Lecture Eight: Trees, part 1

  • Def - a tree is connected graph without circuits

  • Textbook quickly switched to rooted trees: trees with distinguished "root" element; we'll look first at general trees without roots..
  • Prop - If G = (V,E) is is connected, then V <= E+1

  • Prop - If G = (V,E) is a tree, then V = E+1

  • Prop - If G = (V,E) is connected, and V = E+1, then G is a tree!

  • Prop - Every tree has as least 2 vertices of degree 1.

  • Prop - If G = (V,E) is a tree, then there is a unique path between any pair of vertices
  • If we designate a vertex to be root, then we get a rooted tree. Can now view as directed, with all arcs going away from root
  • We then get notions like "level number" (hops from root); internal node (node with children); leaf (node without children); branching factor (number of children). If this is constant for all internal nodes, and if its value is m, we call the tree m-ary.
  • height of tree is highest level number in tree. For m-ary trees, height is logarithmic in # of leaves. This fact makes the world go around in CS.

  • Thm: If N = number of vertices, i = number of internal nodes, L = number of leaves, T is m-ary, then N = mi+1, N = i + L, and a whole bunch of other complicated (but easy to prove) results follow, e.g. L = (m-1)i + 1 -- this is all in textbook

-- RobbieMoll - 03 Oct 2006

Topic revision: r1 - 2006-10-03 - RobbieMoll
 
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