Lecture 10.pptx
- 1. The Theory of Computation االحتسابية النظرية رقم المحاضرة - 10 - Binary Trees الثنائية االشجار إعداد م . حميد حبيب وديان م االساسية التربية كلية / الحاسبات قسم
- 2. Differences Between A Tree and A Binary Tree The subtrees of a binary tree are ordered; those of a tree are not ordered. • Are different when viewed as binary trees. • Are the same when viewed as trees. A B A B
- 3. Data Structure for Binary Trees • A node is represented by an object storing • Element • Parent node • Left child node • Right child node B D A C E
- 4. Arithmetic Expression Tree • Binary tree associated with an arithmetic expression • internal nodes: operators • external nodes: operands • Example: arithmetic expression tree for the expression (2 (a - 1) + (3 b)) + - 2 a 1 3 b
- 5. Decision Tree • Binary tree associated with a decision process • internal nodes: questions with yes/no answer • external nodes: decisions • Example: dining decision
- 6. Maximum Number of Nodes in a Binary Tree • The maximum number of nodes on depth i of a binary tree is 2i, i>=0. • The maximum nubmer of nodes in a binary tree of height k is 2k+1-1, k>=0. Prove by induction. 1 2 2 1 0 - + k k i i
- 7. Full Binary Tree • A full binary tree of a given height k has 2k+1–1 nodes. Height 3 full binary tree.
- 8. Labeling Nodes In A Full Binary Tree • Label the nodes 1 through 2k+1 – 1. • Label by levels from top to bottom. • Within a level, label from left to right.
- 9. Node Number Properties Parent of node i is node i / 2, unless i = 1. Node 1 is the root and has no parent. • Right child of node i is node 2i+1, unless 2i+1 > n, where n is the number of nodes. • If 2i+1 > n, node i has no right child.
- 10. Complete Binary Trees • A labeled binary tree containing the labels 1 to n with root 1, branches leading to nodes labeled 2 and 3, branches from these leading to 4, 5 and 6, 7, respectively, and so on. • A binary tree with n nodes and level k is complete iff its nodes correspond to the nodes numbered from 1 to n in the full binary tree of level k. 1 2 3 7 5 9 4 8 6 Complete binary tree 1 2 3 7 5 11 4 10 6 9 8 15 14 13 12 Full binary tree of depth 3