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Designate ‘a’ the root

b. a. c. A tree is a simple graph satisfying: if v and w are vertices and there is a path from v to w, it is a unique simple path. a. c. b. a. a. c. b. b. g. g. h. h. j. j. i. i. d. d. f. f. e. e. Designate ‘a’ the root.

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Designate ‘a’ the root

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  1. b a c A tree is a simple graph satisfying: if v and w are vertices and there is a path from v to w, it is a unique simple path. a c b

  2. a a c b b g g h h j j i i d d f f e e Designate ‘a’ the root What other node could have been chosen the root? b Then the tree would look:

  3. a Terms: c b g h j i d f e Root level = 0 b,g,h,c Level 1 d,e,f,i,j Level 2 Level of a vertex: Length of the simple path from root to vertex. Height of tree: Maximum level in tree

  4. a c b g h j i d f e More terms: Parent of vn: vn-1 Siblings: nodes (vertices) with the same parent Ancestors: All nodes in the path from the root to the node, except the node itself. Terminal vertex (leaf): A node with no children. Internal vertex (branch vertex): Not a leaf. The graph consisting of x and its descendants and all edges on a path from x to each descendant. Subtree of T rooted at x:

  5. a b g h j i d f e A tree is connected. (by direct proof) A tree does not contain a cycle. (indirect proof) A tree with n vertices has n-1 edges. (mathematical induction)

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