maximum level in the tree.

  10. Tree Terminology (continued) Height = 3

  11. Sorted Binary Trees • In a binary tree, each parent has no more than two children • each node (item in the tree) has a value • a total order (linear order) is defined on these values • left subtree of a node contains only values less than the node's value • right subtree contains only values greater than or equal to the node's value. 21 10 34 5 25 39 2 7 33

  12. Binary Trees • A compiler builds unsorted binary trees while parsing expressions in a program's source code.

  13. Binary Trees (continued) • Each node of a binary tree defines a left and a right subtree. Each subtree is itself a tree. Right child of T Left child of T

  14. Binary Trees (continued) • A recursive definition of a binary tree: • T is a binary tree if T • has no node (T is an empty tree) or • has at most two subtrees.

  15. { -1 if TN is empty 1+max( height(TL), height(TR)) if TN not empty height(N) = height(TN) = Height of a Binary Tree • The height of a binary tree is the length of the longest path from the root to a leaf node • Let TN be the subtree with root N and TL and TR be the roots of the left and right subtrees of N. Then leaf node will always have a height of 0

  16. Height of a Binary Tree (concluded) Degenerate binary tree (least dense)

  17. Density of a Binary Tree • In a binary trees, the number of nodes at each level falls within a range of values. • At level 0, there is 1 node, the root; at level 1 there can be 1 or 2 nodes. • At any level k, the number of nodes is in the range from 1 to 2k. • The number of nodes per level contributes to the density of the tree. • Intuitively, density is a measure of the size of a tree (number of nodes) relative to the height of the tree.

  18. Density of a Binary Tree (continued)

  19. Density of Binary Tree • If degenerate trees allowed: • Problem – search in basic binary tree is O(N) • Value could be anywhere in tree • No better than a list

  20. Density of a Binary Tree (continued) • A complete binary tree of height h has all possible nodes through level h-1, and the nodes on depth h exist left to right with no gaps • Complete binary trees are excellent storage structures due to packing a large number of nodes near the root

  21. Density of a Binary Tree (continued)

  22. Density of a Binary Tree (continued) • Determine the minimum height of a complete tree that holds n elements. • Through first h - 1 levels, total number of nodes is 1 + 2 + 4 + ... + 2h-1 = 2h - 1 • At depth h, the number of additional nodes ranges from a minimum of 1 to a maximum of 2h. • Hence the number of nodes n in a complete binary tree of height h ranges between2h - 1 + 1 = 2h n  2h - 1 + 2h = 2h+1 - 1 < 2h+1

  23. Density of a Complete Binary Tree • After applying the logarithm base 2 to all terms in the inequality, we have h  log2 n < h+1 and conclude that a completebinary tree with n nodes must have height h = int(log2n) search in complete binary tree is O(log2(n))

  24. Binary Tree Nodes • Define a binary tree a node as an instance of the generic TNode class. • A node contains three fields. • The data value, called nodeValue. • The reference variables, left and right that identify the left child and the right child of the node respectively.

  25. Binary Tree Nodes (continued) • The TNode class allows us to constructa binary tree as a collection of TNode objects.

  26. TNode Class // declared as an inner class within the class building the tree

  27. Building a Binary Tree Using previous slide’s TNode class

  28. Scanning a Binary Tree • Next issue: • How will you retrieve the data stored in the tree?

  29. Iterative Level-Order Scan • A level-order scan visits the root, then nodes on level 1, then nodes on level 2, etc.

  30. Iterative Level-Order Scan • A level-order scan is an iterative process that uses a queue as an intermediate storage collection. • Initially, the root enters the queue. • Start a loop ending when the queue is empty • Remove a node from the queue • Perform some action with the node • Add its children onto the queue • Because siblings enter the queue during a visit of their parent, the siblings (on the same level) will exit the queue in successive iterations.Notice in the next example how I'm using Java's Queue interface

  31. Iterative Level-Order Scan (continued)

  32. Iterative Level-Order Scan (continued) Step 1: remove A then add B and C into queue Step 2: remove B then add D into the queue Step 3: remove C then add E into the queue Step 4: remove D Step 5: remove E

  33. Iterative Level-Order Scan (continued) A remove A B C C D remove B D E F remove C E F G H remove D F G H remove E G H I remove F H I remove G I J remove H J remove I remove J Notice the order removed from queue and appended to the output s

  34. Recursive Binary Tree-Scan Algorithms • If current node == null is stopping condition • To scan a tree recursively • Visit and display the node (D) • scan the left subtree (L) and scan the right subtree (R) • The order in which you perform the D, L, R tasks determines the order in which nodes are retrieved • In the following code, t is initially the reference to the root node:

  35. Inorder Scan L D R inorderDisplay method call stack: inorderDisplay(a) inorderDisplay(b) inorderDisplay(d) inorderDisplay(g) inorderDisplay(null) append g to s inorderDisplay(null) append d to s inorderDisplay(h) inorderDisplay(j) inorderDisplay(null) append j to s inorderDisplay(null) append h to s inorderDisplay(null) append b to s inorderDisplay(null) append a to s inorderDisplay(c) inorderDisplay(e) inorderDisplay(null) append e to s inorderDisplay(null) append c to s inorderDisplay(f) inorderDisplay(null) append f to s inorderDisplay(i) inorderDisplay(null) append i to s inorderDisplay(null) • Scan is in order of visits to the left subtree, the node's own value, and visits to the right subtree Inorder Scan: G D J H B A E C F I

  36. Postorder Scan L R D • Scan is in order of visits to the left subtree, visits to the right subtree, and the node's own value Scan order: G J H D B E I F C A Can you write the method call stack for this?

  37. Method Call Stack for Postorder inorderDisplay method call stack: inorderDisplay(a) inorderDisplay(b) inorderDisplay(d) inorderDisplay(g) inorderDisplay(null) inorderDisplay(null) append g to s inorderDisplay(h); inorderDisplay(j); inorderDisplay(null); inorderDisplay(null); append j to s inorderDisplay(null); append h to s // rest left to you… Scan order: G J H D B E I F C A

  38. Preorder Scan D L R • Scan is in order of the node's own value, visits to the left subtree, and visits to the right subtree Scan order: A B D G H J C E F I

  39. More Recursive Scanning Examples Preorder (NLR): A B D G C E H I F Inorder (LNR): D G B A H E I C F Postorder (LRN): G D B H I E F C A

  40. Visitor Design Pattern • When an action is needed on each element of a collection • Don't know in advance what the type of each element will be • Defines the visit() method which denotes what a visitor does • For a specific visitor pattern • Create a Visitor interface • Create a class that implements the Visitor interface • In class scanning a tree • Create an object of the class implementing the Visitor interface • During traversal, call Visitor object's visit() and pass the current value as an argument

  41. Visitor Design Pattern (continued) 1. 2. Another possibility (requires T to be "Comparable"): 2.

  42. 3. scanInorder() • This recursive method scanInorder()provides a generalized inorder traversal of a tree that performs an action specified by a visitor pattern. Uses Visitor object's visit instead of System.out.println

  43. scanInOrder() • If Visitor parameter is VisitorOutput • Prints output to the console • If Visitor parameter is VisitMax • VisitMax parameter stores the tree element with the max value after scanInOrder finishes

  44. Program B_Tree • Illustrates use of all the scanning methods

  45. Computing Tree Height • Recall that the height of a binary tree can be computed recursively. { -1 if T is empty height(T) = 1 + max(height(TL), height(TR)) if T is nonempty F

  46. Computing Tree Height (continued)

  47. Copying a Binary Tree • Simple case – exact duplicate • Duplicate with additional information • Contain nodes with additional field • Possibility - references the parent - this allows a scan up the tree along the path of parents

  48. Copying a Binary Tree (continued) • Copy a tree using a postorder scan • Build the duplicate tree from the bottom up.

  49. Clearing a Binary Tree • Clear a tree with a postorder scan • Remove the left and right subtrees before removing the node.

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