Rooted & binary tree
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Discuss about the graph theory topics like rooted and binary tree and their properties Reference: NarasinghDeo, Graph theory, PHI.
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- 1. Rooted & Binary Tree Dr. Manish T I Associate Professor Dept of CSE Adi Shankara Institute of Engineering & Technology, Kalady [email protected]
- 2. • A Binary tree is defined as a tree in which there is exactly one vertex of degree two, and each of the remaining vertices is of degree one or three. Introduction • A Tree in which one vertex is distinguished from all the others is called a Rooted tree • A special class of rooted trees called Binary Rooted trees.
- 3. VERTICES a - Internal vertex b - Pendant vertex c- Internal vertex d - Pendant vertex e- Internal vertexe- Internal vertex f - Pendant vertex g - Pendant vertex EDGES 1,2,3,4,5,6
- 4. Properties of Binary Tree The number of vertices n in a binary tree is always odd. n No: of Vertices in binary tree p=(n+1)/2 No: of Pendant vertices in binary tree. n-p-1 Number of vertices with degree three. n-1 Number of edges . p-1 Number of internal vertices in binary tree.
- 5. Properties of Binary Tree From the previous diagram n=7 i.e. a odd number Number of edges = 7-1 =6 No: of Pendant vertices in binary tree = (7+1)/2 =4 No: of vertices with degree three = 7-4-1 =2No: of vertices with degree three = 7-4-1 =2 Number of edges = 7-1 =6 Number of internal vertices in binary tree = 4-1 =3
- 6. Properties of Binary Tree 2k Maximum number of vertices possible in k-level binary tree. ⌈ log2 (n+1) -1 ⌉ Minimum possible height of an n – vertex binary tree.⌈ ⌉ (n-1)/2 Maximum possible height of an n – vertex binary tree.
- 7. Properties of Binary Tree From the previous diagram k=4 Max No: of vertices possible in k-level binary tree = 24 = 16 ⌈⌈⌈⌈ ⌉⌉⌉⌉Min possible height of an 7 – vertex binary tree = ⌈⌈⌈⌈ log2 (7+1) -1 ⌉⌉⌉⌉ = ⌈⌈⌈⌈ log2 (8) -1 ⌉⌉⌉⌉ = ⌈⌈⌈⌈ 3 -1 ⌉⌉⌉⌉ = 2 Max possible height of an 7 – vertex binary tree = (7-1)/2 = 4